ON THE BERNOULLI PROPERTY OF PLANARpremat.fing.edu.uy/papers/2014/177.pdfPLANAR HYPERBOLIC BILLIARDS...

ON THE BERNOULLI PROPERTY OF PLANARHYPERBOLIC BILLIARDS

GIANLUIGI DEL MAGNO AND ROBERTO MARKARIAN

Abstract. We consider billiards in non-polygonal domains of theplane with boundary consisting of curves of three different types:straight segments, strictly convex inward curves and strictly convexoutward curves of a special kind. The map of these billiards isknown to have non-vanishing Lyapunov exponents a.e. providedthat the distance between the curved components of the boundaryis sufficiently large, and the set of orbits having collisions onlywith the flat part of the boundary has zero measure. Under a fewadditional conditions, we prove that there exists a full measureset of the billiard phase space such that each of its points has aneighborhood contained up to a zero measure set in one Bernoullicomponent of the billiard map. Using this result, we show thatthere exists a large class of planar hyperbolic billiards that havethe Bernoulli property. This class includes the billiards in convexdomains bounded by straight segments and strictly convex inwardarcs constructed by Donnay.

1. Introduction

A planar billiard is the mechanical system consisting of a point-particle moving freely inside a bounded domain Ω ⊂ R2 with piecewisedifferentiable boundary, and being reflected off ∂Ω so that the angle ofreflection equals the angle of incidence. This paper concerns hyperbolicbilliards, i.e., billiards for which the corresponding map has no vanish-ing Lyapunov exponents. Maps with this property are not necessarilyuniformly hyperbolic, but exhibit a weak form of hyperbolicity callednon-uniform hyperbolicity [27].

The study of hyperbolic billiards was initiated by Sinai. In his semi-nal paper [28], he proved that billiards in 2-dimensional toral domainscontaining finitely many obstacles with strictly convex outward bound-ary are hyperbolic and K-mixing. In fact, Sinai billiards enjoy theBernoulli property as well [17].

Later on, Bunimovich proved that also billiards in domains withboundary formed by strictly convex inward curves and straight seg-ments can be hyperbolic [1, 2]. The most celebrated example of such

Date: December 16, 2014.1991 Mathematics Subject Classification. Primary 37D50; Secondary 37A25,

37D25, 37N05.Key words and phrases. Hyperbolic Billiards, Ergodicity, Bernoulli property.

1

2 DEL MAGNO AND R. MARKARIAN

a domain is the stadium, the region bounded by two semi-circles con-nected by two parallel segments. The only strictly convex inward curvesallowed in Bunimovich billiards were arcs of circles. This limitation waseventually overcome by several researchers. Using new techniques forestablishing the positivity of Lyapunov exponents [23, 32], Wojtkowski,Markarian and Donnay proved independently that besides arcs of cir-cles many other strictly convex inward arcs can be used to constructhyperbolic billiards [16, 23, 25, 32]. Similar results were obtained byBunimovich for a class of strictly convex arcs related to those of Don-nay [5]. All these results showed that billiards in non-polygonal planardomains are hyperbolic if three conditions are fulfilled: B1) the strictlyconvex boundary components of the domain are of a special type, B2)the distance between these components and the other curved boundarycomponents is sufficiently large, and B3) orbits having only collisionswith straight segments of the boundary form a set of zero measure. Aprecise formulation of these conditions is given in Section 5.

In this paper, we address the question whether a hyperbolic billiardhas the Bernoulli property (for short, ‘it is Bernoulli’), i.e., whetherit is isomorphic to a Bernoulli shift. The Bernoulli property is thestrongest among the ergodic properties: it implies K-property, mixingand ergodicity. The Bernoulli property was proved for several billiards,including the Sinai billiards, the Bunimovich billiards, the Wojtkowskibilliards and other special hyperbolic billiards [4, 11, 12, 13, 22, 24, 30].Despite that, there remain many planar hyperbolic billiards for whichnot even the ergodicity has been proved. Notably, among them, thereare the billiards constructed by Donnay [16]. The goal of this paperis to fill this gap: we show that there exists a large class of hyper-bolic billiards, including the Donnay billiards, that have the Bernoulliproperty.

The key ingredient in the proof of this result is a local ergodic theo-rem for hyperbolic planar billiards, which is also the main result of thispaper. Roughly speaking, our local ergodic theorem states that if a pla-nar billiard satisfies conditions B1-B3 and the extra condition B4 (seeSection 5), then there exists a full measure set H in the billiard phasespace with the property that each element of H has a neighborhoodcontained (mod 0) in one ergodic component of the billiard.

Condition B4 regards the singular set of the billiard, the set formedby the elements where the billiard map is not defined or is not oftwice-differentiable. This set corresponds to the trajectories that hit acorner of the billiard domain or have a tangential collision with the itsboundary. Condition B4 requires the elements of the singular set whosetrajectories have eventually collisions only with straight segments toform a negligible subset (in the the measure theoretical sense) of thesingular set.

PLANAR HYPERBOLIC BILLIARDS 3

As a matter of fact, the neighborhood in the conclusion of the lo-cal ergodic theorem belongs (mod 0) to a single Bernoulli componentof the billiard map (for the definition of a Bernoulli component, seeTheorem A.6 in the appendix). As a consequence, every Bernoullicomponent of a billiard satisfying B1-B4 is open (mod 0). We stress,that although this is a remarkable property, it is not enough to yieldthe Bernoulli property of a billiard.

Our local ergodic theorem for billiards is derived from a similar re-sult for general hyperbolic symplectomorphisms with singularities [14],which generalizes work of Liverani and Wojtkowski [22]. The proof ofboth [14] and [22] rely on the method developed by Sinai to prove theergodicity of dispersing billiards [28]. Refinements of Sinai’s methodwere obtained in [9, 21, 22, 29].

This paper is organized as follows. In Section 2, we recall the def-inition of billiard map and its main properties. In Section 3, we givethe definition of the focusing time of a family of billiard trajectories.We also give the definition of a focusing arc introduced by Donnay,and collect the main properties of these arcs. Section 4 is concernedwith the theory of invariant cone fields, and their construction for bil-liards. In this section, we also recall further results on focusing arcs.In Section 5, we give a detailed description of the hyperbolic billiardsconsidered, and state the main results of this paper. In Section 6, weprove Conditions L1-L3 of the local ergodic theorem. Section 7 is en-tirely devoted to the proof of Condition L4. In Section 8, we introducethe class of billiards in polygons with pockets and bumps, and provethat they have the Bernoulli property. As a corollary, we obtain thatthe Donnay billiards have the Bernoulli property. In the Appendix, westate the local ergodic theorem for general hyperbolic symplectomor-phisms with singularities from [14] together with relevant concepts andobservations.

Sections 2, 3 and the beginning of Section 4 are meant to be an in-troduction to basic concepts on billiards and geometric optics, Donnayarcs and the theory of invariant cone fields. The reader already familiarwith these notions may read quickly or skip these sections, and movedirectly to the second part of Section 4.

2. Generalities on billiards

A billiard system can be described either by a flow or a map. Inthis paper, we focus on the billiard map. For the relation between thebilliard map and billiard flow, we refer the reader to the book [10]. Inthis section, we define the billiard map for a 2-dimensional domain, andsingle out its basic properties: regularity, natural invariant probabilityand singular sets.


2.1. Billiard domain. Let 0 ≤ k ≤ ∞. A subset Γ ⊂ R2 is called anarc of class Ck if Γ is the image of a Ck embedding γ : [0, 1]→ R2. Theboundary of an arc Γ is given by ∂Γ = γ(0) ∪ γ(1). A subset Γ ⊂ R2

is called a closed curve of class Ck if Γ is Ck diffeomorphic to the unitcircle S1. Of course, ∂Γ = ∅ if Γ is a closed curve.

Let Ω be an open bounded connected subset of R2 such that ∂Ω isan union of finitely many disjoint closed curves of class C0. We alsoassume that ∂Ω is an union of n > 0 arcs and closed curves Γ1, . . . ,Γnof class C3. The set Ω defines the most general table of the billiardsconsidered in this paper. The conditions imposed on Ω imply that thebilliard tables may contain finitely many obstacles. Denote by ì thelength of Γi, and consider the parametrization γi : [0, ì] → R2 of Γiby arc-length with the property that the interior of Ω remains on theleft of the tangent vector γ′i(s) for s ∈ [0, ì]. We assume that i) thecurvature of each Γi computed with respect to the parametrization γiis either strictly negative, or strictly positive, or identically equal tozero, and ii) Γi ∩ Γj ⊂ ∂Γi ∩ ∂Γj for i 6= j. Note that an arc Γi withzero curvature is just a straight segment.

The arcs and closed curves Γ1, . . . ,Γn are called the components of∂Ω. The union of all components with positive curvature (focusingcomponents), negative curvature (dispersing components) and zero cur-vature (flat components) are denoted by Γ+, Γ−, Γ0, respectively. Apoint of

⋃ni=1 ∂Γi is called a corner of ∂Ω.

2.2. Billiard phase space. For each i = 1, . . . , n, define Mi = [0, ì]×[0, π] ⊂ R2 with the elements (0, θ) and (ì, θ) identified if Γi is aclosed curve. Hence Mi is either a rectangle or a cylinder. Let Mbe the disjoint union of M1, . . . ,Mn. An element x ∈ M is thereforean ordered pair (i, (s, θ)), and is called a collision. We define i(x) = i,s(x) = s and θ(x) = θ for x = (i, (s, θ)) ∈M . To simplify the notation,we identify each x ∈M with the corresponding pair (s, θ) by droppingthe index i from the representation (i, (s, θ)). When we need to specifyi, we will write ‘x ∈ Mi’. We denote by M+, M−, M0 the subsets ofM obtained by taking the disjoint unions of sets Mi with Γi belongingto Γ+, Γ−, Γ0, respectively.

The set M is a smooth manifold with boundary ∂M =⊔ni=1 ∂Mi. We

equip M with the Riemannian metric g = gxx∈M and the symplecticform ω = ωxx∈M given by gx = ds2 +dθ2 and ωx = sin θ(x)ds∧dθ forx ∈M . The norm generated by g is denoted by ‖ · ‖. The Riemannianmetric g induces in the usual way a distance d on each Mi, which canbe extended to the entire set M by setting d(x, y) = 1 whenever x ∈Mi

and y ∈ Mj with i 6= j. We denote by m the volume generated by g.Then µ = (2`)−1 sin θ(x)m is the probability measure generated by ωwith ` :=

∑ni=1 ì being the total length of ∂Ω.


There exists a natural involution I : M → M given by I(s, θ) =(s, π − θ) whenever (s, θ) ∈ Mi. For notational purposes, we write −xinstead of I(x). If B is a subset of M , then −B will denote the set−x : x ∈ B.

2.3. Billiard map. Given x = (s, θ) ∈ Mi, define q(x) = γi(s) ∈ Γiand v(x) to be the unit vector of R2 forming an angle θ with γ′i(s). Weuse the notation (q(x), q(x) + τv(x)) to denote the open segment of R2

with endpoints q(x) and q(x) + τv(x). Consider the set ρ(x) = τ >0: (q(x), q(x) + τv(x)) ⊂ Q, and define t(x) = 0 if ρ(x) is empty, andt(x) = sup ρ(x) otherwise. Also, define q1(x) = q(x) + t(x)v(x). Let

M ′ = x ∈M : q1(x) is not a corner of ∂Ω .If x ∈ M ′, then there exists a unique i1(x) such that q1(x) belongs tothe interior of Γi1(x). Hence, we set s1(x) = γ−1

i1(x)(q1(x)). Now, let

v1(x) = −v(x) + 2〈γ′i1(x)(s1(x)), v(x)〉γ′i1(x)(s1(x)),

where 〈·, ·〉 denotes the usual scalar product in R2. Then, let θ1(x) ∈[0, π] be the oriented angle between γ′i1(x)(s1(x)) and v1(x). The billiard

map for the domain Ω is the transformation T : M ′ →M given by

Tx = (i1(x), (s1(x), θ1(x))) for x ∈M ′.

The regularity (continuity, differentiability, etc.) of T depends onthe regularity of ∂Ω. To clarify this point, we define:

A1 = x ∈M : q(x) is a corner of ∂Ω ,A2 = x ∈M : θ(x) ∈ 0, π ,A3 = x ∈M \ ∂M : q1(x) is a corner of ∂Ω ,A4 = x ∈M \ (∂M ∪ A3) : Tx ∈ A2 .

Note that ∂M is equal to A1∪A2. Now, let S+1 be the closure of A3∪A4,

and let S−1 = −S+1 . Then, define

R+1 = ∂M ∪ S+

1 ,

and for every j ≥ 1, define iteratively

R+j+1 = R+

j ∪ T−1R+j and R−j = −R+

j .

The transformation T is a local Ck−1 diffeomorphism at x ∈M\R+1 [20,

Theorem 4.1]. Thus, the points where T is not continuous or moregenerally Ck−1 are contained in R+

1 . Analogously, R+j (resp. R−j )

contains the points where T j (resp. T−j) is not Ck−1. We call R+j

and R−j the singular set of the billiard map T .

The map T : M \R+1 →M \R−1 is a Ck−1 diffeomorphism preserving

the symplectic form ω and the probability measure µ provided thatk > 1 (see [20, Corollaries 4.1 and 4.4, Part V]). Under proper condi-tions on the components Γ1, . . . ,Γn, which are satisfied by the billiards


considered in this paper (more precisely, for billiards satisfying Condi-tions B1 and B2 in Section 5), the sets R+

j and R−j are union of finitely

many arcs of class C2 [15]. It follows that µ(R+j ) = µ(R−j ) = 0 for

every j ≥ 1. Finally, we observe that T is time-reversible; this meansthat I T = T−1 I on M \R+

1 .

3. Focusing times and focusing arcs

We now introduce the concept of focusing times of an infinitesimalfamily of trajectories. This notion is borrowed from geometrical op-tics, and permits to obtain an intuitive description of the action of thederivative of the billiard map on the projective line.

3.1. Focusing times. Given a tangent vector 0 6= u ∈ TxM withx ∈ M , let (−δ, δ) 3 a 7→ ϕ(a) ∈ M be a differentiable curve forsome δ > 0 such that ϕ(0) = x and ϕ′(0) = u. Next, consider theparametrization

(−δ, δ)× R 3 (a, t) 7→ `+a (t) := q(ϕ(a)) + t · v(ϕ(a)).

The set `+a := `+

a (t) : t ∈ R is a straight line for every a ∈ R, andthe family `+

a of lines forms a variations of the line `+0 . We obtain a

second family `−a of lines by replacing the curve ϕ in the definition of`+a (t) with the curve −ϕ. In geometrical terms, each line `−a is obtained

by reflecting `+a about the tangent of ∂Ω at the point q(ϕ(a)).

All lines `+a intersect in linear approximation at a point q+ ∈ `+

0 .However, if the derivative of v ϕ vanishes at a = 0, then all lines `+

a are parallel in linear approximation, and we define q+ to be the pointat infinity. Similarly, all lines `−a intersect in linear approximation ata point q− ∈ `−0 , but if the derivative of v I ϕ vanishes at a = 0,then all lines `−a are parallel in linear approximation, and we defineq− to be the point at infinity. Note that q+ and q− depend only on thevector u and not on the choice of the curve ϕ. The points q+ and q−are called the forward focal point of u and the backward focal point ofu, respectively.

The distances between q(x) and q+ and between q(x) and q− aredenoted by τ+(x, u) and τ−(x, u), respectively. The first distance iscalled the forward focusing time of u, whereas the second distance iscalled the backward focusing time of u. Let u = (us, uθ) in coordinates(s, θ), and define

m(u) =uθus∈ R,

where R = R ∪ ∞ is the one-point compactification of R. Let κ(x)be the curvature of Γi(x) at q(x). A straightforward computation (for


example, see [32, Section 2]) shows that

τ±(x, u) =

sin θ(x)

κ(x)±m(u)if m(u) 6= ∓κ(x),

∞ otherwise.(1)

Let d(x) = sin θ(x)/κ(x). From (1), one can easily derive the wellknown Mirror Equation of geometrical optics, relating the focusingtimes τ+(x, u) and τ−(x, u):

1

τ+(x, u)+

1

τ−(x, u)=

2

d(x). (2)

Definition 3.1. Let x ∈ M and u ∈ TxM \ 0, and suppose that T j

is a local diffeomorphism at x for some j ∈ Z. Then define τ±j (x, u) =

τ±(T jx,DxTju).

3.2. Focusing arcs. We now recall the notion of a focusing arc intro-duced by Donnay [16].

We start by assuming that∫ `i

0

κ(s(x))ds ≤ π (3)

for every component Γi ⊂ Γ+. The geometrical meaning of this con-dition is that the tangents to Γi at its endpoints form an angle thatis not larger than π. This implies immediately that no components ofpositive curvature can be a closed curve. Condition (3) has anotherimportant consequence: if a billiard orbit is trapped by a componentΓi ⊂ Γ+, i.e. it does not leave Mi, then it is a periodic orbit of period 2,and its trajectory coincides with the segment joining the endpoints ofΓi [16, Lemma 1.1]. Examples of components with such orbits are half-ellipses: the trapped orbits correspond to the semi-axis along whichthe ellipse is cut in half.

Definition 3.2. Let n(x) = supj ≥ 0: x ∈M\R+j and x, Tx, . . . , T jx ∈

Mi(x) for x ∈M+ \ A2, where R+0 := ∅.

The number n(x) is always finite, and represents the number of con-secutive collisions of the trajectory of x with the component Γi(x) ⊂ Γ+

before leaving it.

Definition 3.3. Given Γi ⊂ Γ+, let Ei = x ∈Mi \ A2 : n(−x) = 0 .Also, let E+ be the union of all the Ei’s.

Note that if x ∈ Ei, then x, Tx, . . . , T n(x)x is the longest sequenceof consecutive collisions with Γi; any other collision preceding x has tobelong to Mj with j 6= i.

Now, suppose that Γi ⊂ Γ+. Let x ∈ Ei, and denote by ux ∈ TxMthe tangent vector such that τ−(x, ux) = ∞. In other words, thevariation associated to ux consists of lines parallel to v(x). The vectorv(x) and the number t(x) are defined in Section 2.


Definition 3.4. Suppose that Γi ⊂ Γ+. We say that x ∈ Ei is focusedby Γi if

(1) 0 < τ+i (x, ux) < t(T ix) for 0 ≤ i < n(x),

(2) 0 < τ+n(x)(x, ux) < +∞.

The concept of a focused collision is key in the definition below ofa focusing arc. In words, a collision is focused by Γi if x has a finitenumber n(x) of consecutive collisions with Γi before leaving it, and ifthe infinitesimal family of trajectories associated to ux focuses betweenevery two consecutive collisions with Γi and after the last one.

Definition 3.5. A component Γi ⊂ Γ+ is called focusing if i) Γi is anarc of class C∞, ii) Γi satisfies Condition (3), and iii) if every x ∈ Eiis focused by Γi.

Bunimovich introduced the notion of an absolutely focusing compo-nent [3, 5], which is similar to that of a focusing component. Therelation between these two notions is discussed in [5].

The following results concern the existence of focusing components,their robustness under perturbations, and use in constructing hyper-bolic billiards. They are proved in [16, Theorems 2-4].

Theorem 3.6. Given an arc Γ of class C∞ with positive curvature anda point q ∈ Γ, there exists a neighborhood U of q in Γ such that U isfocusing.

A similar conclusion when Γ is only of class C4, but satisfies also thecondition d2(κ−1/3)/ds2 > 0 was obtained by Markarian [23].

Theorem 3.7. Suppose that Γ is a focusing. Then there exists ε =ε(Γ) > 0 such that every arc Γ′ of class C∞ with positive curvature, ofthe same length as Γ, and ε-close to Γ in the C6 topology is focusing.

Theorem 3.8. The billiard map of a convex region Ω with boundaryconsisting of focusing components connected by straight segments suffi-ciently long has non-zero Lyapunov exponents almost everywhere.

Remark 3.9. We believe that the original proof of Theorems 3.6-3.8,4.2 and Proposition 4.3 remains essentially valid for arcs of class C6

rather than of class C∞. Since the explanation of this claim wouldrequire some time, and this is not the main purpose of this paper, weopted to stick to the original formulation of the results of [16].

Remark 3.10. The important property of a focusing component Γithat makes it suitable for constructing hyperbolic billiards is that thefocusing time τ+

n(x)(x, ux) is uniformly bounded for x ∈ Ei [16, Theo-

rem 4.4]. For this property to hold, an arc with positive curvature neednot be of class C6. Indeed, Wojtkowski and Markarian both discov-ered families of arcs of class C4 with positive curvature that have theprevious property, and used them to construct billiards with non-zero


Lyapunov exponents [23, 32]. However, the curvature of their arcs haveto satisfy some additional conditions.

Examples of focusing arcs are arcs of circles, arcs of cardioids, arcs oflogarithmic spirals and elliptical arcs. An example of a focusing arc notbelonging to the classes of Wojtkowski arcs and Markarian arcs is thehalf-ellipse (x, y) ∈ R2 : x2/a2 + y2/b2 = 1 and x ≥ 0 with a/b <

√2

[16, Theorem 7.1].In this paper, we consider billiard domains that are more general

than those considered in Theorem 3.8. Indeed, the billiard domainsconsidered here are not necessarily convex, and their boundaries areallowed to have components with negative curvature.

4. Cone fields for billiards

In this section, we first recall the concepts of an invariant conefield and a monotone quadratic form. Since we are dealing with pla-nar billiards, we will restrict ourselves to present definitions in the2-dimensional setting [32]. Then, we introduce a family of cone fieldsfor the billiard map T , each of which is eventually strictly invariantfor the billiard domains considered in this paper. This family of conefields plays a fundamental role in our proof of the Bernoulli propertyof T .

4.1. Cone fields. Let V be Consider a two-dimensional vector spaceV . Given two linear independent vectors X1 and X2 of V , we saythat the set C(X1, X2) := a1X1 + a2X2 : a1a2 ≥ 0 ⊂ V is the conegenerated by X1 and X2. We also define intC = a1X1 +a2X2 : a1a2 >0 ∪ 0 and C ′(X1, X2) = C(X1,−X2). The latter set is called theinterior of C(X1, X2), whereas the former set, which is in fact a cone,is called the complementary cone of C(X1, X2)

Now, let U be an open set of M , and suppose that X1 and X2 aretwo measurable vector fields on U such that X1(x) and X2(x) are linearindependent for all x ∈ U . A cone field C on U , denoted by (U,C), isa family of cones C(x)x∈U given by C(x) = C(X1(x), X2(x)) ⊂ TxMfor every x ∈ U . A cone field (U,C) is called continuous if the vectorfields X1 and X2 are continuous on U . A cone field (U,C) is calledinvariant (resp. strictly invariant) if x ∈ U and T kx ∈ U with k > 0implies that DxTC(x) ⊂ C(T kx) (resp. DxT

kC(x) ⊂ intC(T kx)). Acone field (U,C) is called eventually strictly invariant if it is invariant,and for almost every x ∈ U , there exists an integer k(x) > 0 such thatT k(x)x ∈ U and DxT

k(x)C(x) ⊂ intC(T k(x)(x)).

Remark 4.1. A 2-dimensional cone C may be naturally identified witha closed interval I of the projective space P(V ). Accordingly, a conefield (U,C) can be written as follows:

C(x) = u ∈ TxM \ 0 : m(u) ∈ I(x) ∪ 0 for x ∈ U,


where I(x) ⊂ R : x ∈ U is a proper family of closed intervals. Sincethe focusing times τ+(x, ·) and τ−(x, ·) are projective transformations,the cone field (U,C) has two alternative representations:

C(x) = u ∈ TxM \ 0 : τ±(x, u) ∈ I±(x) ∪ 0 for x ∈ U,

where I±(x) ⊂ R : x ∈ U are proper families of closed intervals.

4.2. Quadratic forms. Consider a cone field (U,C) generated by thevector fields X1 and X2. The quadratic form QC = QC(x, ·) : x ∈ Uassociated to (U,C) is defined by

QC(x, u) = ωx(u1, u2) for x ∈ U,

where u1 and u2 are vectors of subspaces generated byX1(x) andX2(x),respectively, such that u = u1 + u2. The form QC is called mono-tone (resp. strictly monotone) if QC(T kx,DxT

ku) ≥ QC(x, u) (resp.QC(T kx,DxT

ku) > QC(x, u)) for all 0 6= u ∈ TxM whenever x ∈ Uand T kx ∈ U with k > 0. The form QC is called eventually strictlymonotone if it is monotone, and for almost every x ∈ U , there existsan integer k(x) > 0 such that QC(T k(x)x,DxT

ku) > QC(x, u) for all0 6= u ∈ TxM .

Following [22], to measure the expansion generated by the action ofDT k with k > 0 on vectors in (U,C) with respect to QC , we define

σC(DxTk) = inf

u∈intC(x)\0

√QC(T kx,DxT ku)

QC(x, u),

and

σ∗C(DxTk) = inf

u∈intC(x)\0

√QC(T kx,DxT ku)

‖u‖.

For k < 0, we define σC and σ∗C similarly by replacing C(x) and QC

in the definition above with the complementary cone C ′(x) and −QC ,respectively.

The cone field (U,C) is invariant (resp. strictly invariant) if andonly if the quadratic form QC is monotone (resp. strictly monotone),and (U,C) is eventually strictly invariant if and only if QC is even-tually strictly monotone. Furthermore, DxT

kC(x) ⊂ C(T kx) (resp.DxT

kC(x) ⊂ intC(T kx)) for x ∈ U such that T kx ∈ U with k > 0 isequivalent to σC(DxT

k) ≥ 1 (resp. σC(DxTk) > 1). See [22], for a de-

tailed proof of these properties. From the definitions of strict invarianceof a cone field, σC and σ∗C , one can easily deduce that if k1, k2, n ∈ Zwith k1 · k2 ≥ 0, then

σC(DxTn) > 1 =⇒ σ∗C(DxT

n) > 0, (4)

and

σ∗C(DxTk1+k2) ≥ σ∗C(DxT

k1) · σC(DTk1xTk2). (5)


4.3. Cone fields for billiards. Previous approaches to the study ofthe ergodicity of billiards used a single continuous invariant cone field.Here we take a slightly different approach. Since cone fields for focusingarcs are piecewise continuous in general, we find it more convenient towork with a family of continuous cone fields (Ux, Cx) : x ∈ E with Uxbeing a neighborhood of x. We stress that Cx is not a single cone at thepoint x, but the family of cones Cx(y) : y ∈ Ux on the neighborhoodUx of x. Note that we restrict ourselves to consider cone fields definedon the subset E rather than on the entire domain M of T . Sincex ∈ E, we will be able to apply the local ergodic theorem stated inthe Appendix A only to neighborhoods of points of E, and so to provethe local ergodicity of T only on E. However, since in a hyperbolicbilliard, almost every orbit of a hyperbolic billiard visits E infinitelymany times, the previous conclusion will be enough to obtain the localergodicity on the entire domain of T .

We define the cone fields (Ux, Cx) separately for x ∈ E+ and forx ∈ E−, i.e., for focusing components and components with negativecurvature, which from now on we will call dispersing components.

4.3.1. Focusing components. For x ∈ E+, the cone field (Ux, Cx) isgiven by Cx(y) = C(y, gx(y)) ∪ 0 for all y ∈ Ux, where

C(y, gx(y)) := u ∈ TyM \ 0 : gx(y) ≤ m(u) ≤ κ(y),and gx : Ux → R is a continuos function such that−κ(y) < gx(y) ≤ κ(y)for all y ∈ Ux. The functions gx are chosen in Theorem 4.2.

Let κi be the maximum of the curvature of the focusing componentΓi.

Theorem 4.2. Given a focusing component Γi, there exist constantst±i , ai, θi > 0, 0 < mi ≤ κi and continuous functions gx : Ux → R withUx ⊂ Mi \ A2 being a neighborhood of x for all x ∈ Ei such that ify ∈ Ux, u ∈ C(y, gx(y)) and 0 ≤ k ≤ n(y), then

(1) −κ(T ky) +mi ≤ m(DyTku) ≤ κ(T ky),

(2) |m(DyTku)| ≤ ai ·minθ(y), π− θ(y) whenever θ(y) ∈ (0, θi)∪

(π − θi, π),(3) d(T ky)/2 ≤ τ+

k (y, u) < t(T ky)−d(T k+1y)/2 whenever k < n(y),(4) infv∈C(y,gx(y)) τ

−(y, v) ≤ t−i and supv∈C(y,gx(y)) τ+n(y)(y, v) ≤ t+i .

Proof. For the proof of Parts (1) and (2), see the proofs of [16, Theo-rems 4.4 and 5.6], whereas for the proof of Parts (3) and (4), see [16,Proposition 4.1 and Theorem 4.4].

Theorem 4.2 has the following geometrical interpretation. Part (1)states that the lower edge of the cone Cx(y) and its iterates alongconsecutive collisions of x with the focusing component Γi are uniformlybounded from −κ(y) for all x ∈ Ei and all y ∈ Ux. This conclusion isstrengthen in Part (2) when θ(y) is uniformly close to 0 or π. In this


case, the iterates of the lower edge of the cone Cx(y) remains uniformlyclose to the horizontal direction. Parts (3) and (4) implies the everyy ∈ Ux is focused (c.f. 3.4), and that the backward and forward focusingtimes of Cx(y) are uniformly bounded by constants that depend onlyon Γi. In fact, the constants t±i are continuous functions of Γi in theC6 topology [16, Theorem 4.4].

The following proposition provides an upper bound on the total num-ber of consecutive collisions along a focusing arc depending on the angleformed by the initial collision with the arc. For its proof see [16, Corol-lary 5.3 and Part (1) of Proposition 6.1].

Proposition 4.3. Let Γi be a focusing component, and let θi be theconstant in Theorem 4.2. Then there exist a constant ci > 0 and afunction Ni : (0, π/2)→ N such that if x ∈Mi\A2, then n(x) ≤ ci/θ(x)whenever θ(x) ∈ (0, θi)∪(π−θi, π), and n(x) < N(θi) whenever θ(x) ∈[θi, π − θi].

4.3.2. Dispersing components. For convenience, we extend Definition 3.3to dispersing components. Given Γi ⊂ Γ−, define Ei = Mi. Also, defineE− to be the union of all Ei such that Γi ⊂ Γ−, and E = E− ∪ E+.

Suppose that Γi ⊂ Γ−. For every x ∈ Ei, choose Ux = Mi and

Cx(y) = u ∈ TyMi \ 0 : m(u) ≤ κ(x) ∪ 0 for y ∈ Ux.

This cone field is clearly continuous. It was introduced by Wotjkowski [32].In geometrical terms, Cx(y) consists of tangent vectors focusing insidethe osculating disk tangent to Γi at q(y) and of radius (4|κ(y)|)−1. In-deed, using (1), we see that if 0 6= u ∈ Cx(y), then d(y)/2 ≤ τ+(y, u) ≤0. Note also that 0 ≤ τ−(y, u) ≤ +∞.

It is useful to introduce a new quantity G±x associated to the conefield (Ux, Cx) for all x ∈ E.

Definition 4.4. For every x ∈ E and every y ∈ Ux, let

G±x (y) =

sin θ(y)

κ(y)±gx(y)if x ∈ E+,

0 if x ∈ E−.

It is easy to check that each cone Cx(y) can be written in terms ofthe projective coordinates τ+ and τ− as follows:

Cx(y) =u ∈ TyMi \ 0 : d(y)/2 ≤ τ+(y, u) ≤ G+

x (y)∪ 0

=u ∈ TyMi \ 0 : G−x (y) ≤ τ−(y, u) ≤ +∞

∪ 0.

Note that for x ∈ E+, the numbers G±x (y) are just the forward andbackward focusing times of the vectors contained in Cx(y) with slopegx(y).


5. Results

In this section, we give a detailed description of the billiards that wewant to study, and formulate the main results of the paper.

The billiards in question are characterized by four conditions calledB1-B4. It is well known that B1-B3 are sufficient to guarantee thehyperbolicity of the billiard map T (see Proposition 5.3). This togetherwith the Spectral Theorem implies that T has at most countably manyergodic components of positive measure with respect to µ, with eachergodic component further decomposed into finitely many Bernoullicomponents cyclically permuted by T (see Theorem A.6).

The main result of this paper is the following: if in addition to B1-B3, a billiard satisfies also Condition B4, then there exists a measurableset H ⊂ M of full measure such that for every x ∈ H, there is aneighborhood of x in M contained up to a set of zero measure in asingle Bernoulli component of T (see Theorem 5.6). This result impliesimmediately that every Bernoulli component of T is open up to a setof zero measure. Results of this type are often called Local ErgodicTheorems.

Local ergodicity alone is not enough to conclude that T is Bernoulli.This is obtained by imposing on T some extra conditions. It is not aneasy task to formulate these conditions for the generality of the billiardsconsidered in this paper. Therefore, rather than trying to formulate theoptimal condition for the Bernoulli property of hyperbolic billiards, welimit ourselves here to give a simple condition, called B5, that yields theBernoulli property of interesting subclasses of hyperbolic billiards (i.e.,billiards with domains without straight boundaries, i.e., Γ0 = ∅). It isunfortunate that B5 does not hold for Donnay billiards. Nevertheless,in Section 8, we prove that these billiards and some generalizations(billiards with pockets and bumps) are Bernoulli, using a proof thatdoes not require B5.

5.1. Important sets. Next, we introduce several sets involved in theformulation of Conditions B1-B5. Recall that R±k are the singular setsdefined in Section 2. Define

• R±∞ =⋃k≥1R

±k ,

• R = R−∞ ∩R+∞,

• N± =x ∈M \R±∞ : ∃k > 0 s.t. T (±)nx ∈M0 for all n ≥ k

,

• N = N− ∩N+,• N ′ = (R−∞ ∩N+) ∪ (R+

∞ ∩N−),• H = M \ (R ∪N ∪N ′).

The geometric meaning of these sets is the following: R+∞ (resp.

R−∞) is the set of collisions with finite positive (resp. negative) semi-orbit; R is the set of collisions with finite orbit; N+ (resp. N−) isthe set of collisions with positive (resp. negative) semi-orbit visiting


eventually only flat components of ∂Ω; N is the set of collisions withboth semi-orbits visiting eventually only flat components of ∂Ω; N ′

is the set of collisions with one semi-orbit being finite and the othersemi-orbit visiting eventually only flat components of ∂Ω; H is the setof collisions with one semi-orbit visiting the curved components of ∂Ωinfinitely many times.

5.2. Hyperbolic billiards. We are ready to formulate Conditions B1-B5.

B1 (Non-polygonal domain): The domain Ω is not a polygon, andits boundary components can only be of the following type: straightsegments, dispersing arcs of class C3 and focusing arcs.

B2 (Distance between boundary components): For each curvedcomponent Γi, we define λ±i = 0 if Γi is dispersing, and λ±i = t±i witht±i as in Theorem 4.2 if Γi is focusing. Given two curved componentsΓi and Γj, denote by tij ≥ 0 the infimum of the Euclidean length ofall finite billiard orbits x0, . . . , xn with n > 0 such that x0 ∈Mi andxn ∈Mj. We assume that

(1) there exists λ > 0 such that if Γi or Γj is focusing, then

tij ≥ λ−i + λ+j + λ,

(2) the distance between each focusing component Γi and the set ofcorners of ∂Ω formed by two straight segments is greater thanλ−i .

B3 (Neutral orbits): We assume that µ(N−) = 0.

B4 (Singular-Neutral orbits): We assume m−(S−1 ∩N+) = 0, wherem− is the measure induced by the Riemann metric g on S−1 .

B5 (Connectedness): The set H ∩Mi is connected for every com-ponent Γi of ∂Ω.

Condition B2 has a couple of obvious consequences for the geome-try of Ω: i) the internal angle between a focusing component and anadjacent curved component is greater than π, and ii) the internal an-gle between a focusing component and an adjacent flat component isgreater than π/2. Also, note that Conditions B1-B4 allow ∂Ω to havecusps formed by two dispersing components or a dispersing and a flatcomponent.

Remark 5.1. Since N+ = −N− and S+1 ∩N− = −(S−1 ∩N+), Condi-

tions B3 and B4 imply that µ(N+) = 0 and m+(S+1 ∩N−) = 0, where

m+ is the measure induced by the Riemann metric g on S+1 .

From Conditions B1 and B2, it follows that cone field (Ux, Cx)x∈Eis strictly invariant along a piece of an orbit connecting two elementsof E. The next lemma is proved in [15, Lemma 5.2].


Lemma 5.2. Suppose that the billiard in Ω satisfies Conditions B1 andB2. Also, suppose that there exist x1, x2 ∈ E and y ∈ E ∩Ux1 \R+

k forsome k > 0 such that T ky ∈ E ∩ Ux2. Then

DyTkCx1(y) ⊂ intCx2(T

ky).

From the previous lemma, one obtains the hyperbolicity of the bil-liard map T provided that Conditions B1-B3 are satisfied. This is wellknown fact, but we give its proof for completeness.

Proposition 5.3. If a billiard in a domain Ω satisfies Conditions B1-B3, then the Lyapunov exponents of the map T are non-zero a.e. onM .

Proof. Let E ′ be the subset of E \ (R+∞ ∪N+) defined by

E ′ =x ∈ E \ (R+

∞ ∪N+) : ∃nk +∞ s.t. T nkx ∈ Vx ∀k > 0,

where (Ux, Cx) is the cone field associated to x. Since (Ux, Cx) is strictlyinvariant for every x ∈ E ′ by Lemma 5.2, results of Wojtkowski [32]imply that the Lyapunov exponents of T are non-vanishing at everypoint of E ′. By µ(R+

∞) = µ(N+) = 0 and the Poincare RecurrenceTheorem, we obtain that µ(E ′) = µ(E) > 0. This fact together withµ(R+

∞) = µ(N+) = 0 gives that the orbit of a.e. point of M visits E ′.Since the Lyapunov exponents are constant along orbits, we can finallyconclude that the Lyapunov exponents of T are non-vanishing at a.e.point of M .

Conditions B1-B3 are sufficient for the hyperbolicity of the map T .In fact, even part (2) of B2 can be dropped if we are only interestedin the hyperbolicity of T . The extra Condition B4 is required to provethat T is locally ergodic. This condition is related to the Sinai-ChernovAnsatz (c.f. Condition L3 of Theorem A.13). Note also that B3 is anecessary condition for the hyperbolicity of T .

Remark 5.4. We do not know whether or not, for a domain Ω satisfy-ing B1 and B2, the conditions B3 or B4 is automatically satisfied. Wealso do not know whether B3 and B4 are independent. These questionsare strictly related to the problem of understanding the distribution oforbits in polygonal billiards.

Lemma 5.5. We have µ(H) = 1 provided that B1 and B3 and aresatisfied.

Proof. Since µ(R+1 ) = 0 for billiards satisfying B1 (see the end of Sub-

section 2.3), we trivially obtain µ(R) = µ(N ′) = 0. From B3, we obtainimmediately µ(N) = 0.


5.3. Main results. The central result of this paper is the followingtheorem. Its proof is given in Section 6.

Theorem 5.6. If a billiard in a domain Ω satisfies Conditions B1-B4, then every point of H has a neighborhood contained (mod 0) in aBernoulli component of T .

We now prove that the map T is Bernoulli if it also satisfies Con-dition B5. As already explained in the introduction to this section,B5 applies only to a small subclass of billiards satisfying B1-B4 (seeTheorem 8.7). We could have weaken considerable B5 for it to includemany more hyperbolic billiards, but at the price of a much more techni-cal formulation. Instead of attempting to give the weakest formulationof B5, we opted for a strong condition but with a simple formulationthat allows for a relatively simple proof of the Bernoulli property forbilliards.

Corollary 5.7. If a billiard in a domain Ω satisfies Conditions B1-B4,then every Bernoulli component of T is open (mod 0).

Proof. Let B be a Bernoulli component. Since µ(B) > 0, we haveµ(B ∩H) > 0. Let x ∈ B ∩H, and let U be the neighborhood of x asin Theorem 5.6. The set V :=

⋃n∈Z T

nU is open. Moreover, since Vis invariant and contained (mod 0) in B, it follows that B = V (mod0).

Corollary 5.8. If the billiard in a domain Ω satisfies Conditions B1-B5, then the map T is Bernoulli.

Proof. By Theorem 5.6, every point of H has a neighborhood containedup to a set of zero measure in a Bernoulli component of T . The sameis true for every connected component of H, and so for every Mi ∩Hsuch that Mi ⊂ M− ∪M+ by the first part of Condition B5. Sinceµ(H) = 1 (see Remark 5.1), we conclude that every set Mi ⊂M−∪M+

is contained (mod 0) in a single Bernoulli component of T .We now show that if Γi and Γj intersect, then Mi and Mj are

contained in the same Bernoulli component of T . First, note thatS−1 \ (R+

∞ ∪ N+) is contained in H. Next, since S−1 ∩ R+k is finite for

every k > 0 (see [15, Propositions 6.17-6.19]), it follows that S−1 ∩ R+∞

is countable. This together with B4 implies that m−-a.e. element ofS−1 is contained in H. Hence, if p ∈ Γi ∩ Γj is a corner of ∂Ω, thenwe can find x ∈ H ∩ S−1 such that the ray emerging from −x is arbi-trarily close to `, the line bisecting p. Now, let U be the neighborhoodof x contained (mod 0) in one Bernoulli component of T as statedby Theorem 5.6. It is clear that T−1U is contained (mod 0) in thesame Bernoulli component. But, if −x is sufficiently close to `, thenMi ∩ T−1U and Mj ∩ T−1U are non-empty open sets so that Mi andMj must belong (mod 0) to the same Bernoulli component.


The previous conclusion implies that q−1(Σ) is contained (mod 0) ina Bernoulli component for every connected component Σ of ∂Ω. SinceΩ is connected, it follows that all sets Mi belongs to the same Bernoullicomponent, i.e., T is Bernoulli.

6. Local ergodicity

In this section, we prove Theorem 5.6. This is accomplished by ap-plying Theorem A.13 to the billiard map T . Because of its length,Theorem A.13 is stated in the Appendix together with all the notionsrequired for its formulation. This theorem is the version for planar bil-liards of a Local Ergodic Theorem for hyperbolic symplectomorphismswith singularities in any dimension proved in [14]. Theorem A.13 hasfour main hypotheses called Conditions L1-L4. In this section, weprove Conditions L1-L3. Section 7 is entirely devoted to the proof ofConditions L4.

Proof of Theorem 5.6. The wanted conclusion follows at once by ap-plying Theorem A.13 to points of H. To do that, we need to show thatevery point of H is sufficient (see Definition A.1), and that for eachof these points, Conditions L1-L4 of Theorem A.13 are satisfied. Thefirst fact is proved in Corollary 6.3, whereas the second fact is provedin Propositions 6.4, 6.5, 6.6 and 7.29.

6.1. Sufficient points. We now prove that every point of H is suffi-cient (see Definition A.1) and that Conditions L1-L3 are satisfied. Asalready mentioned, Condition L4 is proved in Section 7. We start withsome remarks concerning the cone fields used in our arguments.

For technical reasons, which will be clear from the proof of Proposi-tion 6.6, we extend the family of cone fields (Ux, Cx)x∈E introducedin Subsection 4.3 to S−1 \ (R+

∞ ∪N+) and S+1 \ (R−∞ ∪N−). We explain

how to define (Ux, Cx) only for x ∈ S−1 \ (R+∞ ∪N+), the definition for

x ∈ S+1 \ (R−∞ ∪ N−) being similar. Since x /∈ R+

∞ ∪ N+, there existsnk ∞ such that T nkx ∈ E for every k > 0. Write x1 = T n1x, andconsider the cone field (Ux1 , Cx1). Then choose Ux to be a neighbor-hood of x contained in T−n1Ux1 such that Ux ∩ R+

n1= ∅, and define

Cx(y) = Dx1T−n1Cx1(y) for all y ∈ T n1Ux.

Given x ∈M \R+m for some m > 0, we denote by l(x, Tmx) the sum

of the length of the segments [q(T kx), q(T k+1x)] for 0 ≤ k ≤ m − 1.Also, let λ > 0 be the constant in Condition B2.

Given x, T kx ∈ E for some k ∈ N, define

σ(DxTk) = inf

u∈intCx(x)\0

√QC

Tkx(T kx,DxT ku)

QCx(x, u),


and

σ∗(DxTk) = inf

u∈intCx(x)\0

√QC

Tkx(T kx,DxT ku)

‖u‖.

Note the two cone fields Cx and CTkx entering into the previous defini-tion. This is the only difference between σ(DxT

k) and σ∗(DxTk) and

the analogous quantities introduced in Subsection 4.2.

Lemma 6.1. Suppose that x ∈ Ej \ R+m for some m ≥ 1, Tmx ∈ Ek

and T ix /∈ E for every 1 ≤ i < m. Then there exists a constant cindependent of x such that

σ(DxTm) ≥

√1 + cδ +

√cδ,

where δ = λ if Γj or Γk is focusing, and δ = l(x, Tmx) if Γj and Γk aredispersing.

Proof. Note that if m > n(x) + 1, then we must have T ix ∈ M0 forn(x) < i < m. For this reason, the parameter m does not play anyrole in the computation of σ(DxT

m), only the parameter l = l(x, Tmx)matters. We can therefore assume without loss of generality that m =n(x) + 1.

By the definition of (Ux, Cx)x∈E, we can write

CTmx(Tmx) =

u ∈ TTmxM \ 0 : a ≤ τ−(Tmx, u) ≤ b

∪ 0,

DxTmCx(x) =

u ∈ TTmxM \ 0 : a ≤ τ−(Tmx, u) ≤ b

∪ 0,

where

a = G−Tmx(Tmx), b = +∞,

a = l − supτ+m−1(x, u) : 0 6= u ∈ Cx(x)

,

b = l − infτ+m−1(x, u : 0 6= u ∈ Cx(x)

.

Since m− 1 = n(x), Condition B2 implies that

supτ+m−1(x, u) : 0 6= u ∈ Cx(x)

≤ λ+

j .

By the definition of Cx(x) when Γj is dispersing and Part (3) of The-orem 4.2 when Γj is focusing, we deduce that

d(Tm−1x)

2≤ inf

τ+m−1(x, u) : 0 6= u ∈ Cx(x)

.

Also, note that G−Tmx(Tmx) ≤ λ−k . Therefore,

a ≤ λ−k , l − λ+j ≤ a, b ≤ l − d(Tm−1x)

2. (6)


Since DxTmCx(x) ⊂ intCTmx(T

mx) by Lemma 5.2, we can use aformula for σ(DxT

m) proved by Wojtkowski [32, Lemma A.4 and Ap-pendix B], and obtain1

σ(DxTm) =

√1 + w +

√w, w =

a− ab− a

. (7)

From (6), it follows that

w ≥l − λ+

j − λ−kλ+j − d(Tm−1x)/2

.

It is easy to see that d(Tm−1x)/2 < λ+j and d(Tm−1x)/2 ≤ 1/c for some

constant c depending only on Ω. Hence,

w ≥ c(l − λ+j − λ−k ).

The wanted conclusion now follows from (7) once we have observedthat l − λ+

j − λ−k ≥ λ by B2 if Γj or Γk is focusing, and λ+j = λ−k = 0

if Γj and Γk are dispersing.

Proposition 6.2. Let x ∈ E \ R+∞, and suppose that there exists a

strictly increasing (decreasing) sequence of positive (negative) integersnkk∈N such that T nkx ∈ E for every k > 0. Then

limk→+∞

σ(DxTnk) = +∞.

Proof. We prove the proposition only for the case when nkk∈N is astrictly increasing sequence of positive integers. For the other case, theproof is similar in view of the fact that σ(DxT

−n) = σ(DT−nxTn) for

n > 0 (see [22, Section 6]).Without any loss of generality, we can assume that T jx /∈ E for all

nk < j < nk+1. Set n0 = 0, and define xk = T nkx and mk = nk+1 − nkfor k ≥ 0. Also, let lk be the distance between q(T−1xk+1) and q(xk+1).By the supermultiplicativity of σ, we have

σ(DxTnk) ≥

k−1∏i=0

σ(DxiTmi). (8)

By Lemma 6.1, we see that the sequence∏k−1

i=0 σ(DxiTmi) is strictly

increasing in k, and it does not diverge only if T nix ∈ M− for isufficiently large and limi→+∞ li = 0. This means that the positivesemi-trajectory of x is trapped forever inside a cusp formed by twodispersing components or a dispersing and a flat component. Butevery trajectory entering a cusp leaves it after finitely many colli-sions (for instance, see Appendix A1.3 of [7]). Hence, we must have

limn→+∞∏k−1

i=0 σ(DxiTmi) = +∞, which implies the wanted conclusion

by (8).

1In general, we have ρ = b−bb−a ·

a−ab−a

. Here, we have used b = +∞.


For the definitions of sufficient and essential points and relative no-tations, see Definitions A.1 and A.2.

Corollary 6.3. Suppose that x ∈ H. Then there exists l ∈ Z suchthat T lx is essential, and so x is sufficient. If we further assume thatx belongs to S−1 ∪ S+

1 ∪ E, then x is essential.

Proof. Let x ∈ H. By the definition of H, we can find a sequencenk ∞ such that either T nkx ∈ E or T−nkx ∈ E for k > 0. We provethe corollary only for the case T nkx ∈ E for k > 0, the proof for theother case being similar.

Let xk = T nkx for k > 0. By Lemma 5.2, we have σ(Dx1Tn2−n1) > 1,

and so σ∗(Dx1Tn2−n1) > 0 by (4). Fix α > 0. Proposition 6.2 allows us

to find k > 2 such that σ(Dx1Tnk−n1) > α/σ∗(Dx1T

n2−n1). Using (5),we then obtain σ∗(Dx1T

nk−n2) > α. Now, it is easy to see that thereexists a neighborhood V of x1 such that V ⊂ Ux1 , V ∩R+

nk−n1= ∅ and

T nk−n1V ⊂ Uxk . Since σ∗(D·Tnk−n2) is continuous at y, we can choose

V so that σ∗(DzTnk−n2) > α for every z ∈ V . But this means that x1

is u-essential with nx1,α = nk − n1, Ox1,α = V and the cone field Kx1,α

given by Kx1,α = Cx1 on V , and Kx1,α = Cxk on T nk−n1V . By choosingα = 3, we see that x is sufficient with quadruple (l, N,O,K) such thatl = n1, N = nx1,3, O = T nx1,3Ox1,3 and K = Kx1,3.

To prove the last part of the corollary, we observe that if we furtherassume that x ∈ S−1 ∪ S+

1 ∪E, then (Ux, Cx) is defined (if x ∈ H ∩ S±1 ,then x ∈ S±1 \ (R∓∞∪N∓)), and the previous argument can be repeatedverbatim with x1 and n1 replaced by x and 0, respectively.

We can now proceed to prove Conditions L1-L3.

6.2. Proof of Conditions L1-L3. Consider x ∈ H. By Corollary 6.3,it follows that x is sufficient. Let (l, N,O,K) be the quadruple associ-ated to x. From the proof of Corollary 6.3, we see that if z := T lx ∈E, then T−Nz ∈ E, O ⊂ Uz, T

−NO ⊂ UT−Nz and the cone field(O ∪ T−NO,K) is given by K = Cz on O and K = CT−Nz on T−NO.Also, note that by construction O and T−NO do not intersect ∂M .Since E is open (in the topology of M), by taking O sufficiently small,we can assume without loss of generality that O and T−NO are bothcontained in E.

To prove Conditions L1-L3, we use several results concerning thesingulars sets R−k and R+

k proved in [15].

Proposition 6.4. Condition L1 is satisfied.

Proof. For billiards satisfying Conditions B1 and B2, the regularity ofthe singular sets R−k and R+

k was proved in [15, Theorem 2.2].

Proposition 6.5. Every point x ∈ H satisfies Condition L2.

Proof. This proposition follows from [15, Proposition 6.2]. For theconvenience of the reader, we give here a more direct proof. We prove


only the first part of Condition L2, because the second one can beproved similarly. Let Σ be a component of R−k with k > 0. Since∂M ∩ T−NO = ∅, we can assume without loss of generality that Σis not contained in ∂M . It follows from the definition of R−k thatthere exists 0 ≤ i < k such that the rays emerging from the pointsof −T−iΣ focus in linear approximation at a corner of ∂Q or a pointlying on a dispersing component of ∂Q. Therefore, if y ∈ Σ ∩ T−NO,then we have τ−−i(y, u) = t(−T−iy) for every u ∈ T ∗yΣ. Now, it is notdifficult to see that the invariance of (Uz, Cz)z∈E and the second partof Condition B2 imply that τ−−i(y, v) < t(−T−iy) whenever 0 6= v ∈K ′(y). We conclude that TyΣ ⊂ intK(y).


Proof. The proofs of Condition L3 concerning S−1 and S+1 are essentially

the same: one is obtained from the other one by exchanging the symbols+ and −, and replacing T with T−1. Thus, it is enough to prove thepart concerning S−1 .

The set S−1 ∩R+∞ is at most countable by [15, Propositions 6.17-6.19].

This fact and Condition B3 imply that m−(S−1 ∩ (N+ ∪ R+∞)) = 0.

Therefore, it is enough to prove that each element of S−1 \ (N+ ∪R+∞)

is u-essential. This is so because of Corollary 6.3. The second partof L3 is a direct consequence of Lemma 5.2 and the construction ofthe cone fields (O,K) for x ∈ H and (Oy,α ∪ T ny,αOy,α, Ky,α) for y ∈S−1 \ (N+ ∪R+

∞) in the proof of Corollary 6.3.

7. Proof of Condition L4

In this section, we prove Condition L4 for every x ∈ H. This con-dition requires the existence of stable and unstable manifolds a.e. onthe neighborhood O of x, which is guaranteed by Proposition A.3. Wederive L4 from the non-contraction property : there exists β′ > 0 suchthat if z ∈ E \R+

m and Tmz ∈ E ∪ −E with m > 0, then

‖DzTmv‖ ≥ β′‖v‖ for v ∈ Cz(z), (9)

where (Uz, Cz)z∈E is the cone field introduced in Subsection 4.3. Toprove that this property holds true we decompose the billiard orbitsinto special blocks, and study (9) separately for each block. This anal-ysis will be carried out using certain semi-norms defined in terms oftransversal Jacobi fields along billiard orbits.

7.1. Jacobi fields and semi-norms. Let y ∈M . There is a bijectivecorrespondence between a vector u ∈ TyM and a vector (J, J ′) ∈ R2

such that 〈J, v(y)〉 = 〈J ′, v(y)〉 = 0. The pair (J, J ′) defines a Jacobifield along the segment [q(y), q1(y)] of the billiard trajectory of y givenby J(s) = J + sJ ′ for 0 ≤ s ≤ t(y). The property of (J, J ′) impliesthat 〈J(s), v(y)〉 = 0 for 0 ≤ s ≤ t(y). Such a Jacobi field is called


transversal. For more on transversal Jacobi fields and billiards, see [10,33].

In fact that the pair (J, J ′) forms a system of coordinates for thetangent space TyM . The change of coordinates (us, uθ) 7→ (J, J ′) isgiven by

J = sin θ(y)us and J ′ = −κ(y)us − uθ,

where we have chosen the orientation so that the line orthogonal tov(y) makes with the positive tangent line to ∂Ω an angle less thanπ/2. To underline the dependence of J and J ′ on the vector u, wewill write J(u) and J ′(u). When no collision with ∂Ω occurs duringan interval of length t ∈ R, the evolution of (J, J ′) is given by a lineartransformation F (t). At a collision y ∈ M \ ∂M , the pair (J, J ′) istransformed according to a linear transformation R(y). The maps F (t)and R(y) are as follows:

F (t) =

(1 t0 1

)and R(y) =

(−1 0

2d(y)

−1

), (10)

where the entry 2/d(y) has to be understood as zero when y ∈ M0.Hence, the matrix of DyT in coordinates J and J ′ is given by

DyT = R(Ty)F (t(y)) =

( −1 −t(y)2

d(Ty)−1 + 2 t(y)

d(Ty)

). (11)

Note that all the matrices (10) and (11) have determinant equal to ±1.This means that in coordinates J and J ′, the map DyT preserves thestandard symplectic form J ∧ J ′. Finally, we observe that the forwardfocusing time of a tangent vector u ∈ TyM with y ∈ M in terms ofJ(u) and J ′(u) is given by

τ+(y, u) =

− J(u)J ′(u)

if J ′(u) 6= 0,

∞ if J ′(u) = 0.(12)

Definition 7.1. For every y ∈M \ ∂M and every u ∈ TyM , define

‖u‖J =√J2(u) + J ′2(u) and |u|J ′ = |J ′(u)|.

In the rest of this subsection, we prove several relations involvingthe semi-norms ‖ · ‖ and ‖ · ‖J . The goal here is to show that ‖ · ‖ and‖ · ‖J are equivalent on a certain subset of the tangent bundle TM .The cone fields (Uy, Cy)y∈E used in the propositions below are thoseintroduced in Subsection 6.2.

Lemma 7.2. There exists a constant α1 > 1 depending only on Ω andthe family of cone fields (Uy, Cy)y∈E+ such that if y ∈ E+, z ∈ Uyand 0 ≤ k ≤ n(z), then

|DzTku|J ′ ≤ ‖DzT

ku‖J ≤ α1|DzTku|J ′ for u ∈ Cy(z).


Proof. Let y, z, k, u be as in the hypotheses of the lemma. It is clearthat |DzT

ku|J ′ ≤ ‖DzTku‖J . By Part (3) of Theorem 4.2, there exists

a constant f > 0 depending only on (Uy, Cy)y∈E such that

d(T kz)

2≤ τ+

k (z, u) ≤ f.

By (12), we then have |J(DzTku)| ≤ f |J ′(DzT

ku)|, which gives theremaining inequality with α1 = (1 + f 2)1/2.

Lemma 7.3. There exists α2 > 1 such that if y ∈M \ ∂M , then

‖u‖J ≤ α2‖u‖ for u ∈ TyM.

Proof. A straightforward computation gives the wanted inequality withα2 = (2 + κ2

1)1/2, where κ1 = maxz∈M |κ(z)|.

Lemma 7.4. There exists α3 > 0 such that if y ∈ M− and z ∈ Uy,then

‖u‖ ≤ α3‖u‖J for u ∈ Cy(z).

Proof. Let κ2 = minz∈M− |κ|. By the definition of Cy(z), we haveuθ/us ≤ κ(z) < 0 for all 0 6= u ∈ Cy(z). Hence,

‖u‖2J ≥ |J(u)|2 = κ2(z)u2

s + u2θ + 2κ(z)usuθ

≥ κ22u

2s + u2

θ ≥κ2

2

1 + κ22

(u2s + u2

θ) =κ2

2

1 + κ22

‖u‖2.

Lemma 7.5. There exists α4 > 0 such that if y ∈ E+, z ∈ Uy andu ∈ Cy(z), then

‖DzTku‖ ≤ α4‖DzT

ku‖J for 0 ≤ k ≤ n(z).

Proof. Denote by κ3 the maximum of κ on M+, and denote by m thesmallest mi associated to focusing components of ∂Ω as in Theorem 4.2.Let 0 6= u ∈ Cy(z), and write (uk,s, uk,θ) for the vector DzT

ku. FromTheorem 4.2, we know that if z ∈ Mi ⊂ M+, then −κ(T kz) + mi ≤m(uk) ≤ κ(T kz) for 0 ≤ k ≤ n(z). In particular, |uk,θ| ≤ κ(T kz)|uk,s|.Therefore,

‖DzTku‖2

J ≥(κ(T kz)uk,s + uk,θ

)2=(κ(T kz) +m(uk)

)2u2k,s

≥(κ(T kz)− κ(T kz) +mi

)2u2k,s ≥

m2i

1 + κ2(T kz)‖DzT

ku‖2

≥ m2

1 + κ23

‖DzTku‖2.

Lemma 7.6. There exist ε0 > 0 and 0 < θ0 < π/2 such that θ(M0 ∩S−1 (ε0)) ∈ (θ0, π − θ0).


Proof. The lemma is an immediate consequence of the following easy-to-check fact. Consider a flat component Γi of ∂Ω. If the line containingΓi contains also a corner p or is tangent to a dispersing component Γj,then no ray emerging from the elements of Mi ∩ S+

1 contains p or istangent to Γj.

Lemma 7.7. Let ε0 be the constant in Lemma 7.6. There exists α5 > 0such that if y ∈M0 ∩ S−1 (ε0), then

‖u‖ ≤ α5‖u‖J for u ∈ TyM.

Proof. Since κ(y) = 0, we have ‖u‖2J ≥ sin2 θ0u

2s+u2

θ > sin2 θ0‖u‖2.

Corollary 7.8. There exist two constants 0 < A1 < A2 such that ify ∈ E, z ∈ Uy, u ∈ Cy(z) or y ∈M0 ∩ S−1 (ε0), u ∈ TyM , then

A1‖u‖J ≤ ‖u‖ ≤ A2‖u‖J .Proof. The claim follows from Lemmas 7.3-7.7.

Remark 7.9. It can be easily proved that Corollary 7.8 remains validif ‖ ·‖ is replaced by | · |J ′, thus showing that the semi-norms ‖ ·‖, ‖ ·‖J ,| · |J ′ are equivalent in the sense specified in the corollary. However,this general equivalence is not needed for the proof of L4.

7.2. Non-contraction property and block decomposition. Thenon-contraction property is just a slight modification of the conditionwith the same name introduced in [22]. Roughly speaking, this prop-erty states that the contraction of the iterations of vectors from thecones in Uy, Cyy∈E is uniformly bounded below along orbits startingat E and ending at E ∪ −E. This property provides some control onthe accumulation of the expansion along the unstable direction alongbilliard orbits. In particular, it implies no loss of expansion alongarbitrarily long sequences of consecutive collisions with focusing com-ponents or between flat components.

We now show that every sequence of collisions z, . . . , Tmz as inthe non-contraction property can be decomposed into a finite numberof special subsequences called blocks.

Definition 7.10. Let z ∈M \ R+m with m > 0. A sequence of consec-

utive collision z, . . . , Tmz is called a block of type i ∈ 1, . . . , 4 ifthe corresponding condition (i) below is satisfied:

(1) z ∈ E+, T n(z)+1z, . . . , Tm−1z ⊂M0 and Tmz ∈M−,(2) z ∈M−, Tz, . . . , Tm−1z ⊂M0 ∪M− and Tmz ∈M−,(3) z ∈M−, Tz, . . . , Tm−1z ⊂M0 and Tmz ∈ E+,(4) z and Tmz belong to E+.

Definition 7.11. A block is called minimal if it does not contain anyother block of the same type. A block included in a sequence of consec-utive collisions ϕ is called maximal in ϕ if it is not contained in anyother block of the same type in ϕ.


We observe that blocks of type 1 and 3 are always maximal andminimal. Also, we observe that every block of type 2 and 4 is a unionof finitely many minimal blocks of the type 2 and 4, respectively.

Proposition 7.12. Let ϕ = z, . . . , Tmz be a sequence of collisionssuch that both z and Tmz belong to E. Then ϕ = ϕ1 ∪ · · · ∪ ϕn withn ≤ 5 and ϕ1, . . . , ϕn being maximal blocks of type 1-4. Moreover, thisdecomposition is unique.

Proof. First, suppose that ϕ does not contain blocks of type 2 and 4.Then we see that either ϕ is a block of type 1 or 3, or ϕ is the unionof two blocks: a block of type 3 followed by a block of type 1.

Now, suppose that ϕ contains blocks of type 2, but does not containblocks of type 4. A maximal block of type 2 can only be preceded by ablock of type 1 and followed by a block of type 3. Hence, ϕ can containat most two maximal blocks of type 2. Moreover, if ϕ contains exactlytwo maximal blocks of type 2, then ϕ is the union of a maximal blockof type 2, a block of type 1, a block of type 3 and a maximal block oftype 2, following one another in this order. Instead, if ϕ1 contains onlya single maximal block of type 2, then there are three possibilities: ϕis a block of type 2, ϕ is the union of a maximal block of type 2 and ablock of type 1 or 3, and ϕ is the union of a block of type 1, a maximalblock of type 2 and a block of type 3, following one another in thisorder.

Finally, suppose that ϕ contains blocks of type 4. From the definitionof these blocks, we see that there is only one single maximal block oftype 4 in ϕ. Such a block can only be preceded by a block of type 3and followed by a block of type 1. Moreover, blocks of type 1 and 3 inϕ can only be adjacent to blocks of type 2 or 4. Since there is only onemaximal block of type 4 in ϕ, besides this block, ϕ can contain at mosta block of type 1, at most a block of type 3 and at most two maximalblocks of type 2. The blocks of type 1 and 3 are attached to the blockof type 4, whereas one block of type 2 is attached to the block of type1, and the second block of type two is attached to the block of type 3.

It is not difficult to see that the decomposition of ϕ into blocks oftype 1-4 we have just obtained is unique, because the blocks formingit are maximal.

7.3. Proving the non-contraction property. It suffices to showthat (9) holds along each block of type 1-4 with β′ depending only onthe block type.

We start with some preliminary results.

Lemma 7.13. Let 0 ≤ m1 < m2, and suppose that z ∈ E \ R+m2

and0 6= v ∈ Cz(z). Then

|DzTm2v|J ′

|DzTm1v|J ′=

m2∏k=m1+1

∣∣∣∣τ−k (z, v)

τ+k (z, v)

∣∣∣∣ .


Proof. Define zk = T kz and vk = DzTkv for 0 ≤ k ≤ n(z). By the

definition of Cz and its invariance, it is not difficult to see using Theo-rem 4.2 that J ′(vk) 6= 0 for 0 ≤ k ≤ n(z). Using (11) and Condition B2,one can further show that J(vk) 6= 0 for 1 ≤ k ≤ n(z). Therefore,

|vm|J ′|vm1|J ′

=m∏

k=m1+1

∣∣∣∣J ′(vk)J(vk)

∣∣∣∣ · ∣∣∣∣ J(vk)

J ′(vk−1)

∣∣∣∣ .Now, the wanted equality follows from τ+(zk, vk) = −J(vk)/J

′(vk) andτ−(zk, vk) = J(vk)/J

′(vk−1). The first expression for τ+(zk, vk) is just(12), whereas the one for τ−(zk, vk) can be easily derived from (11) and(12).

Proposition 7.14. There exists γ1 > 0 such that if z ∈ E+ and 0 ≤m1 < m2 ≤ n(z), then

‖DzTm2v‖ ≥ γ1‖DzT

m1v‖ for v ∈ Cz(z).

Proof. In virtue of Lemmas 7.2, 7.3 and 7.5, we can prove the propo-sition with the norm ‖ · ‖ replaced by the semi-norm | · |J ′ . Letz ∈ Ei ⊂ E+ for some i, and pick v ∈ Cz(z). Of course, it is enoughto prove the proposition with γ1 depending on the focusing componentΓi. By taking the minimum of such γ1’s over all focusing components,we obtain the proposition in its general form. Let θi, ai, ci, Ni be as inTheorem 4.2 and Proposition 4.3. We define zk = T kz, vk = DzT

kv for0 ≤ k ≤ n(z).

Suppose first that θ = θ(z) ∈ (0, θi)∪(π−θi, π). We consider only thecase when θ ∈ (0, θi), because the argument for θ ∈ (π−θi, π) is similar.By Theorem 4.2, we have m(vk) ≥ −aiθ. Let Ri = maxy∈Mi

1/κ(y).Using (1), we easily obtain

τ−(zk, vk)

τ+(zk, vk)≥ 1−Riaiθ

1 +Riaiθ

By Proposition 4.3, we have n(z) ≤ ci/θ, and so Lemma 7.13 impliesthat

|vm2|J ′|vm1|J ′

≥(

1−Riaiθ

1 +Riaiθ

) ciθ

.

Since the right hand-side of the previous inequality converges to e−2aiciRi >0 as θ → 0+, there exists 0 < θ < θi such that

|vm2|J ′|vm1|J ′

≥ 1

2e−2aiciRi for θ ∈ (0, θ). (13)

Now, suppose that θ(z) ∈ [θ, π − θ]. By Part (3) of Theorem 4.2,there exists a constant fi > 0 such that d(zk)/2 ≤ τ±(zk, vk) ≤ fi for1 ≤ k ≤ m2. Let ri = miny∈Mi

1/κ(y), and let di = ri sin θi. Then

τ−(zk, vk)

τ+(zk, vk)≥ d(zk)

2fi≥ di

2fifor 1 ≤ k ≤ m2.


Using Lemma 7.13 and Proposition 4.3, we conclude that

|vm2 |J ′|vm1 |J ′

≥ min

1,

(di2fi

)Ni(θ). (14)

The wanted conclusion now follows from (13) and (14).

Let λ be as in B2, and let fi be as in the proof of Proposition 7.14.Define f to be the maximum of all fi’s.

Lemma 7.15. Consider a sequence of collisions z, . . . , Tmz withm > 0 such that z, Tmz ∈ E, and T kz ∈ M0 for n(z) < k < m.Then, for every 0 6= v ∈ Cz(z) and every n(z) ≤ k < m, we have

|DzTmv|J ′ > δ|DzT

kv|J ′ ,where δ = λ/f if Tmz ∈M+, and δ = 1 if Tmz ∈M−.

Proof. Let 0 6= v ∈ Cz(z), and define zk = T kz and vk = DzTkv for

0 ≤ k ≤ m. From zn(z)+1, . . . , zm−1 ⊂ M0, it follows that |vk|J ′ =|vm−1|J ′ for all n(z) ≤ k < m. Then, by Lemma 7.13, we have

|vm|J ′|vk|J ′

=|vm|J ′|vm−1|J ′

=τ−(zm, vm)

τ+(zm, vm)for n(z) ≤ k < m.

If zm ∈ M+, then τ−(zm, vm) = l(zn(z), zm) − τ+(zn(z), vn(z)) > λ byCondition B2. Since 0 < τ+(zm, vm) ≤ f (see the proof of Proposi-tion 7.14), we obtain τ−(zm, vm)/τ+(zm, vm) > λ/f . If zm ∈M−, thenwe use the Mirror Formula (2) to relate the focusing times τ−(zm, vm)and τ+(zm, vm). A simple computation using d(zm) < 0 shows thatτ−(zm, vm) < τ+(zm, vm) < 0, and so τ−(zm, vm)/τ+(zm, vm) > 1.

Lemma 7.16. Consider a sequence of collisions z, . . . , Tmz withm > 0 such that z ∈ M−, Tmz ∈ E and T kz ∈ M0 for n(z) < k < m.Then, for every 0 6= v ∈ Cz(z) and every 0 ≤ k < m, we have

|J(DzTmv)| > |J(DzT

kv)|.

Proof. Let 0 6= v ∈ Cz(z). By the definition of Cz (see Subsec-tion 4.3), it is easy to see that J(v)J ′(v) ≥ 0 and J ′(v) 6= 0. SinceTz, . . . , Tm−1z ⊂M0, we have |J ′(DzT

kv)| = |J ′(v)| for 0 < k < m.Using (11), we obtain

|J(DzTkv)| = |J(v)|+ l(z, T kz)|J ′(v)| for 0 ≤ k ≤ m,

where l(z, T kz) is the length of the piece of trajectory starting at zand ending at T kz (see the beginning of Subsection 6.1). The wantedconclusion follows immediately from the previous equality.

Lemma 7.17. There exists a constant δ1 > 0 such that for every se-quence of consecutive collisions z, . . . , Tmz with z ∈ E+, Tmz ∈ Eand T kz ∈M0 for n(z) < k < m, we have

‖DzTmv‖ ≥ δ1‖DzT

n(z)v‖ for v ∈ Cz(z).


Proof. If we replace ‖ · ‖ with | · |J ′ , then the wanted conclusion withδ1 = δ follows from Lemma 7.15. To obtain the actual conclusion, usethe obvious fact that ‖ · ‖J ≥ | · |J ′ , apply Lemma 7.3 to ‖DzT

mv‖J ,and finally apply Lemmas 7.2 and 7.5 to |DzT

n(z)v|J ′ and ‖DzTn(z)v‖J ,

respectively.

Lemma 7.18. There exists β′1 > 0 such that every sequence of consecu-tive collisions z, . . . , Tmz such that z ∈ E+, T n(z)+1z, . . . , Tm−1z ⊂M0 and Tmz ∈ E satisfies (9) with β′ = β′1. In particular, the previousconclusion is true for every block of type 1.

Proof. Let ϕ = z, . . . , Tmz be a block of type 1, and choose 0 6= v ∈Cz(z). By Proposition 7.14, we have ‖DzT

n(z)v‖ ≥ γ1‖v‖. To completethe proof, use Lemma 7.17.

Lemma 7.19. There exists β′2 > 0 such that every block of type 2satisfies (9) with β′ = β′2.

Proof. Note first that every block of type 2 consists of finitely manyminimal blocks of type 2. Next, suppose that z, . . . , Tmz is a minimalblock of type 2, and let 0 6= v ∈ Cz(z). In this case, we have T kz ∈M0

for 1 ≤ k ≤ m − 1. It follows that we have |J ′(DzTmv)| > |J ′(v)|

by Lemma 7.15, and |J(DzTmv)| > |J(v)| by Lemma 7.16. Therefore,

‖DzTmv‖J > ‖v‖J . Of course, the same conclusion extends to a general

block of type 2. To complete the proof, apply Lemmas 7.3 and 7.4 to‖DzT

mv‖J and ‖v‖J , respectively.

Lemma 7.20. There exists β′3 > 0 such that every block of type 3satisfies (9) with β′ = β′3.

Proof. Suppose that z, . . . , Tmz is a block of type 3, and let 0 6=v ∈ Cz(z). Since T kz ∈ M0 for 1 ≤ k < m, we have |J ′(DzT

mv)| >|J ′(v)|λ/f by Lemma 7.15, and |J(DzT

mv)| > |J(v)| by Lemma 7.16.Hence, ‖DzT

mv‖J > minλ/f, 1‖v‖J . To complete the proof, applyLemmas 7.3 and 7.4 to ‖DzT

mv‖J and ‖v‖J , respectively.

Let z, . . . , Tmz be a block of type 4. As before, we define zj = T jzfor all 0 ≤ j ≤ m. We will use this notation throughout the restof this section. Note that every block of type 4 consists of finitelymany minimal blocks of type 4. In other words, there exist N > 0integers 0 = i0 < · · · < iN = m such that zik , . . . , zik+1

is a minimalblock of type 4 for each 0 ≤ k ≤ N − 1 (see Fig. 1). Now, recallthat n(zik) denotes the number of consecutive collisions of zik with thefocusing component Γi before leaving it (see Definition 3.2). Definejk = n(zik) + ik. Then

DzTm = DzjN−1

T iN−jN−1 Dzj0T jN−1−j0 Dz0T

j0 .


Proposition 7.21. If N > 1, then the matrix of Dzj0T jN−1−j0 in co-

ordinates (J, J ′) is given by

Dz0TjN−1−j0 = F−1

N−1AN−1BN−2 · · ·B1A1B0F0,

where Ak, Bk, Fk are matrices of the form

Ak = ±(

1/ζk 0ηk ζk

), Bk = ±

(1 + ak bkck 1 + dk

), Fk =

(1 fk0 1

)(15)

with ζk, ηk, fk > 0, ak, ck, dk ≥ 0 and bk > λ.

Proof. For every 0 ≤ k ≤ N − 1, define

s−k = G−zik(zik) and s+

k = supτ+jk−ik(zik , v) : 0 6= v ∈ Czik (zik)

,

where G±zikare defined in Definition 4.4. Recall the definitions of the

matrices F and R in (10). If we define

Ak = F (s+k )Dzik

T jk−ikR(zik)F (s−k ),

Bk = F−1(s−k )R−1(zik+1)Dzjk

T ik+1−jkF−1(s+k ),

Fk = F (s+k ),

then we easily see that

DzjkT jk+1−jk = F−1

k+1Ak+1BkFk.

Therefore,

Dz0TjN−1−j0 =

N−2∏k=0

DzjkT jk+1−jk = F−1

N−1AN−1BN−2 · · ·B1A1B0F0.

The form of the matrices Fk can be immediately derived from (10).Since fk = s+

k , the positivity of fk follows from s+k > 0. We now de-

termine the form of the matrices Ak and Bk. Let vk ∈ TzikM such

that τ+(zik , vk) = G+zik

(zik), and denote by q−k and q+k the points along

the trajectory of zik where the vector vk focuses backward, and thevector Dzik

T jk−ikvk focuses forward, respectively. The matrix Ak de-scribes the dynamics of transversal Jacobi fields along the trajectorystarting at q−k and ending at q+

k (see Fig. 1). In other words, if (J0, J′0)

is a transversal Jacobi field at q−k , then (J1, J′1) := Ak(J0, J

′0) is its

evolution at q+k .

By construction the property of the billiard cone field (see Theo-rem 4.2), every transversal Jacobi field focusing at q−k focuses again atq+k . Since by the Mirror Formula, τ+ is a strictly decreasing function

of τ−, it follows that every transversal Jacobi field consisting of raysparallel to the billiard trajectory passing through q−k focuses beforereaching q+

k . Thus, there exist ξk, ζk, ηk > 0 such that

Ak

(10

)= ±

(ξkηk

), Ak

(01

)= ±

(0ζk

).


zi0 = z

zj0

ziN= zm

q+0

q+1

q1

zi1

zj1

qN

Figure 1. Decomposition of a block of type 4. The col-lisions zik ∈ E+ and zjk ∈ −E+ are the first and the lastin a sequence of consecutive collisions with a focusing arc,respectively. The sequence of collisions zik , . . . , zik+1

isa minimal block of type 4. The dashed piece of trajec-tory between the points q+

1 and q−N represents a sequenceof collisions with finitely many boundary components ofthe billiard domain.

Since Ak is product of finitely many matrices as in (10), we havedetAk = 1, and so ξk = ζ−1

k . We conclude that

Ak = ±(

1/ζk 0ηk ζk

).

To derive the form ofBk, we argue similarly. The matrixBk describesthe dynamics of transversal Jacobi fields along the piece of the billiardtrajectories starting at q+

k and ending at q−k+1 (see Fig. 1). By thedefinition of q±k , we see that along this piece of trajectory, there areonly collisions with flat or dispersing components. Hence, it easilyfollows from (11) that Bk is a product of finitely many matrices of theform

P = ±(

1 + a bc 1 + d

)(16)

with a, b, c, d ≥ 0 and detP = 1. All the matrices P forms a semi-group with respect to the standard matrix multiplication. For thisreason, there are ak, bk, ck, dk ≥ 0 such that

Bk = ±(

1 + ak bkck 1 + dk

).

Since F and R are upper and lower triangular, respectively, we seethat the entry bk cannot be smaller than the length of the piece of thetrajectory connecting q+

k and q−k+1. This observation combined withCondition B2 gives bk > λ.


The notation used in the following proposition is as in Proposi-tion 7.21 and the paragraph before it.

Proposition 7.22. There exists β′′4 > 0 such that

|FN−1DzTjN−1v|J ′ ≥ β′′4 |F0DzT

j0v|J ′ for v ∈ Cz(z).

Proof. The inequality holds trivially if N = 1. Therefore, we assumethat N > 1. Given a matrix L, denote by |L| the matrix obtained byreplacing each entry of L with its absolute value. Given two squarematrices L1 and L2 of the same order, we write L1 ≥ L2 if each entryof L1 is greater than or equal to the corresponding entry of L2. Letζ = ζ1 · · · ζN−1 > 0. By the properties the matrices Ak and Bk, itfollows that

|AN−1BN−2 · · ·A1B0| = |AN−1||BN−2| · · · |A1||B0|≥ |AN−1||AN−2| · · · |A1||B0|

≥(

1/ζN−1 00 ζN−1

)· · ·(

1/ζ1 00 ζ1

)(1 λ0 1

)=

(1/ζ λ/ζ0 ζ

).

Let v ∈ Cz(z). Also, define vk = DzTkv for 0 ≤ k ≤ m, w0 = F0vj0

and w1 = FN−1vjN−1. By Proposition 7.21, we have

w1 = AN−1BN−2 · · ·B1A1B0w0.

From the construction of the cone field (Uy, Cy)y∈E and the definitionof Fk, we easily see that 0 ≤ J(wi)/J

′(wi) ≤ f for i = 0, 1, where f isthe positive constant in the proof of Lemma 7.2. The fact that J(wi)and J ′(wi) have the same sign implies that

|J(w1)| ≥ 1

ζ|J(w0)|+ λ

ζ|J ′(w0)| and |J ′(w1)| ≥ ζ|J ′(w0)|. (17)

From 0 ≤ J(wi)/J′(wi) ≤ f and the inequality on the left hand-side

of (17), we obtain

|J ′(w1)| ≥ 1

f|J(w1)| ≥ λ

ζf|J ′(w0)|,

which together with the inequality on the right-hand sides of (17) gives

|J ′(w1)| ≥ 1

2

(ζ +

λ

ζf

)|J ′(w0)|.

But x+ c2/x ≥ 2c for every x > 0 and every c ∈ R, and so

|J ′(w1)| ≥(λ

f

)1/2

|J ′(w0)|.

The wanted inequality holds with β′′4 = (λ/f)1/2.


Lemma 7.23. Let β′′4 > 0 be the constant in Proposition 7.22. Then

‖DzTjN−1v‖ ≥ β′′4‖DzT

j0v‖ for v ∈ Cz(z).

Proof. By Lemmas 7.2, 7.3 and 7.5, it suffices to prove the inequalitywith ‖ · ‖ replaced by | · |J ′ . From the form of the matrices Fk, we seeat once that |vj0|J ′ = |w0|J ′ and |vjN−1

|J ′ = |w1|J ′ . The notation here isas in the proof of Proposition 7.22. The wanted conclusion now followsfrom Proposition 7.22.

Proposition 7.24. There exists β′4 > 0 such that every block of type 4satisfies (9) with β′ = β′4.

Proof. The notation is an in the proof of Proposition 7.21 and theparagraph before it. Accordingly, a block ϕ of type 4 is given by ϕ =zi0 , . . . , ziN. We write ϕ = ϕ1 ∪ ϕ2, where ϕ1 = zi0 , . . . , zjN−1

andϕ2 = zjN−1

, . . . , ziN. It is easy to see that there are 2 integers jN−1 <k1 ≤ k2 ≤ iN such that ϕ2 = ψ1 ∪ ψ2 ∪ ψ3 with ψ1 = zjN−1

, . . . , zk1being an orbit starting at zjN−1

∈ −E+ and ending at M− after asequence of collisions with M0, ψ2 = zk1 , . . . , zk2 being a block oftype 2, and ψ3 = zk2 , . . . , ziN being a block of type 3.

Proposition 7.14 and Lemma 7.23 give ‖vj0‖ ≥ γ1‖vi0‖ and ‖vjN−1‖ ≥

β′′4‖vj0‖, respectively. By Lemma 7.17, we obtain ‖vk1‖ ≥ δ1‖vjN−1‖

with δ1 independent of ψ1 and v0. Since ψ2 and ψ3 are blocks of type 2and 3, we have ‖vk2‖ ≥ β′2‖vk1‖ and ‖viN‖ ≥ β′3‖vk2‖ by Lemmas 7.19and 7.20. Putting all together, we obtain the wanted conclusion withβ′4 = β′2β

′3β′′4γ1δ1.

Corollary 7.25. If Tmx ∈ E, then (9) is satisfied.

Proof. If Tmx ∈ E, (9) follows from Lemmas 7.18-7.20 and Proposi-tion 7.24.

Proposition 7.26. There exists a constant γ2 > 0 such that if z ∈E \ R+

m with m > 1, T n(z)+1z, . . . , Tm−1z ⊂ M0, T jz ∈ S−1 (ε0) forsome n(z) < j < m, and Tmz ∈ E, then

‖DzTmv‖ ≥ γ2‖DzT

jv‖ for v ∈ Cz(z).

Proof. Let 0 6= v ∈ Cz(z), and define zk and vk for 0 ≤ k ≤ m as in theproof of Lemma 7.13. We study separately the two cases: J(vj)J

′(vj) ≥0 and J(vj)J

′(vj) < 0. By Lemmas 7.3 and 7.7, it is enough to provethe desired inequality with ‖ · ‖ replaced by ‖ · ‖J .

Suppose that J(vj)J′(vj) ≥ 0. By Lemma 7.15, we have |J ′(vm)| >

δ|J ′(vj)|, and by arguing as in the proof of Lemma 7.16, one can easilyprove that |J(vm)| ≥ |J(vj)|. It follows at once that there exists c > 0independent of the sequence z, . . . , Tmz as in the hypotheses of theproposition and v ∈ Cz(z) such that ‖vm‖J ≥ c‖vj‖J .

Suppose now that J(vj)J′(vj) < 0. This means that vj is focus-

ing (τ+(zj, vj) > 0), and so z ∈ M+. By Theorem 4.2, we have


τ+(zn(z), vn(z)) ≤ f . Recall that f = maxi fi (see the paragraph be-fore Lemma 7.15). From zk ∈ M0 for n(z) < k ≤ j, it followsthat |J ′(vj)| = |J ′(vn(z))|, and it is not difficult to see that |J(vj)| ≤|J(vn(z))| ≤ f |J ′(vj)|. Thus ‖vj‖J ≤ δ1|vj|J ′ , where δ1 = (1+f 2)1/2. Onthe other hand, by applying Lemma 7.15 to z, . . . , Tmz, we obtain|vm|J ′ ≥ δ|vj|J ′ . Therefore, using the trivial fact ‖vm‖J ≥ |vm|J ′ , weobtain ‖vm‖J ≥ δδ−1

1 ‖vj‖J . This completes the proof, because δ andδ1 do not depend on the sequence z, . . . , Tmz and v ∈ Cz(z).

Proposition 7.27. The non-contraction property is satisfied.

Proof. Let ϕ = z, . . . , Tmz with z ∈ E and Tmz ∈ −E. SinceM− ⊂ E and −M− = M−, it is enough to prove the lemma for thecase Tmz ∈ −E+. If Tmz ∈ −E+∩E+, then there is nothing to prove.So, suppose that Tmz /∈ E+. There are two cases: ϕ ⊂ Mi ⊂ M+

for some i, and ϕ 6⊂ Mi for every i such that Mi ⊂ M+. In the firstcase, we have z ∈ Ei and m = n(z). By Proposition 7.14, it followsthat (9) is satisfied with β′ = γ1. In the second case, we can writeϕ = ϕ1 ∪ ϕ2, where ϕ1 = z, . . . , T kz with z, T kz ∈ E, and ϕ2 =T kz, . . . , Tmz with T kz ∈ E+ and Tmz ∈ −E+. Since ϕ2 is an orbitof the type considered in the first case, we have ‖DTkzT

m−kv‖ ≥ γ1‖v‖for v ∈ CTkz(T kz). Note that ϕ1 is a piece of orbit as in (9) so thatby Corollary 7.25, there exists β′5 > 0 independent of ϕ1 such that‖DzT

kv‖ ≥ β′5‖v‖ for v ∈ Cz(z). Hence, we see again that (9) issatisfied with constant β′5γ1. This together with the fact that β′5 andγ1 do not depend on ϕ completes the proof.

7.4. Conclusion of the proof of Condition L4. Since the billiardmap T is time-reversible, it is easy to check that the stable part of L4for x is equivalent to the unstable part of L4 but with O replaced by−O. Also, note that by Proposition 5.3, the invariant set Λ ⊂M whereT admits a local stable and an unstable manifold has full measure. Forthe definition of these manifolds, see Proposition A.3. The local stablemanifold (resp. local unstable manifold) of y ∈ Λ is denoted by V s(y)(resp. vu(y)).

In view of the time-reversibility of T , we have −V s(y) = V u(−y)and −V u(y) = V s(−y) for every y ∈ Λ. Hence Λ = −Λ. From theseconsiderations, it follows that the unstable part of L4 with O replacedby E ∪ −E, and Λx replaced by Λ implies the full L4 (stable andunstable parts).

We will also need the following technical lemma. Its proof is exactlyas the one of [14, Lemma 3.6]. The sets W s(y) and W u(y) are definedin Definition A.5.

Lemma 7.28. The set Λ can be chosen so that it satisfies the follow-ing property: if y ∈ Λ ∩ E and z ∈ W u(y) (resp. z ∈ W s(y)), then


TzWu(y) ⊂ Cy(y) (resp. TzW

u(y) ⊂ C ′y(y)), where (Uy, Cy)y∈E isthe family of cone fields defined in Subsection 4.3.


Proof. Suppose that x ∈ H, and let (l, N,O,K) be the quadruple ofx as specified at the beginning of Subsection 6.2. Also, let ε0 as inLemma 7.6. We will prove the unstable part of L4 with O replaced byE ∪ −E, Λx replaced by Λ and ε = ε0. As explained at the beginningof this subsection, this implies L4.

Suppose that y ∈ Λ∩(E∪−E), z ∈ (E∪−E)∩W u(y)∩T kS−1 (ε0) forsome k > 0. We study separately the cases T−kz ∈ E and T−kz /∈ E.For simplifying the notation, we will write Di,j for the restriction ofDT izT

j to the tangent subspace of W u(T iy) at T iz with i, j ∈ Z.Before continuing with the proof, we need to make a remark. First,

that the cone field (O,K) is equal to (Uy, Cy) for a proper y ∈ E.Then, note that by Lemma 7.28 and the invariance of the cone fields(Uy, Cy)y∈E, the tangent space of any unstable manifold consideredin this proof at a point z ∈ E is always contained in the cone Cz(z).This property is essential for applying Propositions 7.26 and 7.27, andwill be used implicitly in this proof every time that one of the twopropositions is applied.

We can now resume the proof. First, suppose that T−kz ∈ E. UsingProposition 7.27, we immediately obtain that ‖D−k0 ‖ ≤ 1/β′, where β′

is the constant appearing in (9).Now, suppose that T−kz /∈ E. Then, there are 2 possibilities, either

T−kz ∈ M0 ∩ S−1 (ε0) or T−kz ∈ M+ \ E. We analyze each possibilityindividually. First, define m∓ = infi > 0: T−k∓iz ∈ E.

Assume that T−kz ∈ M0 ∩ S−1 (ε0). Applying Proposition 7.26 toT−k−m−z with m = m−+m+ and j = m−, we obtain ‖D−k+m+,−m+‖ ≤1/γ2. Also, note that ‖D0,−k−m+‖ is trivially equal to 1 if m+ = k, andis not larger than 1/β′ if m+ < k by Proposition 7.27. Combining theprevious observations, we conclude that ‖D0,−k‖ ≤ 1/(γ2 min1, β′).

Finally, assume that T−kz ∈M+\E. Let n = n(T−kz). By breakingthe sequence of consecutive numbers −k, . . . , 0 into the sequences ofconsecutive numbers −k, . . . ,−k + n, −k + n, . . . ,−k + m+ and−k + m+, . . . , 0, we see that D0,−k = D−k+n,−n D−k+m+,−m++n D0,−k+m+ . From Proposition 7.27, Lemma 7.17 and Proposition 7.14,

we obtain ‖D−k+m+

0 ‖ ≤ 1/β′, ‖D−k+m+,−m++n‖ ≤ 1/δ1 and ‖D−k+n,−n‖ ≤1/γ1, respectively. Combining these inequalities, we conclude that‖D0,−k‖ ≤ 1/(β′γ1δ1).

To complete the proof, we observe that the upper bounds of ‖D0,−k‖for the different cases studied above do not depend on the data y, z, kof Condition L4.


Figure 2. Polygon with pockets and bumps (solidcurve) and original polygon (dashed curve).

8. Donnay billiards and their generalizations

We now apply Theorem 5.6 to a large class of hyperbolic billiards.The purpose of this section is to provide concrete examples of newbilliards with the Bernoulli property. For the definition of Bernoullicomponent, see Appendix A.

Definition 8.1. A polygon with pockets and bumps is a planar domainΩ obtained from a polygon P by replacing a sufficiently small neighbor-hood of each vertex with internal angle less than π with a focusing arc(pocket) or a dispersing arc (bump), and of each vertex with internalangle greater than π by a dispersing arc so that the vertex lies outsidethe closure of Ω (see Fig. 2).

Stadium-like domains with either parallel or non-parallel segmentscan be considered degenerate polygons with pockets. These domainscorrespond to quadrilaterals with a pocket or a bump replacing twovertexes instead of one. The results proved in this subsection applyalso to billiards in stadium-like domains.

Note that polygons with pockets and bumps satisfy B1 by definition.In the rest if this section, we will assume that billiards in polygonswith pockets and bumps satisfies Condition B2. This can achieved, forexample, by using sufficiently short focusing arcs as pockets.

Proposition 8.2. If a billiard in a polygon with pockets and bumpssatisfies Condition B2, then it satisfies Conditions B3 and B4 as well.In particular, T has non-zero Lyapunov exponents a.e. on M .

Proof. A general result for polygonal billiards states that every semi-orbit of a polygon billiard is either periodic or accumulates at least atone vertex of the polygon [18]. From this result and the assumptionon the geometry of Ω, it follows that if x ∈ N− ∪N+, then x has to beperiodic. Hence, N− = N+ = N consists of periodic orbits. In a polyg-onal billiard, every periodic orbit is contained in a family of parallelorbits having the same period. This family consists of finitely many


segments contained in horizontal segments (θ = const) with endpointbelonging to R. Since the number of distinct strips in a polygonalbilliard is countable, we immediately obtain µ(N) = 0. Hence, B3 issatisfied. The fact that N consists of periodic orbits implies also thatN ′ = ∅. Thus, S−1 ∩N+ ⊂ N ′ = ∅, and so Condition B4 is satisfied aswell.

The second part of the proposition follows from Proposition 5.3.

Chernov and Troubetzkoy proved that a billiard in convex polygonswith pockets Ω is ergodic if the pockets are arcs of circles with the fullcircles contained in Ω [11]. Here, we consider the more general situationof polygons with pockets and bumps. The condition in Definition 8.1that each vertex with internal angle greater than π is replaced by adispersing arc so that the vertex lies outside the closure of the polygonplays a crucial role in the proof of the next theorem.

Theorem 8.3. The map T of a billiard in a polygon with pockets andbumps that satisfies Condition B2 has the Bernoulli property.

Proof. Conditions B3 and B4 are satisfied by Proposition 8.2. By The-orem 5.6, every point of H has neighborhood contained (mod 0) in oneBernoulli component. Now, suppose that Mi ⊂ M− ∪M+. Since R iscountable (see [15, Propositions 6.17-6.19]), N ′ = ∅ and N ∩Mi = ∅,the set H ∩Mi is connected. Hence, every set Mi ⊂M− ∪M+ is con-tained (mod 0) in one Bernoulli component. We cannot claim the samefor sets Mi ⊂ M0, because in this case2, N may disconnect H ∩Mi.Thus, to prove that T is Bernoulli, we cannot use Corollary 5.8.

The alternative approach that we take is quite simple: we provethat the entire set M− ∪ M+ is contained (mod 0) in one Bernoullicomponent. This together with µ(N) = 0 implies that T is Bernoulli.Indeed, by Theorem A.6, the number of Bernoulli components of T isequal to the minimum integer n > 0 such that T nB = B for someBernoulli component B. By the geometry of Ω, it is always possible tofind two distinct curved components Γi and Γj such that µ(Mj∩TMi) >0. Thus, if B is the Bernoulli component containing M− ∪M+, thenTB = B, and so n = 1.

To prove that M− ∪M+ is contained (mod 0) in a single Bernoullicomponent, we start by observing that the polygon P decomposes intofinitely many triangles with pairwise disjoint interiors. The fact thatthis decomposition is not unique is irrelevant for our purposes. Let∆ be one of triangles of the decomposition. By construction of Ω,every vertex with internal angle greater than π lies outside Ω. Then,we claim that the union of the sets Mi corresponding to the pocketsand bumps attached to the vertexes of ∆ are contained (mod 0) in the

2This is indeed the case whenever the polygon P has two parallel sides facingeach other.


same Bernoulli component. In fact, let Γi, Γj, Γk be the pockets orbumps attached at the vertexes of ∆. If B is the Bernoulli componentcontaining Mi, then by Theorem A.6 states that there exists a Bernoullicomponent B′ such that TB = B′. It is easy to see that µ(Mj ∩TMi) > 0 and µ(Mk ∩ TMi) > 0. Thus, Mj and Mk must belongto B′. By symmetry of the configuration, also Mi and Mj belong thesame Bernoulli component. We can then conclude that Mi, Mj, Mk

belong to the same Bernoulli component. In a stadium-like domain,the previous argument does not work, because there are only 2 curvedcomponents Γi and Γj attached to the 3 vertexes of ∆. However, inthis case, we have i) µ(Mj ∩ TMi) > 0 and µ(Mj ∩ TMi) > 0, andii) µ(Mj ∩ T 2Mi) > 0. Claim i) is obvious. Claim ii) follows fromthe fact that there is a positive measure set of orbits starting at Mi

and reaching Mj after one collision with flat components of ∂Ω. Sinceeach Mi and Mj is contained in a Bernoulli component, and all theBernoulli components are cyclically permuted with the same periodn > 0 by Theorem A.6, claim i) implies that n ≤ 2, whereas claim ii)implies that n is either 3 or one of its divisors. Hence n = 1. Thesame conclusion is clearly true for all the sets Mi corresponding topockets and bumps attached to vertexes of two adjacent triangles ∆1

and ∆2. Hence, the entire set M−∪M+ belongs (mod 0) to a Bernoullicomponent.

Donnay billiards (see Theorem 3.8) are billiard in convex polygonswith pockets. The next result then follows directly from Theorem 8.3.

Corollary 8.4. Donnay billiards have the Bernoulli property.

The conclusion of Theorem 8.3 is robust for sufficiently small per-turbations of the pockets and bumps.

Definition 8.5. Two polygons with pockets and bumps Ω1 and Ω2 areε-close if Ω2 is obtained from Ω1 by replacing each pocket (resp. bump)with another pocket (resp. bump) of the same length, ε-close in theC6-topology (resp. C3-topology) and ε-close in the Hausdorff distance.

Proposition 8.6. Suppose that the billiard in a polygon with pocketsand bumps Ω1 satisfies B2. Then there exists ε > 0 such that if Ω2 isε-close to Ω1, then the billiard in Ω2 is Bernoulli.

Proof. Condition B2 is an open condition. Thus, if ε is sufficientlysmall, then the billiard in Ω2 satisfies B2, and so it is Bernoulli byTheorem 8.3.

We conclude this section with a result concerning hyperbolic bil-liards with domains for which Γ0 = ∅, i.e., without straight boundarycomponents. For these billiards, the Bernoulli property follows quitedirectly from Corollary 5.8. We observe that Sinai billiards [28] – thosewith domains bounded only by strictly convex outwards arcs – belong


to this class of billiards. For them, the Bernoulli property was firstproved in [17].

Theorem 8.7. Let Ω be a billiard domain without straight boundarycomponents, and suppose that its map T satisfies Conditions B1-B4.Then T is Bernoulli.

Proof. For billiards satisfying the hypotheses of the theorem, the setR is countable (see [15, Propositions 6.17-6.19]), and we trivially haveN = NR = ∅. Thus, each set H ∩Mi is connected, and B5 is satisfied.The wanted conclusion now follows from Corollary 5.8.

Appendix A. Local Ergodic Theorem

We state the Local Ergodic Theorem proved in [14], and recall therelevant definitions. The formulation of this theorem in its generalform requires a series of technical definitions, which are not neededfor 2-dimensional billiards. Thus, to avoid unnecessary technicali-ties, we specialize the presentation of the Local ergodic Theorem to2-dimensional billiards. Accordingly, T will denote the billiard map forsome planar domain Ω throughout this appendix.

The definition of a cone field and related notions are given in Sec-tion 4.

Definition A.1. A point x ∈M \ ∂M is called sufficient if there exist

(i) an invariant continuous cone field K on O ∪ T−NO such thatσK(DyT

N) > 3 for every y ∈ T−NO.(ii) a neighborhood O of T lx and an integer N > 0 such that O and

R−N are disjoint,(iii) an invariant continuous cone field K on O ∪ T−NO such that

σK(DyTN) > 3 for every y ∈ T−NO.

Every time we need to emphasize the role of the data l, N,O,K inthis definition, we will write that x is a sufficient point with quadruple(l, N,O,K).

Definition A.2. A point x ∈M \ ∂M is called u-essential if for everyα > 0, there exist nx,α ∈ N, a neighborhood Ox,α of x with Ox,α ∩R+nx,α = ∅ and a continuous invariant cone field (Ox,α∪T nx,αOx,α, Kx,α)

such that σ∗Kx,α(DyTnx,α) > α for every y ∈ Ox,α. Analogously, a point

x ∈ M \ ∂M is called s-essential if, in the previous definition, T andR+nx,α are replaced by T−1 and R−nx,α, respectively.

The cone field (O,K) in Definition A.1 is eventually strictly invari-ant. By a well-known result [23, 31, 32], it follows that all the Lya-punov exponents of T are non-zero a.e. on the set

⋃k∈Z T

kO. This factcombined with the Katok-Strelcyn theory [20] gives Proposition A.3below (Part (3) is proved in [14, Proposition 5.3]). For the definitionof absolute continuity of a foliation, we refer the reader to [10, 20].


Proposition A.3. Let x ∈M \∂M be a sufficient point with quadruple(l, N,K,O). Then there exist an invariant set Λx ⊂

⋃k∈Z T

kO withµ(Λx) = µ(

⋃k∈Z T

kO) > 0 and two families V s = V s(y)y∈Λx andV u = V u(y)y∈Λx consisting of C2 submanifolds such that for everyy ∈ Λx, we have

(1) V s(y) ∩ V u(y) = y,(2) V s(y) and V u(y) are embedded open intervals,(3) TyV

s(y) ⊂ K ′(y) and TyVu(y) ⊂ K(y) provided that y ∈ O ∪

T−NO,(4) TV s(y) ⊂ V s(Ty) and T−1V u(y) ⊂ V u(T−1y),(5) d(T ny, T nz) → 0 exponentially as n → +∞ for every z ∈

V s(y), and the same is true as n→ −∞ for every z ∈ V u(y),(6) V s(y) and V u(y) vary measurably with y ∈ Λx,(7) the families V s and V u have the absolute continuity property.

Definition A.4. The submanifolds forming the families V s and V u arecalled local stable manifolds and local unstable manifolds, respectively.

Definition A.5. Let x be a sufficient point of M \ ∂M , and let Λx bethe set in Proposition A.3. For every y ∈ Λx, we denote by W u(y) theconnected component of ⋃

k≥0

T kV u(T−ky)

containing y. Analogously, denote by W s(y) the set obtained by replac-ing T with T−1 and V u with V s in the definition of W u(y).

We now recall the Spectral Decomposition Theorem. It applies to alarger class of hyperbolic system with singularities than billiards, buthere we formulate it only for billiards. For its proof, one has to combinetwo results: [20, Theorem 13.1, Part II], which extends Pesin’result forsmooth systems [27] to systems with singularities, and [9, Theorem 3.1].See also [26], for results similar to those of [9].

Theorem A.6. Suppose that the billiard map T has non-vanishingLyapunov exponents a.e. on M . Then there exist countably manypairwise disjoint measurable subsets E0, E1, . . . of M such that

(1) M =⋃∞i=0Ei,

(2) µ(E0) = 0, and µi(Ei) > 0 for i ∈ N,(3) TEi = Ei, and (T |Ei , µ|Ei) is ergodic for i ∈ N,(4) for each i ∈ N, there exist mi ∈ N pairwise disjoint mea-

surable subsets Bi,1, . . . , Bi,mi , Bi,mi+1 = Bi,1 of M such thatEi =

⋃mij=1Bi,j, TBi,j = Bi,j+1 and (Tmi |Bi,j , µ|Bi,j) is Bernoulli

for every j = 1, . . . ,mi.

The sets Ei and Bi,j are called an ergodic component of T and aBernoulli component of T , respectively. These sets are uniquely de-fined up to a set of zero measure.


We need one last definition before formulating the Local ErgodicTheorem.

Definition A.7. Let (O1, C1) and (O2, C2) be two cone fields. Wesay that (O1, C1) and (O2, C2) are jointly invariant if DxT

kC1(x) ⊂C2(T kx) for every x ∈ O1 and k > 0 such that T kx ∈ O2, andDxT

kC2(x) ⊂ C1(T kx) for every x ∈ O2 and k > 0 such that T kx ∈ O1.

Note that in the previous definition, we neither require that the setsO1 and O2 are disjoint nor that the cone fields C1 and C2 are invariant.However, it is easy to see that C1 and C2 are invariant in the followingsense: if x ∈ O1 and k2 > k1 > 0 such that T k1x ∈ O2 and T k2x ∈ O1,then DxT

k2C1(x) ⊂ C1(T k2x). The same is true for C2, once O1 hasbeen replaced by O2.

Definition A.8. A subset Σ ⊂ M is called regular if it is a unionof finitely many arcs Σ1, . . . ,Σk of class C2 that can only intersect attheir boundaries. The arcs Σ1, . . . ,Σk are called the components of Σ.

Definition A.9 (Regularity). We say that T satisfies Condition L1 ifthe singular sets R+

k and R−k are regular for every k > 0.

In the rest of this subsection, we assume that x ∈M \ ∂M is a suffi-cient point with quadruple (l, N,O,K). Let Λx be the subset associatedto x as in Proposition A.3.

Definition A.10 (Alignment). We say that x satisfies Condition L2 ifthe sets O∩R+

k and O∩R−k are regular for every k > 0, and the tangentsubspace3 TyΣ is contained in K(y) (resp. K ′(y)) for every k > 0, everycomponent Σ of O∩R−k (resp. O∩R+

k ) and every y ∈ Σ∩T−NO (resp.Σ ∩O).

Definition A.11 (Ansatz). We say that x satisfies Condition L3 if theset of u-essential points of S−1 (resp. s-essential points of S+

1 ) has fullm−-measure (resp. m+-measure), and if y is any of such points, thenthe cone fields (O,K) and (Oy,α, Ky,α) are jointly invariant for everyα > 0.

Given A ⊂ M and ε > 0, we call the set A(ε) = x ∈ M : d(x,A) <ε the ε-neighborhood of A.

Definition A.12 (Contraction). We say that x satisfies Condition L4if there exist β > 0 and ε > 0 such that∥∥DzT

−k|TzWu(y)

∥∥ ≤ β(resp.

∥∥DzTk|TzW s(y)

∥∥ ≤ β)

for every y ∈ O ∩ Λx and every z ∈ O ∩ W u(y) ∩ T kS−1 (ε) (resp.O ∩W s(y) ∩ T−kS+

1 (ε)) with k > 0.

3In the general version of the theorem, the tangent space TyΣ has to be replacedby its Lagrangian skew-orthogonal complement with respect to the symplectic formω.


Theorem A.13 (Local Ergodic Theorem). Suppose that L1 is satisfied,and that x ∈M \∂M is a sufficient point satisfying L2-L4. Then thereexists a neighborhood of x contained up to a set of zero µ-measure in aBernoulli component of T .

Acknowledgements

The first author was supported by Fundacao para a Ciencia e a Tec-nologia through the Program POCI 2010 and the Project ‘Randomnessin Deterministic Dynamical Systems and Applications’ (PTDC-MAT-105448-2008). The first author gratefully acknowledge the hospital-ity and the financial support of CEMAPRE (ISEG, Universidade deLisboa). The second author wishes to thank Grupo de Investigacion‘Sistemas Dinamicos’ (CSIC, UdelaR, Uruguay).

References

[1] L. A. Bunimovich, Billiards that are close to scattering billiards, Mat. Sb.(N.S.) 94(136) (1974), 49–73, 159.

[2] L. A. Bunimovich, On the ergodic properties of nowhere dispersing billiards,Comm. Math. Phys. 65 (1979), no. 3, 295–312.

[3] L.A. Bunimovich, Many dimensional nowhere dispersing billiards with chaoticbehavior, Physica D33, 58 64 (1988)

[4] L. A. Bunimovich, A theorem on ergodicity of two-dimensional hyperbolicbilliards, Comm. Math. Phys. 130 (1990), no. 3, 599–621.

[5] L. A. Bunimovich, On absolutely focusing mirrors, in Ergodic theory andrelated topics, III (Gustrow, 1990), 62–82, Lecture Notes in Math., 1514Springer, Berlin.

[6] L. A. Bunimovich and G. Del Magno, Track billiards, Comm. Math. Phys.288 (2009), no. 2, 699–713

[7] L. A. Bunimovich, Ya. G. Sinaı and N. I. Chernov, Uspekhi Mat. Nauk 46(1991), no. 4(280), 43–92, 192; translation in Russian Math. Surveys 46(1991), no. 4, 47–106.

[8] L. Bussolari and M. Lenci, Hyperbolic billiards with nearly flat focusingboundaries. I, Phys. D 237 (2008), no. 18, 2272–2281

[9] N. I. Chernov and C. Haskell, Nonuniformly hyperbolic K-systems areBernoulli, Ergodic Theory Dynam. Systems 16 (1996), no. 1, 19–44

[10] N. Chernov and R. Markarian, Chaotic billiards, Mathematical Surveys andMonographs, 127, Amer. Math. Soc., Providence, RI, 2006.

[11] N. Chernov and S. Troubetzkoy, Ergodicity of billiards in polygons with pock-ets, Nonlinearity 11 (1998), no. 4, 1095–1102.

[12] G. Del Magno, Ergodicity of a class of truncated elliptical billiards, Nonlin-earity 14 (2001), no. 6, 1761–1786.

[13] G. Del Magno and R. Markarian, Bernoulli elliptical stadia, Comm. Math.Phys. 233 (2003), no. 2, 211–230.

[14] G. Del Magno and R. Markarian, A local ergodic theorem non-uniformly hy-perbolic symplectic maps with singularities, Ergodic Theory Dynam. Systems,33 (2013), 83–107

[15] G. Del Magno and R. Markarian, Singular sets of hyperbolic planar billiardsare regular, Regul. Chaotic Dyn. 18 (2013), no. 4, 425-452

[16] V. J. Donnay, Using integrability to produce chaos: billiards with positiveentropy, Comm. Math. Phys. 141 (1991), no. 2, 225–257.


[17] G. Gallavotti and D. S. Ornstein, Billiards and Bernoulli schemes, Comm.Math. Phys. 38 (1974), 83–101.

[18] G. Galperin, T. Kruger and S. Troubetzkoy, Local instability of orbits inpolygonal and polyhedral billiards, Comm. Math. Phys. 169 (1995), no. 3,463–473.

[19] E. Hopf, Statistik der geodatischen Linien in Mannigfaltigkeiten negativerKrummung, Ber. Verh. Sachs. Akad. Wiss. Leipzig 91 (1939), 261–304.

[20] A. Katok et al., Invariant manifolds, entropy and billiards; smooth maps withsingularities, Lecture Notes in Mathematics, 1222, Springer, Berlin, 1986.

[21] A. Kramli, N. Simanyi and D. Szasz, A “transversal” fundamental theorem forsemi-dispersing billiards, (see also Erratum) Comm. Math. Phys. 129 (1990),535-560.

[22] C. Liverani and M. P. Wojtkowski, Ergodicity in Hamiltonian systems, inDynamics reported, 130–202, Dynam. Report. Expositions Dynam. Systems(N.S.), 4 Springer, Berlin.

[23] R. Markarian, Billiards with Pesin region of measure one, Comm. Math. Phys.118 (1988), no. 1, 87–97.

[24] R. Markarian, New ergodic billiards: exact results, Nonlinearity 6 (1993),no. 5, 819–841.

[25] R. Markarian, Non-uniformly hyperbolic billiards, Ann. Fac. Sci. ToulouseMath. (6) 3 (1994), no. 2, 223–257.

[26] D. Ornstein and B. Weiss, On the Bernoulli nature of systems with somehyperbolic structure, Ergodic Theory Dynam. Systems 18 (1998), no. 2, 441–456.

[27] Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory,Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287.

[28] Ja. G. Sinaı, Dynamical systems with elastic reflections. Ergodic propertiesof dispersing billiards, Uspehi Mat. Nauk 25 (1970), no. 2 (152), 141–192.

[29] Ya. G. Sinaı and N. I. Chernov, Ergodic properties of some systems of two-dimensional disks and three-dimensional balls, Uspekhi Mat. Nauk 42 (1987),no. 3(255), 153–174, 256.

[30] D. Szasz, On the K-property of some planar hyperbolic billiards, Comm.Math. Phys. 145 (1992), no. 3, 595–604.

[31] M. Wojtkowski, Invariant families of cones and Lyapunov exponents, ErgodicTheory Dynam. Systems 5 (1985), no. 1, 145–161.

[32] M. Wojtkowski, Principles for the design of billiards with nonvanishing Lya-punov exponents, Comm. Math. Phys. 105 (1986), no. 3, 391–414.

[33] M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball prob-lem, J. Differential Geom. 40 (1994), no. 1, 155–164.

Universidade Federal da Bahia, Instituto de Matematica, AvenidaAdhemar de Barros, Ondina, 40170110 - Salvador, BA - Brasil

E-mail address: [email protected]

Instituto de Matematica y Estadıstica ‘Prof. Ing. Rafael Laguardia’(IMERL), Facultad de Ingenierıa, Universidad de la Republica, Mon-tevideo, Uruguay

E-mail address: [email protected]

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