A sharp formula for the essential spectral radius of the

Ruelle transfer operator on smooth and Holder spaces∗

V. M. Gundlach

Institut fur Dynamische Systeme

Universitat, Postfach 330 440

28334 Bremen, Germany

email: Matthias.Gundlach@DePfa.com

Y. Latushkin

Department of Mathematics

University of Missouri – Columbia

Columbia, MO 65211, USA

email: yuri@math.missouri.edu

Abstract

We study Ruelle’s transfer operator L induced by a Cr+1–smooth expanding map ϕ of

a smooth manifold and a Cr–smooth bundle automorphism Φ of a real vector bundle E .

We prove the following exact formula for the essential spectral radius of L on the space

Cr,α of r-times continuously differentiable sections of E with α-Holder r-th derivative:

ress(L;Cr,α) = exp

(supν∈Erg

{hν + λν − (r + α)χν}

),

where Erg is the set of ϕ-ergodic measures, hν the entropy of ϕ with respect to ν,

λν the largest Lyapunov exponent of the cocycle induced by Φ, and χν the smallest

Lyapunov exponent for the differential Dϕ.

∗Ergodic Theory & Dynamical Systems, to appear.

1

1 Introduction and Results

The transfer operator is important in many questions of dynamical systems and statistical

mechanics. It is particularly helpful for studying the mixing and statistical properties of

measures, investigations of zeta functions and Fredholm determinants, piecewise monotone

transformations, etc. We refer to the books [B, KS, PP, R4] and recent papers [Ba, F, K, Rg],

and the bibliography therein.

Estimates from above for the essential spectral radius of the transfer operator are obtained

in [R2, R3], see also [CL, ChL]. Results on the calculation of the essential spectral radius in

the scalar case are obtained in [CI, Ke] for dimension one and in [H], see also [BH], for higher

dimensions. In the current paper we derive an estimate from below, and, as a result, an exact

formula in terms of Lyapunov–Oseledets exponents for the essential spectral radius of the

matrix coefficient transfer operator for the multidimensional case in the following setting,

which is due to Ruelle [R2].

Let ϕ be a Cr+1–smooth expanding map (small distances are increased by a factor ρex > 1)

of a smooth compact d-dimensional connected manifold Θ. We assume that ϕ is not one-to-

one, that is N = card{ϕ−1{θ}} > 1. Let E be a smooth real `-dimensional vector bundle over

Θ, Eθ ' R`, and Φ be a Cr+1–smooth bundle automorphism over ϕ, that is Φ(θ) : Eθ → Eϕθ

and det Φ(θ) 6= 0. For r = 0, 1, . . . and α ∈ [0, 1] let Cr,0 = Cr denote the space of r-times

continuously differentiable sections f of E , and Cr,α, 0 < α ≤ 1, denote the space of r-times

continuously differentiable sections f with r-th derivative that satisfies a (global) Holder

condition with exponent α. On the space Cr,α, α ∈ [0, 1], consider the matrix coefficient

2

transfer operator, L, defined as follows:

(Lf)(θ) =∑

η∈ϕ−1θ

Φ(η)f(η), θ ∈ Θ, f ∈ Cr,α.

Let Erg = Erg(ϕ,Θ) denote the set of all ϕ-invariant ergodic Borel probability measures

on Θ, and hν denote the entropy of ϕ with respect to ν ∈ Erg. For each ν ∈ Erg, let λν

denote the largest Lyapunov-Oseledets exponent of the cocycle Φk(θ) = Φ(ϕk−1θ) · . . . ·Φ(θ),

generated by Φ and ϕ, and let χν denote the smallest Lyapunov-Oseledets exponent of the

differential Dϕk(θ), θ ∈ Θ, k = 1, 2, . . ..

In the current paper we give the proof of the following result announced in [GL].

Theorem 1.1.

ress(L;Cr,α) = ρ#(r, α), where ρ#(r, α) := exp(

supν∈Erg{hν + λν − (r + α)χν}

). (1.1)

Theorem 1.1 gives the following formula for the scalar case ` = 1, Φ : Θ→ R \ {0}:

ress(L;Cr,α) = exp(

supν∈Erg{hν +

∫Θ

log |Φ(θ)| dν − (r + α)χν}), (1.2)

and for the one-dimensional case d = 1, Θ = S1:

ress(L;Cr,α) = exp(

supν∈Erg{hν +

∫Θ

log |Φ(θ)| dν − (r + α)

∫Θ

log |ϕ′(θ)| dν}). (1.3)

The transfer operator L shares many common properties with the evolution operator

T defined for f ∈ Cr,α by the formula (Tf)(θ) = Φ(ϕ−1diffeoθ)f(ϕ−1

diffeoθ), where ϕdiffeo is

a diffeomorphism on Θ (see [ChL] for a systematic discussion and many applications of

evolution operators, and also related topics in [L]). In particular, using the techniques of the

current paper, one can prove that if the set of the aperiodic points of ϕdiffeo is dense in Θ,

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then

ress(T ;Cr,α) = exp(

supν∈Erg{λν − (r + α)χν}

). (1.4)

Formulas of the type (1.1)–(1.4) have a fairly long history that we will now briefly review.

For the scalar case ` = 1, Φ : Θ → R and r = α = 0, and for the spectral radius rsp(·) the

following formula was known, probably, since [R1]:

rsp(L, C0) = exp(

supν∈Erg{hν +

∫Θ

log |Φ| dν}). (1.5)

Its counterpart, rsp(T ;C0) = exp(

supν∈Erg{∫

Θlog |Φ| dν}

)for the evolution operator T was

obtained in [AL, CS, Ki1]. In the case d = 1, Θ = S1, a formula for the essential spectral

radius ress(T ;Cr,α), similar to (1.3), that is, with no entropy hν , can be also obtained, cf.,

e.g., [An]. In the important paper [CI] in the case ` = d = 1 and α = 0 it was proved that

the essential spectrum of L on Cr,0 is a disk and a formula for ress(L;Cr,0) was given (cf.

Lemma 2.1 below); this formula can be modified to the form (1.3).

For the matrix case ` > 1 the transfer operator on Cr,α was first studied in [T] and [R2].

For r = α = 0 the formula

rsp(L, C0) = exp(

supν∈Erg{hν + λν}

)(1.6)

was obtained in [CL, Theorem 3] (see also [LS, Theorem 4.15], where another type of transfer

operators was considered). Also, it was proved in [LS] that rsp(T,C0) = exp(

supν∈Erg{λν})

.

We remark that a comparison of the right-hand side of (1.5) and (1.6) shows that the

expression supν∈Erg{hν + λν} should play the role of topological pressure P (log |Φ|) [W] in

the case of matrix-valued Φ’s. A simple proof of the inequality ress(L;Cr,0) ≤ ρ#(r, 0) was

4

given in [CL, Theorem 2]; unfortunately, the technique of [CL] does not work for α 6= 0. Note

that the inequality ress(L;Cr,α) ≤ ρ#(r, α) in Theorem 1.1 improves the following estimate

due to Ruelle [R2]:

ress(L;Cr,α) ≤ exp(

supν∈Erg{hν +

∫Θ

log ‖Φ(θ)‖ dν − (r + α) log ρex})

and hence, see Remark 3.2 for r = 0 below, yields also an improvement of Ruelle’s estimate

for the radius of convergence of the power series for ζ–function.

For the scalar case ` = 1 and ΦPF(θ) = | detDϕ(θ)|−1, the transfer operator L is the

Perron-Frobenius operator (LPFf)(θ) =∑

η∈ϕ−1{θ} | detDϕ(η)|−1f(η), θ ∈ Θ, f : Θ → R.

We remark that in the case of the isolated rate of decay [BY, p.358] the right-hand side of

(1.2) with this ΦPF gives an estimate from below for the rate of decay of correlations τ0(r, α).

The quantity τ0(r, α) is defined (see, e. g., [BY, p.357]) as the smallest number such that

the following holds: for each τ > τ0(r, α) and each pair of test functions f1, f2 ∈ Cr,α we

have

∣∣∣ ∫Θ

(f1 ◦ ϕk) · f2 dm− (

∫Θ

f1 dm) · (∫

Θ

f2 dm)∣∣∣ ≤ c(τ, ‖f1‖r,α, ‖f1‖r,α) τ k, k ∈ N.

Here m is the unique ϕ–invariant probability measure on Θ which is absolutely continuous

with respect to Lebesgue measure on Θ. Note that for the topological pressure we have

P (log ΦPF) = hm −∫

Θlog |Dϕ(θ)| dm = 0. Also, in this case 1 = rsp(LPF) is a simple

eigenvalue for LPF, and the spectrum σ(LPF;Cr;α) = {1} ∪ σ0, where |σ0| := sup{|λ| :

λ ∈ σ0} is strictly smaller than 1, see [R2, Thm.3.6]. It is proved in [BY, p.358] that if

|σ0| > ress(LPF;Cr,α) (the case of the isolated rate of decay) then τ0(r, α) = |σ0|. Thus, in

this case τ0(r, α) is bounded from below by the right-hand side of (1.2) as claimed.

5

Our strategy of the proof of Theorem 1.1 is as follows. We obtain the inequality

ress(L;Cr,α) ≤ ρ#(r, α) by refining the Ruelle’s technique from [R2]. The idea is to keep

track of the “local” contraction rate of the inverses for the iteration ϕk, and use Nussbaum’s

formula. The actual implementation of this idea is rather technical (see Lemmas 2.1 and 3.1

below). For the inequality ress(L;Cr,α) ≥ ρ#(r, α) the difference between the matrix ` > 1

and the scalar case becomes more substantial, cf. [R3, Rem.1.2(b)]. To prove this inequality

we develop further the operator-theoretical approach from [CL, ChL]. We study yet an-

other transfer operator, Kr,α, acting on the space of continuous sections over an extended

compact space Θr,α, and induced by an extension ψr,α of ϕ and Ψr,α of Φ. We give a gen-

eralization of [ChL, Thm. 8.56] for α > 0, and prove that ress(L;Cr,α) ≥ rsp(Kr,α;C0(Θr,α))

(Lemma 3.3). The method behind this proof goes back to Mather [M], see also [ChL] for

a detailed discussion of the “Mather localization”, and papers [BJL, Ke, Ki2] where simi-

lar ideas have been used for the scalar ` = 1 case. Note that the presence of the Holder

seminorm makes our proof somewhat different from [ChL, Thm.8.56]. Next, we show that

rsp(Kr,α;C0(Θr,α)) ≥ ρ#(r, α) (Lemma 3.4). This proof is based on the strategy of Bowen’s

proof of the variational principle [Bo] and the proof of [ChL, Thm. 8.5]. A critical new

moment here is the use of our Lemma 2.1 for the estimate for ρ#(r, α) from above.

Notation: c(a, b) - generic constant depending on parameters a, b; k ∈ N; dist(·, ·) -

distance in Θ; | · | = ‖ · ‖Rd ; ‖ · ‖ = ‖ · ‖R` ; ‖ · ‖r,α = ‖ · ‖Cr,α ; Br – the set of r–multilinear

operators; σap(A) – the approximate point spectrum of A, that is, σap(A) = {z ∈ C : for

each ε > 0 there exists gε such that ‖(z − A)gε‖ ≤ const ·ε · ‖gε‖}. We use bold face to

denote r-tuples of unit vectors, e.g., v = (v1, . . . , vr), and bars to denote (r + 1)-tuples of

unit vectors, e.g., v = (v0,v) = (v0, v1, . . . , vr).

6

Acknowledgment. We would like to express our warm thanks to Ludwig Arnold for many

suggestions and to Lai-Sang Young for a stimulating discussion. Our special thanks go to the

referee who suggested many excellent ideas for improving the exposition and simplification

of the proofs. Y. L. was supported by the Research Board of the University of Missouri. He

also thanks Konstantin Makarov for suggesting ([Ma]) a simple proof of Lemma 2.3.

2 Preliminaries

Fix ω > 0 small and consider a Markov partition {Θ1, . . . ,Θs} of Θ with diam Θj < ω (see,

e.g., [R2, Prop. 2.1, 2.2]). Define, for i, j ∈ {1, . . . , s}, πij = 1 if ϕ(Θi) ⊃ Θj and πij = 0

otherwise. Fix open sets Uj ⊃ Θj with diamUj < ω such that ϕ(Ui) contains the closure

of Uj if and only if πij = 1 and Θi0 ∩ Θi1 ∩ . . . ∩ Θik = ∅ implies Ui0 ∩ Ui1 ∩ . . . ∩ Uik = ∅

(see, e.g., [R2, Prop. 2.3]). For πij = 1 there is a unique local inverse ϕ−1ij : Uj → Ui of the

map ϕ and dist(ϕ−1ij θ, ϕ

−1ij η) ≤ ρ−1

ex dist(θ, η) whenever θ, η ∈ Uj. We will assume that ω is

so small that each Ui belongs to a single coordinate chart on the manifold Θ, thus, we will

view Ui as a subset of Rn. We will write iθ to denote the ϕ-preimage of θ ∈ Θ that is in

Ui. A k-tuple ik = i1 . . . ik ∈ {1, . . . , s}k is called admissible if πi1i2 = . . . = πik−1ik = 1.

For each admissible k-tuple, k > 1, we set Uik = ϕ−1i1i2◦ ϕ−1

i2i3◦ . . . ◦ ϕik−1ik(Uik) and let

ikθ = i1 . . . ikθ ∈ Uikj denote the corresponding preimage of θ ∈ Uj under ϕk. Since ϕ is

expanding, we have diam Uik ≤ ωρk−1ex .

We will define the extended compact space Θr,α and the extensions ψr,α of the map ϕ

and {Ψkr,α} of the cocycle {Φk} as follows. Let Θr,α = {(θ,v) : θ ∈ Θ, vj ∈ T 1

θ , j = 0, . . . , r}

and Θr,0 = {(θ,v) : θ ∈ Θ, vj ∈ T 1θ , j = 1, . . . , r}, where T 1

θ is the fiber in the unit tangent

7

bundle to Θ. We define maps ψr,α : Θr,α → Θr,α and ψr,0 : Θr,0 → Θr,0 by

ψr,α : (θ, v0, v1, . . . vr) 7→(ϕθ,

Dϕ(θ)v0

|Dϕ(θ)v0|,Dϕ(θ)v1

|Dϕ(θ)v1|, . . . ,

Dϕ(θ)vr

|Dϕ(θ)vr|

),

ψr,0 : (θ, v1, . . . vr) 7→(ϕθ,

Dϕ(θ)v1

|Dϕ(θ)v1|, . . . ,

Dϕ(θ)vr

|Dϕ(θ)vr|

).

Throughout, we use letter η to denote ϕ–preimages of θ, and letter u to denote the corre-

sponding component in ψ–preimages of (θ,v). In particular, if (θ0,v0), (θ1,v

1), and (θ2,v2)

are any given points in Θr,α, and ik is admissible, then we denote

ηl = ikθl, ul ={

[Dϕk(ikθl)]−1vlj/|[Dϕk(ikθl)]−1vlj|

}r

j=0, (2.1)

such that ψ−kr,α(θl,vl) = {(ηl,ul) : ik} and ψ−kr,0 (θl,v

l) = {(ηl,ul) : ik} for l = 0, 1, 2.

We define cocycles {Ψkr,α}k∈N over ψr,α by

Ψkr,α(θ, v0, v1, . . . vr) = Φk(θ)|Dϕk(θ)v0|−α

r∏j=1

|Dϕk(θ)vj|−1,

and, similarly, cocycles {Ψkr,0}k∈N over ψr,0 setting α = 0. On the space C0(Θr,α) of contin-

uous sections over Θr,α we define the extended transfer operator, Kr,α, by the rule

(Kr,αF )(θ,v) =∑

(η,u)∈ψ−1r,α(θ,v)

Ψr,α(η,u)F (η,u),

where v = (v0, v1, . . . , vr). Similarly, Kr,0 is defined on C0(Θr,0) by ψr,0 and Ψr,0 with

v = (v1, . . . , vr). The operator Kr,0 appears naturally when we differentiate Lf (see [CI] and

[ChL, Lemma 8.57], and (2.12) below). Using (2.1), it is easy to see that

(Kkr,αF )(θ,v) =∑ik

Φk(ikθ)|[Dϕk(ikθ)]−1v0|αr∏j=1

|[Dϕk(ikθ)]−1vj|F (ikθ,u). (2.2)

8

We will need the growth rates R(r, α), ρ(r, α) and s(r, α), defined as follows:

Rk(r, α) = sup(θ,v)∈Θr,α

∑η∈ϕ−kθ

‖Φk(η)‖ |[Dϕ(η)]−1v0|αr∏j=1

|[Dϕ(η)]−1vj|, (2.3)

R(r, α) = limk→∞

Rk(r, α)1/k;

ρk(r, α) =∑ik

supθ‖Φk(ikθ)‖ · ‖[Dϕk(ikθ)]−1‖r+α,

ρ(r, α) = limk→∞

ρk(r, α)1/k;

sk(r, α) = maxq=1,...,r

∑ik

supθ1,θ2

‖Φk(ikθ1)‖‖[Dϕk(ikθ1)]−1‖q‖[Dϕk(ikθ2)]−1‖r−q (2.4)

×[dist(ikθ1, i

kθ2)/dist(θ1, θ2)]α, s(r, α) = lim

k→∞sk(r, α)1/k.

Here the supremum in (2.4) is taken over all θ1, θ2 ∈ Θ if α = 0 and θ1 6= θ2 if α > 0.

We note that ψr,α is a covering, but, generally, is not expanding. A general formula for the

spectral radius of a transfer operator induced by a covering map ([ChL, Prop. 8.50]) gives:

rsp(Kr,α;C0(Θr,α)) = limk→∞

(sup

(θ,v)∈Θr,α

∑(η,u)∈ψ−kr,α(θ,v)

‖Ψkr,α(η,u)‖

)1/k

. (2.5)

If (θ,v) ∈ Θr,α and v = (v0, v1, . . . , vr), then for each (η,u) ∈ ψ−kr,α(θ,v) we have η ∈ ϕ−k(θ)

and u = {[Dϕk(η)]−1vj/|[Dϕk(η)]−1vj|}rj=0 as in (2.1). By the definition of Ψkr,α we then

have:

Ψkr,α(η,u) = Φk(η)

∣∣∣Dϕk(η)[Dϕk(η)]−1v0

|[Dϕk(η)]−1v0|

∣∣∣−α r∏j=1

∣∣∣Dϕk(η)[Dϕk(η)]−1vj|[Dϕk(η)]−1vj|

∣∣∣−1

= Φk(η)∣∣∣[Dϕk(η)]−1v0

∣∣∣α r∏j=1

∣∣∣[Dϕk(η)]−1vj

∣∣∣.We substitute this expression in (2.5), and observe that the summation over all (η,u) ∈

ψ−kr,α(θ,v) in (2.5) is the same as the summation over all η ∈ ϕ−k(θ). Using (2.3), we see that

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the right-hand side of (2.5) is equal to limk→∞(Rk(r, α))1/k. Therefore, we have proved that

rsp(Kr,α;C0(Θr,α)) = R(r, α). (2.6)

We will prove below that, in fact, ress(L;Cr,α) = R(r, α).

Lemma 2.1. R(r, α) = ρ(r, α) = s(r, α).

Proof. By standard calculus, Rk(r, α) ≤ ρk(r, α) ≤ sk(r, α). To see that s(r, α) ≤ R(r, α),

for each ε > 0 we choose a δ = δ(ε) > 0 such that if dist(η1, η2) ≤ δ then

‖Φ(η2)‖ ≤ ‖Φ(η1)‖(1 + ε) and ‖[Dϕ(η2)]−1‖ ≤ ‖[Dϕ(η1)]−1‖(1 + ε). (2.7)

Consider θ1 and θ2 with dist(θ1, θ2) ≤ δ as vectors in Rn (up to a constant factor), and let

γi(t) = iγ(t), t ∈ [0, 1], be the i-th preimage under ϕ of the point γ(t) = (θ1 − θ2)t + θ2

that belongs to the segment connecting θ1 and θ2. Taking the inf over all smooth curves χ

that connect iθ1 = γi(1) and iθ2 = γi(0), we have dist(iθ1, iθ2) ≤∫ 1

0|[Dϕ(γi(t))]

−1γ(t)| dt ≤

max0≤t≤1 ‖[Dϕ(γi(t))]−1‖·|θ1−θ2|. Since ϕ is an expanding map, we have dist(γi(t), γi(1)) ≤

δ. By (2.7), we have the inequality max0≤t≤1 ‖[Dϕ(γi(t))]−1‖ ≤ ‖[Dϕ(iθ1)]−1‖(1+ ε). There-

fore,

[dist(ikθ1, i

kθ2)/dist(θ1, θ2)]α ≤ ‖[Dϕ(ikθ1)]−1‖α . . . ‖[Dϕ(ikθ1)]−1‖α(1 + ε)αk.

Fix m so large that diamUim ≤ δ. Using the last inequality, we have

sk+m(r, α) ≤ maxq=1,...,r

∑im

c(im)∑ik

supθ1,θ2∈Uim

‖Φ(ikθ1)‖ . . . ‖Φ(ikθ1)‖(1 + ε)αk× (2.8)

‖[Dϕ(ikθ1)]−1‖q+α. . . ‖[Dϕ(ikθ1)]−1‖q+α‖[Dϕ(ikθ2)]−1‖r−q. . . ‖[Dϕ(ikθ2)]−1‖r−q.

10

For each im fix θ = θ(im) ∈ Uim independent of ik. Since ϕ expands, we have the inequality

dist(ip . . . ikθ, ip . . . ikθl) ≤ δ for p = 1, . . . , k, l = 1, 2. Now (2.7) and (2.8) imply

sk+m(r, α) ≤∑im

c(im)∑ik

. . .∑i1

(1 + ε)k+αk+rk‖Φ(ikθ)‖ . . . ‖Φ(ikθ)‖

×‖[Dϕ(ikθ)]−1‖r+α . . . ‖[Dϕ(ikθ)]−1‖r+α.

Since θ does not depend on ik, we have sk+m(r, α) ≤ c(m)(1 + ε)k(1+α+r)(R1)k, where we

denote R := limn→∞R1/n

n and Rn = supθ∈Θ

∑in ‖Φn(inθ)‖‖[Dϕn(inθ)]−1‖r+α, n = 1, 2, . . ..

The same argument applied for ϕn and Φn gives skn+m(r, α) ≤ c(m)(1 + ε)k(1+α+r)(Rn)k,

which implies s(r, α) ≤ R.

It remains to show that R(r, α) ≥ R. Fix ε ∈ (0, 1) and for each θ ∈ Θ let Gθ =

{vh}Nh=1 denote an ε-net in T 1θ . For each θ ∈ Θ and ik choose w = w(ik, θ) ∈ T 1

θ such that

‖[Dϕk(ikθ)]−1‖ = |[Dϕk(ikθ)]−1w|, and find v = vh ∈ Gθ, h = h(ik, θ), such that |v−w| < ε.

Then

Rk ≤ supθ

∑ik

‖Φk(ikθ)‖∣∣∣∣[Dϕk(ikθ)]−1 w − v

|w − v|

∣∣∣∣r+α

|v − w|r+α +

+ supθ

∑ik

‖Φk(ikθ)‖|[Dϕk(ikθ)]−1v|r+α ≤ εr+αRk +Rk(r, α).

Thus, (1− εr+α)Rk ≤ Rk(r, α), and the lemma is proved.

For F ∈ C0,α(Θr,0) we denote ‖F ||α = supθ1 6=θ2 supv ‖F (θ1,v) − F (θ2,v)‖ dist(θ1, θ2)−α,

such that ‖f‖r,α = ‖Drf‖C0(Θr,0) + ‖Drf‖α.

11

Lemma 2.2. If f ∈ Cr,α(Θ), then

‖Lkf‖r,α ≤ ‖Kkr,0Drf‖α + c(k)‖f‖r,0, (2.9)

‖Lkf‖r,α ≥ ‖Kkr,0Drf‖α − c(k)‖f‖r,0, (2.10)

‖Lkf‖r,α ≤ sk(r, α)‖Drf‖α + c(k)‖f‖r,0. (2.11)

Proof. Similarly to [ChL, Lemma 8.57], the chain rule implies

(DrLkf)(θ,v) = (Kkr,0Drf)(θ,v) + (Lkf)(θ,v), (2.12)

where Lkf contains only the derivatives of f up to the order r − 1. Thus, ‖Lkf‖r,0 ≤

c(k)‖f‖r,0 and ‖Lkf‖α ≤ c(k)‖f‖r,0. Since ‖Lkf‖r,α ≥ ‖DrLkf‖α, we have (2.9) and (2.10).

To finish the proof of (2.11), for θ1 6= θ2 and any v we estimate

‖(Kkr,0Drf)(θ1,v)− (Kkr,0Drf)(θ2,v)‖ ≤ a(θ1, θ2) + b(θ1, θ2),

where, recalling (2.1), we denote

a(θ1, θ2) =∑ik

‖Φk(η1)‖‖[Dϕk(η1)]−1‖r‖Drf(η1)(u1)−Drf(η2)(u2)‖,

b(θ1, θ2) =∑ik

∥∥∥Φk(η1)r∏j=1

|[Dϕk(η1)]−1vj| − Φk(η2)r∏j=1

|[Dϕk(η2)]−1vj|∥∥∥ · ‖f‖r,0.

Since Φk and Dϕk are α-Holder, we have that b(θ1, θ2) ≤ c(k) dist(θ1, θ2)α‖f‖r,0. Since

Drf(ηl), l = 1, 2, are multilinear operators, ‖Drf(η1)(u1)−Drf(η2)(u2)‖ is dominated by

‖Drf(η1)−Drf(η2)‖Br + ‖Drf(η2)‖Br

r∑j=1

(∏p<j

|u1p|)|u1

j − u2j |(∏p>j

|u2p|).

Since Dϕk is α-Holder, we have the inequality |u1j−u2

j | ≤ c(k) dist(η1, η2)α. Also, ‖Drf(η1)−

12

Drf(η2)‖Br ≤ ‖Drf‖α dist(η1, η2)α. Using the definition of sk(r, α), we have that ‖Kkr,0Drf‖α

is dominated by the right-hand side of (2.11).

Lemma 2.3. There exists a constant a = a(d, r, α), a ∈ (0, 1), such that for each η0 ∈ Θ,

each neighborhood U 3 η0, and each u = (u0, u1, . . . , ur), |uj| = 1, j = 0, 1, . . . r, one can

find a scalar Cr,α-smooth function β such that the following holds: supp β ⊂ U , ‖β‖r,0 ≤ δ,

‖β‖r,α ≤ 1, and

limτ→0

τ−αDrβ(η0 + τu0)(u1, . . . , ur) = a. (2.13)

Proof. Without loss of generality we will assume that U is so small that U ⊂ Rd, and that

η0 = 0. We will choose β(θ) = β0(θ)β1(|θ|), θ ∈ Rd. Here β1 : R+ → [0, 1] is a “flat”

Cr+1-cutoff function such that β1(τ) = 1 for τ ∈ [0, b] and β1(τ) = 0 for τ ≥ 2b, and

sup0≤τ≤2b |Dkβ(τ)| = O(b−k) as b → 0, k = 0, . . . , r + 1. Parameter b will be selected small

to make sure that supp β ⊂ U , ‖β‖r,0 ≤ δ, and ‖β‖r,α ≤ 1. The Cr,α-smooth function

β0 : Rd → R will be selected as follows to satisfy (2.13).

First, we claim that for a sufficiently small a1 = a1(d, r, α) ∈ (0, 1) and each choice of u

there exists a vector w = w(u) ∈ Rd, |w| = 1, such that |u0 · w|α∏r

j=1 |uj · w| ≥ a1. Here ·

denotes the scalar product in Rd. For instance, if r = 0 then w = u0; if r = 1 then w bisects

the smallest angle between u0 and ±u1. To prove the claim for any r, let us suppose that

for each k ∈ N there exists u(k) = (u0(k), . . . , ur(k)), |uj(k)| = 1, j = 0, . . . , r, such that for

all w, |w| = 1, we have |u0(k) · w|α∏r

j=1 |uj(k) · w| ≤ k−1. Using compactness of the unit

sphere in Rd, we may assume that {u(k)}∞k=1 converges to some u = (u0, . . . , ur), |uj| = 1.

Passing to the limit, we have that for each w, |w| = 1, at least one of the r + 1 unit vectors

13

uj is perpendicular to w. Since each of the sets {w : |w| = 1, w⊥uj}, j = 0, . . . , r, has zero

surface measure on the unit sphere in Rd, we have a contradiction, and the claim is proved.

Next, take any a ∈ (0, 1), and for a given u pick w = w(u) as indicated in the claim

above. Define

β0(θ) = a[(r + α) · · · (1 + α)|u0 · w|α

r∏j=1

|uj · w|]−1

|θ · w|r+α, θ ∈ Rd.

We will show that for sufficiently small a and b, independent of u, the function β(θ) =

β0(θ)β1(|θ|) is as required.

Note, that β0 is constant along the hyperplanes orthogonal to w, and ∇β0 is parallel to

w. Also, we have |β0(θ)| ≤ c|θ|r+α, where c does not depend on u by the claim above. For

each v = (v1, . . . , vr), |vj| = 1, we have:

Drβ0(θ)(v1, . . . , vr) = a

[|u0 · w|α

r∏j=1

|uj · w|

]−1 r∏j=1

|vj · w||θ · w|α.

In particular, Drβ0(τu0)(u1, . . . , ur) = aτα. Since β1 is identically one in the ball |θ| ≤ b, we

have (2.13). Using the claim above, we have sup|θ|≤2b ‖Dkβ0(θ)‖Bk = O(br−k+α) as b→ 0 for

k = 0, 1, . . . , r. Then, using the choice of β1 and product rule, we have:

‖β‖r,0 =r∑

k=0

sup|θ|≤2b

‖Dk(β0(θ)β1(|θ|))‖Bk = O(bα) as b→ 0.

Thus, ‖β‖r,0 ≤ δ for sufficiently small b. Since ‖Dr(β0β1)‖α ≤ ‖β1Drβ0‖α + ‖β2‖α, where

β2 contains derivatives of β0 up to the order r− 1, we can choose sufficiently small a and b,

independent of u, to satisfy ‖β‖r,α ≤ 1.

14

3 Proof of Theorem 1.1

The inequality rsp(Kr,α;C0(Θr,α)) ≤ ρ#(r, α) can be obtained as in [ChL, Thm 8.60]. Thus,

by (2.6) and Lemma 2.1, the inequality ress(L;Cr,α) ≤ ρ#(r, α), yielding one direction of the

proof of Theorem 1.1, is implied by the following lemma, whose proof is an adaptation of

Ruelle’s argument in [R2, p.248-249].

Lemma 3.1. ress(L;Cr,α) ≤ s(r, α).

Proof. By identifying Cr,α(Θ) with⊕

iCr,α(Ui) one can define operatorsM andM(k), which

are equivalent, respectively, to L and Lk (see [R2, p. 246–247]):

(Mf)j(θ)=∑i

Φ(ijθ)f(ijθ), (M(k)f)ik(θ)=∑

i0...ik−1

Φk(i0 . . . ik−1ikθ)f(i0 . . . ik−1ikθ), (3.1)

where fj = f |Uj . We have ress(L) ≤ ress(M). By Nussbaum’s formula, it suffices to show

that

lim infk→∞

‖M(k) −K(k)‖1/k ≤ s(r, α), (3.2)

where K(k) has finite rank for each k. Following the proof of [R2, Thm. 3.2, p.249], choose

θ = θ(i0, . . . , ik) ∈ Ui0...ik and define K(k)f = M(k)Frf , where Frf is the Taylor expansion

of f of order r at θ. Let h denote the remainder of the Taylor expansion. To see (3.2), we

need to estimate the Cr,α-norm of the function

(M(k) −K(k))f(θ) =∑

i0...ik−1

Φk(i0 . . . ik−1θ)h(i0 . . . ik−1θ), θ ∈ Uik .

15

As in [R2, p. 249] we have

‖Dr−qh(θ)‖ ≤ c‖f‖r,α dist(θ, θ)q+α for q = 0, . . . , r and θ ∈ Uik . (3.3)

Since limk→∞ sk(0, r +α)1/k ≤ s(r, α), and h(θ) = Dqh(θ) = 0, q = 1, 2, . . . , r, for each ε > 0

we have

‖(M(k) −K(k))f‖C0 ≤ c‖f‖r,α supik

supθ∈Uik

∑i0...ik−1

‖Φk(i0 . . . ik−1θ)‖[

dist(i0 . . . ik−1θ, θ)

dist(θ, ϕkθ)

]r+α

≤ c(ε)(s(r, α) + ε)k‖f‖r,α.

In the remainder of the proof we show that similar estimates (up to a polynomial in k) hold

for the C0-norms of the derivatives and Holder seminorm of Lkh = (M(k)−K(k))h. We will

work with the r-th derivative; lower order derivatives can be considered similarly. By (2.12)

we have DrLkh = Kkr,0Drh + hr, where hr contains a variety of terms with derivatives of

order r− 1, but no derivatives of order r of h.

To estimate ‖Kkr,0Drh‖C0(Θr,0), we let θ1 = θ, θ2 = ϕk(θ), ik = i0 . . . ik−1. By the definition

of Kr,0 and the bound (3.3) on ‖(Drh)(θ)‖, we have that ‖Kkr,0Drh‖C0(Θr,0) is dominated by

max(θ,v)

∑ik

‖Φk(ikθ)‖r∏j=1

|[Dϕk(ikθ)]−1vj| dist(ikθ, θ)α · ‖f‖r,α ≤ c(ε)(s(r, α) + ε)k‖f‖r,α.

Similarly for the other derivatives and, hence, ‖hr‖C0(Θr,0) and ‖DrLkh‖C0(Θr,0) are dominated

by the expression c(ε)p(k)(s(r, α)+ε)k‖f‖Cr,α for a polynomial p(·). The same estimate holds

for the Holder seminorm of hr. Thus, to finish the proof, we need to show that

supθ1 6=θ2

maxv‖(Kkr,0Drh)(θ1,v)− (Kkr,0Drh)(θ2,v)‖ dist(θ1, θ2)−α ≤ c(ε)p(k)‖f‖r,α(s(r, α) + ε)k.

Consider r-linear operators A = (Drh)(ikθ1), B = (Drh)(ikθ2) and split (for fixed θ1, θ2 ∈ Θ,

16

v ∈ (T 1)r) the difference (Kkr,0Drh)(θ1,v)− (Kkr,0Drh)(θ2,v) = S1 + S2 + S3, where

S1 =∑ik

[Φk(ikθ1)− Φk(ikθ2)]A({[Dϕk(ikθ1)]−1vj}rj=1),

S2 =∑ik

Φk(ikθ2)[A({[Dϕk(ikθ1)]−1vj}rj=1)−B({[Dϕk(ikθ1)]−1vj}rj=1)],

S3 =∑ik

Φk(ikθ2)[B({[Dϕk(ikθ1)]−1vj}rj=1)−B({[Dϕk(ikθ2)]−1vj}rj=1)].

Since Φk(ikθ2)− Φk(ikθ1) can be written as

k−1∑p=0

Φp(ik−p+1 . . . ikθ1)[Φ(ik−p . . . ikθ2)− Φ(ik−p . . . ikθ1)]Φk−p−1(i1 . . . ikθ2)

and ‖A({[Dϕk(ikθ1)]−1vj}rj=1)‖ ≤ ‖f‖r,α∏r

j=1 |[Dϕk(ikθ1)]−1vj|, we have

‖S1‖ ≤ c‖f‖r,αk−1∑p=0

∑ik−p...ik

‖Φp(ik−p+1 . . . ikθ1)‖r∏j=1

|[Dϕp+1(ik−p . . . ikθ1)]−1vj| ×

× dist(ik−p . . . ikθ1, ik−p . . . ikθ2)α∑

i1...ik−p−1

‖Φk−p−1(ikθ2)‖r∏j=1

|[Dϕk−p−1(ikθ1)]−1vj|.

Thus ‖S1‖ ≤ c(ε)p(k)(s(r, α) + ε)k‖f‖r,α. A similar, but longer argument works for ‖S2‖

and ‖S3‖.

Remark 3.2. For r = 0 let d(·) be the generalized ζ–function, see [R2, Thm. A1], defined as

the formal power series d(z) = exp(−∑∞

k=1 zkk−1

∑θ∈Fix ϕk Tr Φk(θ)

), where Tr denotes

the trace. Combining the proof of [R2, Thm. A1] and Lemma 3.1 we have that the power

series converges in {z ∈ C : |z|s(0, α) < 1} (that is, in fact, for |z| < 1/ ress(L;C0,α)). Indeed,

arguing as in the proof of Lemma 3.1, we can replace in the estimates (A.3) and (A.6) of

[R2] the quantity θαeP+ε by the quantity s(0, α) + ε; the rest of the proof of [R2, Thm. A1]

remains unchanged. We suspect that for r > 0 similar changes could be made in (1.5) and

Proposition 3.2 of [R3].

17

We split the proof of the inequality ress(L;Cr,α) ≥ ρ#(r, α) in two lemmas.

Lemma 3.3. ress(L;Cr,α) ≥ rsp(Kr,α;C0(Θr,α)).

Proof. For α = 0 the proof is given in [ChL, Thm. 8.56, p. 328]. We will describe how to

extend this proof for α > 0. As in [ChL], passing to the operator z−1Kr,α for some z with

|z| = rsp(Kr,α), we will assume without loss of generality that rsp(Kr,α;C0(Θr,α)) = 1 and

1 ∈ σap(Kr,α;C0). We will prove that 1 ∈ σap(Kr,α;C0(Θr,α)) implies 1 ∈ σap(L;Cr,α). This

will be done by an explicit construction for any ε > 0 of a g = gε ∈ Cr,α satisfying

‖g − Lg‖r,α ≤ c(d, r, α) · ε · ‖g‖r,α. (3.4)

Fix ε < 1/8, and choose a large N = N(ε) ∈ N, and small δ = δ(N, ε), which will be specified

later. Since 1 ∈ σap(Kr,α;C0(Θr,α)), we can, as in [ChL, p.328], pick an F = Fε,N ∈ C0(Θr,α)

and (θ0,v0) ∈ Θr,α such that

‖F‖C0(Θr,α) = 1, ‖KNr,αF − F‖C0(Θr,α) ≤ ε, ‖F (θ0,v0)‖ ≥ 1/2. (3.5)

Using (3.5) we have that for each N there exist an aperiodic point (θ0,v0) and vectors

F (η0,u0), for (η0,u

0) ∈ ψ−Nr,α (θ0,v0), such that

‖F (η0,u0)‖ ≤ 1 and

∥∥∥ ∑η0∈ϕ−Nθ0

ΨNr,α(η0,u

0)F (η0,u0)∥∥∥ ≥ 3/4. (3.6)

We also recall that rsp(Kr,α;C0(Θr,α)) = R(r, α) = 1 by (2.6). Using (3.6) we will construct

g as in (3.4), as follows.

Choose a small open set B0 in Θ such that θ0 ∈ B0 and the components of ϕk−N(B0),

k = 0, . . . , 2N + 1 are disjoint. Take a smaller ball B ⊂ B0 such that θ0 ∈ B. Let χ :

18

Θ→ [−2, 2] be any (r, α)-smooth bump-function such that: χ(θ) = 1 for θ ∈⋃Nk=0 ϕ

k−N(B),

χ(θ) = (ε− 1)−1 for θ ∈⋃2Nk=N+1 ϕ

k−N(B), and χ(θ) = 0 for θ 6∈⋃2Nk=0 ϕ

k−N(B0). Note that

‖χ‖r,α grows as ε→ 0 and N →∞.

For a sufficiently small δ > 0, use Lemma 2.3 to choose a function β : Θ → [0, 1]

such that supp β ⊂ ϕ−N(B), ‖β‖r,0 ≤ δ, ‖β‖r,α ≤ 1. In addition, we select β such that

limτ→0 τ−αDrβ(η0 + τu0

0)(u0) = a for each η0 = iNθ0. Here v0 = (v00,v

0) is as in (3.6),

notation (2.1) is used, and a = a(d, r, α) ∈ (0, 1) is the constant from Lemma 2.3. Then

Drβ(η0)(u0) = 0 and

limt→0

t−αDrβ(ϕ−NiN

(θ0 + tv00))(u0) = a|[DϕN(η0)]−1v0

0|α. (3.7)

Define h(η) = β(η)F (η0,u0) for η ∈ ϕ−N

iN(B) that belong to the same component of ϕ−N

iN(B)

that contains η0 (for each iN), and let h(θ) = 0 for θ /∈ ϕ−N(B). Let

g(θ) = (1− ε)|N−k|(Lkh)(θ) for θ ∈ ϕk−N(B), k = 0, . . . , 2N,

and g(θ) = 0 for θ /∈⋃2Nk=0 ϕ

k−N(B), cf. [ChL, (8.140),(8.141)]. Note that if k = 0, . . . , 2N+1,

then suppLkh ⊂ ϕk−N(B) ⊂ ϕk−N(B0).

Since supports of Lkh, k = 0, . . . , 2N , are disjoint, we can re-write g(θ) for all θ ∈ Θ as

the following sum: g(θ) =∑2N

k=0(1− ε)|N−k|(Lkh)(θ). Then

(Lg)(θ) =2N∑k=1

(1− ε)|N−k+1|(Lkh)(θ) + (1− ε)N(L2N+1h)(θ) for θ ∈ Θ.

For k = 0, . . . , 2N we denote, for brevity, εk = (1− ε)|N−k| − (1− ε)|N−k+1|. Again using the

fact that supports of Lkh are disjoint, we conclude that if θ ∈ ϕ−N(B) then g(θ)− (Lg)(θ) =

(1 − ε)Nh(θ); if k = 1, . . . , 2N and θ ∈ ϕk−N(B) then g(θ) − (Lg)(θ) = εk(Lkh)(θ); if

19

θ ∈ ϕN+1(B) then g(θ)− (Lg)(θ) = −(1− ε)N(L2N+1h)(θ); and if θ /∈⋃2N+1k=0 ϕk−N(B) then

g(θ)− (Lg)(θ) = 0. For θ ∈ ϕ−N(B) we rewrite (1− ε)Nh(θ) = ε0(L0h)(θ) + (1− ε)N+1h(θ).

If k = 0, . . . , N then εk = ε(1− ε)|N−k|. If k = N +1, . . . , 2N then εk = ε(ε−1)−1(1− ε)|N−k|.

Thus, if k = 0, . . . , 2N and θ ∈ ϕk−N(B), then εk(Lkh)(θ) = εχ(θ)g(θ). We stress, that

supp(χg) belongs to the set⋃2Nk=0 ϕ

k−N(B) and, therefore, χg does not depend on the choice

of χ outside of this set. Finally, we conclude that for all θ ∈ Θ the following identity holds:

g(θ)− (Lg)(θ) = (1− ε)N+1h(θ) + εχ(θ)g(θ)− (1− ε)N(L2N+1h)(θ). (3.8)

First, we estimate ‖L2N+1h‖r,α from above. Since R(r, α) = 1, by Lemma 2.1 there

exists c(ε) such that s2N+1(r, α) ≤ c(ε)(1 + ε2)2N+1. Apply (2.11) from Lemma 2.2 for

f = h and k = 2N + 1. Since ‖h‖r,0 ≤ δ and ‖Drh‖α ≤ 1 by the choice of β, we have

‖L2N+1h‖r,α ≤ c(ε)(1 + ε2)2N+1 + c(N)δ. For the middle term in (3.8), using the product

rule, we have that ‖χg‖r,α ≤ ‖χ‖C0‖g‖r,α + c(r)‖χ‖r,α‖g‖r,0 ≤ 2‖g‖r,α + c(ε,N)δ. Using

these estimates and the inequality ‖h‖r,α ≤ 1 in (3.8), we have:

‖g − Lg‖r,α ≤ 2ε‖g‖r,α + (1− ε)N+1[1 + c(ε)(1 + ε2)2N+1

]+ c(ε,N)δ. (3.9)

We claim, that for a sufficiently small δ = δ(ε,N) we have

1 ≤ 2a−1‖g‖r,α. (3.10)

Recall that a = a(d, r, α) does not depend on N nor ε. As soon as the claim is proved, we

use it in (3.9) and, first, choose N sufficiently large, and, next, choose δ = δ(ε,N) sufficiently

small to see (3.4).

To prove the claim, we use the inequality ‖h‖r,0 ≤ δ and (2.10) from Lemma 2.2 for

20

f = h and k = N to obtain the inequality ‖g‖r,α ≥ ‖LNh‖r,α ≥ ‖KNr,0Drh‖α − c(N)δ. Note

that (KNr,0Drh)(θ0,v0) = 0 by the choice of β. Thus,

‖KNr,0Drh‖α ≥ limt→0

t−α‖(KNr,0Drh)(θ0 + tv00,v

0)‖

=

∥∥∥∥∥∑iN

ΦN(iNθ0)r∏j=1

|[DϕN(iNθ0)]−1v0j |F (iNθ0,u

0) limt→0

t−αDrβ(iN(θ0 + tv00))(u0)

∥∥∥∥∥ ,see notations (2.1). By (3.7) the last expression is equal to

a

∥∥∥∥∥∑iN

ΦN(iNθ0)N∏j=1

|[DϕN(iNθ0)]−1v0j | · |[DϕN(iNθ0)]−1v0

0|αF (iNθ0,u0)

∥∥∥∥∥= a‖(KNr,αF )(θ0,v

0)‖ ≥ 3a/4,

where we have used the choice of F as indicated in (3.6). Thus, ‖g‖r,α ≥ 3a/4−c(N)δ > a/2

for all sufficiently small δ = δ(N, ε).

By Lemma 3.3, (2.6) and Lemma 2.1 we have ress(Kr,α;Cr,α) ≥ ρ(r, α). Thus, the

inequality ress(Kr,α;Cr,α) ≥ ρ#(r, α) is implied by the following lemma.

Lemma 3.4. ρ(r, α) ≥ ρ#(r, α).

Proof. We will make use of the Oseledec Multiplicative Ergodic Theorem (see, e.g., [Ar,

Thm. 3.4.1]). This theorem gives the existence of exact Oseledec-Lyapunov exponents and

corresponding subbundles almost everywhere whith respect to ergodic measures. We will

apply the multiplicative ergodic theorem for three cocycles: {Dϕk}, {Φk}, and {Ψkr,α}. First,

we will select sets of full measure such that for each point from these sets the conclusions

of the multiplicative ergodic theorem hold. This will allow us to recalculate the largest

Lyapunov exponent, Λµ(ψ,Ψ), for the cocycle {Ψkr,α} via the largest Lyapunov exponent for

the cocycle {Φk} and the smallest Lyapunov exponent for the cocycle {Dϕk}. Next, we

21

will use Furstenberg-Kesten formula for Λµ(ψ,Ψ). To finish the proof of the lemma, we will

estimate Λµ(ψ,Ψ) following the plan of the Bowen’s proof of the variational principle [Bo],

thus generalizing the proof of [CL, Thm. 8.53].

Let ν ∈ Erg(ϕ,Θ). Recall that the conclusions of the multiplicative ergodic theorem

hold for both cocycles {Dϕk} and {Φk} on a subset of Θ of full ν-measure. Let Θ(ν) denote

this subset. The conclusions that we will need are as follows. For each point θ ∈ Θ(ν)

and each vector v ∈ Tθ there exists exact limit χν(θ, v) = limk→∞ k−1 log |Dϕk(θ)v|, the

Oseledec-Lyapunov exponent. There are only p, p ≤ d, distinct values χ(1)ν , . . . χ

(p)ν of the

limits. They depend on the choice of v ∈ Tθ, but independent of θ ∈ Θ(ν). We order the

Oseledec-Lyapunov exponents as follows: χ(1)ν > . . . > χ

(p)ν . For the smallest Oseledec-

Lyapunov exponent we abbreviate χν = χ(p)ν . Also, there are p subbundles V1, . . . , Vp such

that if v belongs to the fiber Vi(θ) over θ ∈ Θ(ν) of a subbundle Vi, then χν(θ, v) = χ(i)ν , i =

1, . . . , p. We stress, that the value of χν(θ, v) does not depend on v as long as v belongs

to Vi(θ) with the same number i. That is, for any choice of vectors vj ∈ Vp(θ) we have

χν = limk→∞ k−1 log |Dϕk(θ)vj|, j = 0, . . . , r. Since the conclusions of the multiplicative

ergodic theorem for the cocycle {Φk} hold on Θ(ν), we also have that for each θ ∈ Θ(ν) there

exists exact limit limk→∞ k−1 log ‖Φk(θ)‖. Here ‖ · ‖ denotes the norm of (` × `) matrices.

This limit coincides with λν = λ(1)ν , the largest Oseledec-Lyapunov exponent for the cocycle

{Φk} over ϕ.

Let V p(θ), θ ∈ Θ(ν), p ≤ d, denote the trace in T 1θ of the Oseledets subbundle Vp that

corresponds to the smallest Oseledets–Lyapunov exponent χν for the cocycle {Dϕk} over

ϕ. Let Θr,α ⊂ Θr,α denote the bundle with the base Θ(ν) and fibers V p(θ) × . . . × V p(θ).

Take any measure µ ∈ Erg(ψr,α,Θr,α) such that ν = prµ and that µ is supported in Θr,α

22

(cf. [Ar, Thm.6.2.3]). Select a full µ-measure subset Θ(µ)r,α ⊂ Θr,α such that for each (θ,v) ∈

Θ(µ)r,α the conclusions of the multiplicative ergodic theorem hold for the cocycle {Ψk

r,α} over

ψr,α. In particular, for each point (θ,v) ∈ Θ(µ)r,α there exists the exact limit Λµ(ψ,Ψ) =

limk→∞ k−1 log ‖Ψk

r,α(θ,v)‖, the largest Oseledec-Lyapunov exponent for the cocycle {Ψkr,α}

over ψr,α. Again, by the multiplicative ergodic theorem the limit is independent of (θ,v) ∈

Θ(µ)r,α. Fix a point (θ,v) ∈ Θ

(µ)r,α with v = (v0, . . . , vr). Since θ ∈ Θ(ν) and each vj ∈ V p(θ),

we have that χν = limk→∞ k−1 log |Dϕk(θ)vj|, j = 0, . . . , r and λν = limk→∞ k

−1 log ‖Φk(θ)‖

by the multiplicative ergodic theorem for the cocycles {Dϕk} and {Φk}, respectively. Using

the definition of Ψkr,α, we have that Λµ(ψ,Ψ) = λν − (r + α)χν . Indeed,

Λµ(ψ,Ψ) = limk→∞

k−1 log ‖Ψkr,α(θ,v)‖

= limk→∞

k−1 log ‖Φk(θ)‖ − α limk→∞

k−1 log |Dϕk(θ)v0| −r∑j=1

limk→∞

k−1 log |Dϕk(θ)vj|

= λν − (r + α)χν .

On the other hand, using the Furstenberg-Kesten Theorem ([Ar, Thm.3.3.3] or [W,

Cor.10.1.2]) for µ and {Ψkr,α}, we can also compute Λµ(ψ,Ψ) as follows:

Λµ(ψ,Ψ) = limk→∞

1

k

∫Θr,α

log ‖Ψkr,α(θ,v)‖ dµ.

Using the disintegration µθ(·) of µ with respect to ν and the definition of Ψkr,α, we have:

Λµ(ψ,Ψ) = limk→∞

1

k

∫Θ

∫(T 1θ )r+1

log‖Φk(θ)‖

|Dϕk(θ)v0|α∏r

j=1 |Dϕk(θ)vj|dµθ(v) dν(θ)

≤ limk→∞

1

k

∫Θ

log(‖Φk(θ)‖ · ‖[Dϕk(θ)]−1‖r+α

)dν(θ).

Let A = {A1, . . . , A|A|} denote a finite Borel partition of Θ such that each θ ∈ Θ belongs to

23

the closure of at most M elements of A. As in Step 1 of the proof of Theorem 8.53 in [ChL],

it follows from the definition of entropy and a standard variational lemma [W, Lem.9.9] that

Hν(k−1∨i=0

ϕ−iA) +

∫Θν

log(‖Φk(θ)‖ · ‖[Dϕk(θ)]−1‖r+α

)dν (3.11)

≤ log∑

B∈∨k−1i=0 ϕ

−iA

supθ∈B

(‖Φk(θ)‖ · ‖[Dϕk(θ)]−1‖r+α

)= log

∑B∈∨k−1i=0 ϕ

−iA

(‖Φk(θB)‖ · ‖[Dϕk(θB)]−1‖r+α

)

for suitable θB in the closure of B.

First, we remark that there is an n = n(A) such that each Uin intersects no more

than M sets A for A ∈ A. Indeed, if not, then there is a nested sequence U(m)in and θ =

limm→∞ θ(m), θ(m) ∈ U

(m)in , such that θ belongs to at least M + 1 sets A for A ∈ A. The

contradiction to the assumptions on A proves the remark. Next, for each θB ∈ B, with

B = Aj0 ∩ ϕ−1(Aj1) ∩ . . . ∩ ϕ−(k−1)(Ajk−1), choose ik+n = i1 . . . ikik+1 . . . ik+n such that

θB ∈ Uik+n . Define f(B) := Uik+n for B ∈∨k−1i=0 ϕ

−iA. We claim that the multiplicity

of the function B 7→ f(B) does not exceed Mk. Indeed, if f(B) = f(B′) then, for each

m = 0, . . . , k − 1, we have that ϕmθB ∈ Ajm and ϕmθB′ ∈ Aj′m , but also that ϕmθB and

ϕmθB′ belong to Ui1+m...ik...ik+n. By the remark above, there are no more than M choices for

jm 6= j′m for each m = 0, . . . , k − 1, and the claim is proved.

Therefore, (3.11) is dominated by

log(Mk

∑in+k

supθ∈U

ik+n

(‖Φk(θ)‖ · ‖[Dϕk(θ)]−1‖r+α

))≤ log

(Mk

∑in

∑ik

supη∈Uin

(‖Φk(ikη)‖ · ‖[Dϕk(ikη)]−1‖r+α

))≤ log

(Mkc(n)ρk(r, α)

).

Since Λµ(ψ,Ψ) = λν − (r + α)χν , we conclude that hν(ϕ,A) + λν − (r + α)χν ≤ logM +

24

log ρ(r, α). Now, passing to the higher iterates of ϕ, the proof can be completed along the

lines of [ChL], see Steps 2 and 3 on p. 320.

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