THERMODYNAMIC FORMALISM FOR RANDOM COUNTABLE · 2017-09-07 · erties of Markov chains with random...

THERMODYNAMIC FORMALISM FOR RANDOM COUNTABLEMARKOV SHIFTS

MANFRED DENKER, YURI KIFER, AND MANUEL STADLBAUER

Dedicated to Yakov Pesin on the occasion of his sixtieth birthday.

Abstract. We introduce a relative Gurevich pressure for random countabletopologically mixing Markov shifts. It is shown that the relative variational

principle holds for this notion of pressure. We also prove a relative Ruelle-Perron-Frobenius theorem which enables us to construct a wealth of invariant

Gibbs measures for locally fiber Holder continuous functions. This is accom-

plished via a new construction of an equivariant family of fiber measures usingCrauel’s relative Prohorov theorem. Some properties of the Gibbs measures

are discussed as well.

1. Introduction

Thermodynamic formalism for deterministic topological Markov shifts with finitenumber of states originates from the work in [25], [27] and [3]. More recently, thetheory has been extended in [28]–[30] and [22] to countable Markov shifts. In thenoncompact situation there is no variational principle with respect to topologicalpressure defined in the standard way. Another quantity, called the Gurevich pres-sure, turns out to be more appropriate as it satisfies the variational principle and thethermodynamic formalism can be adapted to it, as well. Relative thermodynamicformalism for random dynamical systems on compact sets has been developed in[14], [20] and [4] (see also [19] and references in there), and in a slightly differentcontext in [7]. In particular, random Markov shifts with a finite number of statesare included.

In this paper we continue to develop the theory. First, the relative (fiber) Gure-vich pressure for random countable topologically mixing Markov shifts is introducedas the exponential growth rate of the partition functions on cylinders at times ofreturns to the cylinder. It turns out that this number is independent of cylinders,thus the relative Gurevich pressure ΠG(φ) is well defined. This construction issummarized in Theorem 3.2.

ΠG(φ) can be alternatively obtained as the supremum over the relative topo-logical pressure of finite state random subshifts (cf. [17]). This result is Theorem4.1, which in turn is used in the proof of Theorem 4.2 on the relativized variationalprinciple. Here it is shown that the relative Gurevich pressure is the supremum of

Date: July 15, 2007.2000 Mathematics Subject Classification. Primary: 37D35 Secondary: 37H99 .Key words and phrases. random countable shifts, variational principle, thermodynamic for-

malism, random transformations.The authors acknowledge support by ‘Deutsch-Israelische Forschungskooperation, Land Nieder-

sachsen’ and the DFG-project ‘Ergodentheoretische Methoden in der hyperbolischen Geometrie’.

1

2 M. DENKER, YU. KIFER, AND M. STADLBAUER

h(r)µ (T ) +

∫φdµ, where the supremum extends over all T -invariant measures µ and

where the first summand denotes the relative entropy of µ with respect to T .Following [29], a potential φ is called positively recurrent if the partition functions

behave like the product of the relativized eigenvalues. These eigenvalues are definedby the Perron-Frobenius operator and are functions on the base probability space.For such potentials it is shown in Proposition 5.1 that the Gurevich pressure hasanother representation as the expected logarithm of this random eigenvalue. Theeigenvalues are related to the relative Ruelle-Perron-Frobenius operator assuring theexistence of generalized eigenfunctions. These eigenfunctions will produce invariantfiber Gibbs measures provided there are eigenmeasures for the dual operator. Themain theorem on the existence of eigenmeasures and eigenfunctions is 5.2. Its proof,including uniqueness, is carried out in the following sections.

A major part of the present paper is devoted to the construction of these eigen-measures, called random Gibbs measures. Here we do not follow the randomizedversion of the deterministic construction principle in [28], but prefer Patterson’smethod, first developed in [26]. We give a randomized version of the Pattersonconstruction and combine it with a new argument to show relative tightness of thefiber measures. Here we use Crauel’s refinement of Prohorov’s theorem (see [5])to the random case. The method of proof, adapted to the deterministic case, alsoprovides some new insight into the proofs in [28]–[30].

These two facts, the existence of Gibbs measures and eigenfunctions, can becalled the starting point of thermodynamic formalism for random countable Markovshifts. In the case of random finite shifts the thermodynamic formalism was usedin [16] to study random fractals, in particular, for computation of Hausdorff di-mensions of random attractors and random Gibbs measures, and so it is naturalto apply the results of this paper to constructions of random fractals via infiniterandom iterated function systems which in the deterministic case was done in [13]and [21]–[23]. Another application we have in mind is the study of recurrence prop-erties of Markov chains with random stationary changing transition probabilitiesand countable state space beyond the random Doeblin property framework whichwas considered in [17]. Recurrence properties of such Markov chains were studiedby several authors (see [24] and references there) but from a completely differentviewpoint.

Section 6 contains the construction of the fiber Gibbs measures (see Theorem6.1). The key point in the construction is Proposition 6.4, which ensures the ex-istence conditioned on a tightness condition for approximating measures on fibers.The tightness itself is shown in Proposition 6.7 using Crauel’s random Prohorovtheorem in [5].

In a last section we investigate uniqueness properties of (relative) Gibbs measuresand eigenfunctions. It turns out that Guivar’ch’s definition of relative exactness(see [12]) plays a crucial role. We first show that the random Markov chain isconservative and ergodic (Proposition 7.3). Next it is shown that it is relativelyexact under any Gibbs measure with (relatively) locally Holder continuous potential(Proposition 7.4). The proof is adapted from [1]. The uniqueness of the randomGibbs measure is a consequence of the general principle that under weak distortionevery two Gibbs measures (for the same potential) have to be equivalent, henceuniqueness follows from ergodicity (cf. [32]). This is shown in Theorem 7.5 and

THERMODYNAMIC FORMALISM FOR RANDOM SHIFTS 3

its proof. In particular, the characterization of relative exactness (Proposition 7.1)then implies that the random eigenfunction is unique.

Finally it should be pointed out that we are dealing with random dynamicalsystems where the base transformation is invertible and ergodic. It is straightforward to drop the assumption of ergodicity; in order to drop the other assumption,one needs to overcome some unpleasant new phenomena as explained in [7], whereit is shown that Gibbs measures can only exist for certain potentials (althoughequilibrium measures exist).

Due to the random setting the notation becomes a bit overloaded, for which weapologize, but we still hope that the reader will find his way through. We shall setup the notation in Section 2, Section 3 contains the proof of the existence of theGurevich pressure. The variational principle is obtained in Section 4 and the relativeversion of the Ruelle-Perron-Frobenius theorem in Section 5. The construction ofGibbs measures is carried out in Section 6, and exactness and uniqueness are provedin Section 7.

2. Preliminaries

Let (Ω,F , P ) be a probability space and θ : Ω → Ω be a P -preserving ergodicinvertible transformation. Our setup consists also of a N ∪ ∞-valued randomvariable ` = `(ω) > 1, the sets S(ω) = j ∈ N : j < `(ω) and measurablydepending on ω matrices Aω =

(αij(ω), i ∈ S(ω), j ∈ S(θω)

)with entries αij(ω) ∈

0, 1. Introduce the shift spaces

Xω = x = (x0, x1, ...) : αxixi+1(θiω) = 1 ∀i = 0, 1, ...,

which we assume to be nonempty, together with the left shifts Tω : Xω → Xθω

acting by Tω : (x0, x1, x2...) = (x1, x2, ...). We view Xω as measurable spaces takingthem together with the Borel σ-algebras Bω with respect to the product topology onthe space of sequences when on each factor the discrete topology is chosen. Denotealso by T the skew product transformation acting on X = (ω, x) : x ∈ Xω(which becomes naturally a measurable space, as well) by T (ω, x) = (θω, Tωx).The pair (X, T ) will be called a countable random Markov shift. Observe thatTn(ω, x) = (θnω, Tn

ω x) where Tnω = Tθn−1ω · · ·TθωTω. Set [a0, a1, ..., an−1]ω = x ∈

Xω : xi = ai, i = 0, 1, ..., n − 1 which is called a cylinder set of length n and theset of such cylinders we denote by γn−1

ω . Remark that [a]ω, a ∈ S(ω) forms anatural partition of Xω into the 1-cylinders. We assume throughout this paper thatΩa = ω : `(ω) > a = ω : [a]ω 6= ∅ and that P (Ωa) > 0 for all a ∈ S which is acountable or finite set whose elements are successively numbered and without lossof generality we can disregard all other a with P (Ωa) = 0.

We say that the shifts Tω are topologically mixing if for any a, b ∈ N there existsan N-valued random variable Nab = Nab(ω) such that if n ≥ Nab(ω), a ∈ S(ω)and b ∈ S(θnω) then [a]ω ∩ (Tn

ω )−1[b]θnω 6= ∅, i.e. for such a and b the ab-elementα

(n)ab (ω) of the matrix An

ω = Aθn−1ω · · ·AθωAω is positive.For any x, y ∈ Xω set ιω(x, y) = minj : xj 6= yj if x 6= y and ιω(x, x) = ∞.

Fix r ∈ (0, 1) and set dω(x, y) = dωr (x, y) = rιω(x,y) which provides a metric on Xω

compatible with the product topology. For each function φ : X → R, φ = φω(x) =φ(ω, x), (ω, x) ∈ X set V ω

n (φ) = sup|φω(x) − φω(y)| : xi = yi, i = 0, 1, ..., n − 1.We say that φ is locally fiber Holder continuous if there exists a random variableκ = κ(ω) ≥ 1 such that

∫log κdP < ∞ and for any n ≥ 1, V ω

n (φ) ≤ κ(ω)rn.


Introduce a random version of the Ruelle operator by

Lωφf(θω, x) =

∑y∈Xω,Tωy=x

eφω(y)f(y)

and set φn =∑n−1

k=0 φ T k. Then

Lω,nφ f(θnω, x) = Lθn−1ω

φ · · · Lθωφ Lω

φf(θnω, x)

=∑

y∈Xω,T nω y=x eφω

n(y)f(y) =∑

a∈Rnω(x) eφω

n(ax)f(ax)

where Rnω(x) = a = (a0, ..., an−1) : ax ∈ Xω and ax = a0...an−1x0x1... is the

concatenation of a word a and x.For every pair of cylinders [a0, ..., an]ω ⊂ Xω and [b0, ..., bm]θnω ⊂ Xθnω with

an = b0 set

[a0, ..., an] · [b0, ..., bm] = [a0, ..., an−1, b0, ..., bm]ω ⊂ Xω.

For every subset E of Xω let E ∩ γk−1ω denotes the set of all cylinders of length k

that are included in E. If Γ1 ⊂ (Tnω )−1[a]θnω ∩ γn

ω and Γ2 ⊂ [a]θnω ∩ γmθnω set

Γ1 · Γ2 = ∪[a]ω⊂Γ1,[b]θnω⊂Γ2 [a] · [b].

We will need the following simple result.

2.1. Lemma. Let φ : X → R be locally fiber Holder continuous as above and set

Bωn = Bω

n (φ) = exp∑k≥n

V θ−kωk+1 (φ).

Then for P -almost all ω (P -a.a. ω),

(2.1) Bωn < ∞

and for any n ≤ m,

(2.2) V ωm(φn) ≤

n−1∑l=0

V θlωm−l(φ) =

m∑k=m−n+1

V θm−kωk (φ) ≤ log Bθm−1ω

m−n (φ).

Set φωn [a] = supφω

n(x) : x ∈ [a]ω. Then

(2.3) |φωn(x)− φω

n [x0, ..., xm−1]| ≤ V ωm(φn) ≤ log Bθm−1ω

m−n (φ).

Let Γ1 ⊂ (Tnω )−1[a]θnω ∩ γn

ω and Γ2 ⊂ [a]θnω ∩ γmθnω. Set

Cn,mω (Γ1,Γ2) =

( ∑[a]ω∈Γ1∩γn

ω

eφωn [a])( ∑

[b]θnω∈Γ2∩γmθnω

eφθnωm [b]

)and

Cn+mω (Γ1 · Γ2) =

∑[c]ω∈(Γ1·Γ2)∩γn+m

ω

eφωn+m[c].

Then

(2.4) Cn+mω (Γ1 · Γ2) ≤ Cn,m

ω (Γ1,Γ2) ≤ Bθnω1 Bθn+mω

1 Cn+mω (Γ1 · Γ2).

If in addition κ is integrable, then∫log Bω

ndP < ∞.


Proof. Since log κ is integrable then κ(θ−kω) may only grow in k subexponentiallywhich together with

(2.5)∑k≥n

V θ−kωk+1 (φ) ≤

∑k≥n

κ(θ−kω)rk+1

yield (2.1). The inequalities (2.2) follow immediately from the definitions and (2.3)follows from (2.2). In order to obtain (2.4) observe that if [a]ω ∈ Γ1 ∩ γn

ω and[b]θnω ∈ Γ2 ∩ γm

θnω then [a]ω · [b]θnω = [c]ω ∈ (Γ1 ·Γ2)∩ γn+mω , and so if x ∈ [a]ω and

Tnω x ∈ [b]θnω then x ∈ [ab]ω. Hence,

φωn [a] + φθnω

m [b] ≥ φωn+m[ab]

and the left hand side of (2.4) follows. In order to derive its right hand side observethat if [c] ∈ (Γ1 · Γ2)∩ γn+m

ω then there exist [a]ω ∈ Γ1 ∩ γnω and [b]θnω ∈ Γ2 ∩ γm

θnω

such that c = ab. Now if x ∈ [c]ω then x ∈ [a]ω and Tnω x ∈ [b]θnω. By (2.3),

φωn+m[c] ≥ φω

n+m(x) = φωn(x)+φθnω

m (Tnω x) ≥ φω

n [a]−log Bθnω1 +φθnω

m [b]−log Bθn+mω1

and the right hand side of (2.4) follows. The remaining assertion then follows byapplication of the monotone convergence theorem to (2.5).

3. Relative Gurevich pressure

Let (X, T ) be a topologically mixing countable random Markov shift and φ :X → R be a locally fiber Holder continuous function as defined in the previoussection. Introduce a random partition function by

Zωn (φ, a, b) =

∑eφω

n [c] : c = (c0, c1, ..., cn−1), c0 = a, [cb]ω ∈ γn+1ω

which is defined provided a ∈ S(ω), b ∈ S(θnω) and n ≥ Nab(ω) and if there areno words c satisfying the above conditions in braces then we set Zω

n (φ, a, b) = 0.Denote also Zω

n (φ, a) = Zωn (φ, a, a). Observe that Zω

n (φ, a, b) > 0 for all n ≥ Na,b(ω)whenever a < `(ω) and b < `(θnω).

3.1. Lemma. Let X, T, θ and φ be as above. Then for any m,n ∈ N and a, b, c ∈ S,

(3.1) Zωn (φ, a, b)Zθnω

m (φ, b, c) ≤ Bθnω1 Bθm+nω

1 Zωm+n(φ, a, c).

Proof. If the right hand side of (3.1) is zero then it is easy to see that one of thefactors in its left hand side is zero too. If all three terms in (3.1) are not zero then(3.1) follows from (2.4).

For each N ∈ N denote ΓabN = ω ∈ Ω : Nab(ω) ≤ N and Γa

N = ΓaaN . Set

Ωa,N = Ωa ∩ ΓaN and for all N so large that P (Ωa,N ) > 0 define n

(0)a,N (ω) = 0 and

inductivelyn

(k+1)a,N (ω) = minn > n

(k)a,N (ω) : θnω ∈ Ωa,N.

Then by the Kac lemma (see §5 in Ch.1 of [6]),∫Ωa,N

n(1)a,NdP = 1

and, in particular, n(1)a,N < ∞ Pa,N -almost surely (Pa,N -a.s.). In fact, n

(1)a,N < ∞

P −a.s. in view of the ergodicity of θ which implies that P (∪j≥0θ−jΩa,N ) = 1. For

each function ϕ = ϕω(x) = ϕ(ω, x) on X set ‖ϕω‖ = supx∈Xω|ϕ(ω, x)|.


3.2. Theorem. Let (X, T ) be a topologically mixing countable random Markov shiftand φ be a locally fiber Holder continuous function such that

(3.2)∫

log Bω1 dP,

∫‖φω‖dP (ω) < ∞ and

∫ ∣∣ log ‖Lωφ1‖∣∣dP (ω) < ∞.

Denote by Pa,N the normalized restriction of P to Ωa,N . Then for Pa,N -a.a. ω ∈Ωa,N the limit

(3.3) ΠG(φ) = limj→∞

(n(j)a,N (ω))−1 log Zω

n(j)a,N (ω)

(φ, a)

exists and it is constant Pa,N -a.s. Furthermore, the limit in (3.3) is the same forall N large enough, it does not change if we replace a by any b ∈ S and it will becalled the relative (fiber) Gurevich pressure of φ. Moreover, for any a, b ∈ S and Nlarge enough for Pa,N -almost all ω ∈ Ωa,N ,

(3.4) ΠG(φ) = limj→∞

(n(j)b,N (ω))−1 log Zω

n(j)b,N (ω)

(φ, a, b).

Proof. Observe, first, that by the last assertion of Lemma 2.1 the integrability oflog Bω

1 follows if we assume the integrability of κ. Next, set Ω(m)a,N = ω : n

(1)a,N (ω) =

m. Then θjΩ(m)a,N , j = 0, 1, ...,m − 1; m = 1, 2, ... are disjoint. Since θ is ergodic

we have that

P (Ω \ (∪∞m=1 ∪m−1j=0 θjΩ(m)

a,N )) = P (Ω \ ∪∞j=0θjΩa,N ) = 0.

Hence, by (3.2), ∫IΩa,N

∑0≤j<n

(1)a,N (ω)

‖φθjω‖dP (ω)(3.5)

=∑∞

m=1

∑m−1j=0

∫IΩ

(m)a,N

‖φθjω‖dP (ω) =∫‖φω‖dP (ω) < ∞,

where IΓ(ω) = 1 if ω ∈ Γ and = 0, otherwise. Similarly,

(3.6)∫

IΩa,N

∑0≤j<n

(1)a,N (ω)

∣∣ log ‖Lθjωφ 1‖

∣∣dP (ω) =∫ ∣∣ log ‖Lθjω

φ 1‖∣∣dP (ω) < ∞.

For M ≥ N set

qωN,M (j) = − log Zω

n(jM)a,N (ω)

(φ, a) and Θa,Nω = θn(1)a,N (ω)ω

so that Θa,N is the induced transformation of θ on the set Ωa,N (see §5, Ch. 1 in[6]). Then by Lemma 3.1,

(3.7) qωN,M (j) + q

ΘjMa,N ω

N,M (l) ≥ qωN,M (j + l)− log B

ΘjMa,N ω

1 − log BΘ

(j+l)Ma,N ω

1 .

SetF (ω) =

∑0≤i<n

(1)a,N (ω)

‖φθiω‖ and Q(ω) =∑

0≤i<n(1)a,N (ω)

log ‖Lθiωφ 1‖.

Then for any m ≥ N and each ω ∈ Ωa,N ,

−∑m−1

j=0 F (Θja,Nω) ≤ log Zω

n(m)a,N (ω)

(φ, a)(3.8)

≤ log ‖Lω,n(m)a,N (ω)

φ 1‖ ≤∑m−1

j=0 Q(Θja,Nω).


Since n(M)a,N (ω) ≥ N it follows from (3.2), (3.5), (3.6) and (3.8) that for all l = 1, 2, ...,

qN,M (l) ∈ L1(Ωa,N , Pa,N )

and

inf l≥1(1/l)∫

qωN,M (l)dPa,N (ω) ≥ −M

∫Q(ω)dPa,N (ω)

= − NP (Ωa,N )

∫ ∣∣ log ‖Lωφ1‖∣∣dP (ω) > −∞.

Since log Bω1 is integrable by (3.2), we have also that Pa,N -a.s.,

l−1(| log Bω

1 |+ | log BΘlM

a,N ω

1 |)→ 0 as l →∞.

Hence, by Theorem 4 in [11] (see also Theorem 2 in [31]) the limit

(3.9) liml→∞

l−1qωN,M (l) = qa,N,M (ω)

exists Pa,N -a.s. and in L1(Ωa,N , Pa,N ) and the function qa,N defined on Ωa,N isΘM

a,N -invariant. Since θ is ergodic and Θa,N is its induced transformation thenΘa,N is ergodic too (see §5, Ch. 1 in [6]), and so Pa,N -a.s. the limit

(3.10) na,N,M = liml→∞

l−1n(lM)a,N (ω) = M lim

k→∞k−1n

(k)a,N (ω)

exists and it is not random. Therefore, P -a.s.,

(3.11) liml→∞

(n

(lM)a,N (ω)

)−1 log Zω

n(lM)a,N (ω)

(φ, a) = Πa,N,M (ω)

where Πa,N,M (ω) = −n−1a,N,M qa,N,M (ω) is ΘM

a,N -invariant. Observe that for any

integers M1,M2 ≥ N the sequence n(lM1M2)a,N (ω), l = 1, 2, ... is a subsequence of

both n(lM1)a,N (ω), l = 1, 2, ... and of n(lM2)

a,N (ω), l = 1, 2, ..., and so Pa,N -a.s.,

Πa,N,M1(ω) = Πa,N,M1M2(ω) = Πa,N,M2(ω).

Hence, the limit in (3.11) does not depend on M ≥ N with Pa,N -probability one,and so we can denote it by Πa,N (ω). But Πa,N is ΘM

a,N -invariant for all M ≥ N

which implies that Πa,N is Θa,N -invariant. Since Θa,N is ergodic as an inducedtransformation of θ we obtain that Πa,N is constant Pa,N -a.s.

Let lN ≤ j < (l + 1)N . Then by (3.1),

log Zω

n((l+2)N)a,N (ω)

(φ, a)− log ZΘj

a,N ω

n((l+2)N)a,N (ω)−n

(j)a,N (ω)

(φ, a)(3.12)

+ log BΘj

a,N ω

1 + log BΘ

(l+2)Na,N ω

1 ≥ log Zω

n(j)a,N (ω)

(φ, a) ≥ log Zω

n((l−1)N)a,N (ω)

(φ, a)

− log BΘj

a,N ω

1 − log BΘ

(l−1)Na,N ω

1 + log ZΘ(l−1)N ω

n(j)a,N (ω)−n

((l−1)N)a,N (ω)

(φ, a).

Since n((l+2)N)a,N (ω)− j ≥ N, j − n

((l−1)N)a,N (ω) ≥ N , Θj

a,Nω ∈ Ωa,N and Θ(l−1)Na,N ω ∈

Ωa,N we obtain similarly to (3.8) that

(3.13) − log ZΘj

a,N ω

n((l+2)N)a,N (ω)−n

(j)a,N (ω)

(φ, a) ≤2N∑i=0

F (Θj+ia,Nω)


and

(3.14) log ZΘ(l−1)N ω

n(j)a,N (ω)−n

((l−1)N)a,N (ω)

(φ, a) ≥ −2N∑i=0

F (Θ(l−1)N+ia,N ω).

We conclude from (3.2), (3.5) and (3.11)–(3.14) that Pa,N -a.s.,

(3.15) limj→∞

(n

(j)a,N (ω)

)−1 log Zω

n(j)a,N (ω)

(φ, a) = Πa,N

where we use the standard fact that if∫|ϕ|dPa,N < ∞ then limj→∞ j−1ϕ(Θj

a,Nω) =

0 for Pa,N -a.a. ω ∈ Ωa,N . Now observe that for N ′ > N the sequence n(j)a,N ′(ω),

j = 1, 2, ... contains the sequence n(j)a,N (ω), j = 1, 2, ... for Pa,N -a.a. ω ∈ Ωa,N . It

follows that Πa,N = Πa,N ′ for N ′ > N . Hence, Πa,N does not depend on N and weobtain that Pa,N -a.s. the limit in (3.15) is equal to some constant Πa independentof N .

Next, we prove that Πa = Πb for any b ∈ S. Indeed, by (3.1),

log Zωk (φ, a, b) + log Zθkω

l (φ, b) + log Zθk+lωm (φ, b, a)(3.16)

≤ log Bθkω1 + 2 log Bθk+lω

1 + log Bθk+l+mω1 + log Zω

k+l+m(φ, a).

Set Ωa,b,N = Ωa,N ∩ ΓabN and Ωb,a,N = Ωb,N ∩ Γba

N assuming that N is large enoughso that

P (Ωa,b,N ) > 0 and P (Ωb,a,N ) > 0.

Then by ergodicity of θ we can choose k ≥ N and m ≥ N so that

P (Ωa,b,N ∩ θ−kΩb,a,N ) > 0 and P (Ωb,a,N ∩ θ−mΩa,N ) > 0.

Set n(0)b,a,N (ω) = 0 and inductively

(3.17) n(j+1)b,a,N (ω) = minn > n

(j)b,a,N (ω) : θnω ∈ Ωb,a,N ∩ θ−mΩa,N.

Then for all ω ∈ Ωa,b,N ∩ θ−kΩb,a,N except may be for a set of ω’s having zeroP -probability the sequence k + n

(j)b,a,N (θkω) + m, j = 1, 2, ... is a subsequence of

n(j)a,N (ω), j = 1, 2, ... and the sequence n(j)

b,a,N (θkω), j = 1, 2, ... is a subsequence

of n(j)b,N (θkω), j = 1, 2, .... Take l = lj = n

(j)b,a,N (θkω) in (3.16), divide both parts

of (3.16) by lj and let j → ∞. Then taking into account (3.2), (3.5), (3.6), (3.8)and the fact that (3.15) holds true and the limit there does not depend on N andω both for a and b we obtain from (3.16) that Πb ≤ Πa. By the symmetry of a andb we have also the inequality in the other direction concluding that Πa = Πb whichenables us now to denote this common value by ΠG(φ).

It remains to derive (3.4). By (3.1),

(3.18) log Zωn (φ, a, b)+ log Zθnω

m (φ, b, a) ≤ log Bθnω1 +log Bθn+mω

1 +log Zωn+m(φ, a)

and

(3.19) log Zωn (φ, a)+log Zθnω

m (φ, a, b) ≤ log Bθnω1 +log Bθn+mω

1 +log Zωn+m(φ, a, b).

As above we choose N large enough so that

P (Ωb,a,N ) > 0 and P (Ωa,N ) > 0.

Then by ergodicity of θ there exists m ≥ N such that

P (Ωb,a,N ∩ θ−mΩa,N ) > 0.


Let n(j)b,a,N (ω), j = 1, 2, ... be as in (3.17). Then n(j)

b,a,N (ω), j = 1, 2, ... and

n(j)b,a,N (ω) + m, j = 1, 2, ... are subsequences of n(j)

b,N (ω), j = 1, 2, ... and of

n(j)b,N (ω), j = 1, 2, ..., respectively, provided ω ∈ Ωa,N . Since m ≥ N then for any

ω ∈ Ωb,a,N ∩ θ−mΩa,N ,

(3.20) −m−1∑i=0

‖φθiω‖ ≤ log Zωm(φ, b, a) ≤

m−1∑i=0

∣∣ log ‖Lθiωφ 1‖

∣∣.Similarly to above, dividing (3.18) by n, taking there n = n

(j)b,a,N (ω) and letting

j →∞ we obtain in view of (3.2), (3.3), (3.18) and (3.20) that Pa,N -a.s.,

(3.21) lim supj→∞

(n

(j)b,a,N (ω)

)−1 log Zω

n(j)b,a,N (ω)

(φ, a, b) ≤ ΠG(φ).

Now, let i = i(j) be such that n(j)b,a,N (ω) ≤ n

(i)b,N (ω) < n

(j+1)b,a,N (ω). Then by (3.1) for

any ω ∈ Ωa,N ,

log Zω

n(i)b,N (ω)

(φ, a, b) ≤ log Zω

n(j+N)b,a,N (ω)

(φ, a, b)(3.22)

− log ZΘj+N

b,a,N ω

n(j+N)b,a,N (ω)−n

(i)b,N (ω)

(φ, b) + BΘj+N

b,N ω

1 + BΘj+N

b,a,N ω

1

where Θb,a,Nω = θn(1)b,a,N (ω)ω. Since

n(N)b,a,N (Θj

b,a,Nω) ≥ n(j+N)b,a,N (ω)− n

(i)b,N (ω) ≥ N and Θj+N

b,a,Nω ∈ Ωb,N

then similarly to (3.8),

(3.23) − log ZΘj+N

b,a,N ω

n(j+N)b,a,N (ω)−n

(i)b,N (ω)

(φ, b) ≤N∑

l=0

F (Θj+lb,a,Nω)

whereF (ω) =

∑0≤k<n

(1)b,a,N (ω)

‖φθkω‖.

Similarly to (3.5) we obtain that

(3.24)∫

Ωb,a,N∩θ−mΩa,N

F dP =∫

Ω

‖φω‖dP (ω) < ∞.

Observe that n(i)b,N (ω) ≥ i ≥ j, and so we derive from (3.2) that P -a.s.,

(3.25) limi→∞

(n

(i)b,N (ω)

)−1(B

Θib,N ω

1 + BΘj+N

b,a,N ω

1

)= 0.

By (3.24) we have also that for all ω ∈ Ωb,a,N ∩θ−mΩa,N except may be for a subsetof ω’s from Ωb,a,N ∩ θ−mΩa,N having zero P -probability,

(3.26) limj→∞

j−1F (Θj+lb,a,N ω) = 0

for l = −1, 0, ..., N . Observe that for any subset U ⊂ Ωa,N with Pa,N (U) > 0 wehave also that P (Θb,a,NU) > 0. Indeed, the former implies that there exists k suchthat Pa,N (Uk) > 0 for Uk = ω ∈ U : n

(1)b,a,N (ω) = k. But Θb,a,Nω = θkω for each

ω ∈ Uk, and so0 < P (Uk) = P (Θb,a,NUk) ≤ P (Θb,a,NU).


It follows that (3.26) holds true for Pa,N -a.a. ω ∈ Ωa,N . Now, dividing (3.22) byn

(i)b,N (ω) and letting i → ∞ we derive from (3.21)–(3.26) (the latter considered for

Pa,N -a.a. ω) that Pa,N -a.s.,

(3.27) lim supi→∞

(n

(i)b,N (ω)

)−1 log Zω

n(i)b,N (ω)

(φ, a, b) ≤ ΠG(φ).

Next, we show that Pa,N -a.s.,

(3.28) lim infi→∞

(n

(i)b,N (ω)

)−1 log Zω

n(i)b,N (ω)

(φ, a, b) ≥ ΠG(φ)

which together with (3.27) yields (3.4). First, we choose N large enough so thatP (Ωa,b,N ) > 0 and P (Ωb,N ) > 0 and then by ergodicity of θ we pick up m ≥ N sothat

P (Ωa,b,N ∩ θ−mΩb,N ) > 0.

Define n(j)a,b,N (ω), j = 1, 2, ... by (3.17) exchanging a and b. Then n(j)

a,b,N (ω), j =

1, 2, ... and n(j)a,b,N (ω)+m, j = 1, 2, ... are subsequences of n(j)

a,N (ω), j = 1, 2, ...and of n(j)

b,N (ω), j = 1, 2, ..., respectively, provided ω ∈ Ωa,N . Dividing both parts

in (3.19) by n(j)a,b,N (ω) + m, letting j → ∞ and arguing as above we obtain, first,

that Pa,N -a.s.,

(3.29) ΠG(φ) ≤ lim infj→∞

(n

(j)a,b,N (ω) + m

)−1 log Zω

n(j)a,b,N (ω)+m

(φ, a, b).

Then, choosing i = i(j) so that n(j)a,b,N (ω) + m ≤ n

(i)b,N (ω) < n

(j+1)a,b,N (ω) + m and

proceeding as above in (3.22)–(3.27) we arrive at (3.28), completing the proof ofTheorem 3.2.

4. Variational principle

Let L` be the set of random variables ℘ with values in N and such that ℘ ≤ `

and ℘ < ∞ P -a.s. Set S(℘)ω = j ∈ N : j < ℘(ω) and A

(℘)ω =

(αij(ω), i ∈ S

(℘)ω , j ∈

S(℘)θω

), which is the (℘(ω)− 1)× (℘(θω)− 1) upper left block in the matrix Aω, and

denote

X(℘)ω = x = (x0, x1, ...) : xi ∈ S

(℘)θiω and αxixi+1(θ

iω) = 1 ∀i = 0, 1, ....

Employing again the left shift Tω : X(℘)ω → X

(℘)θω we arrive at a random subshift

of finite type with compact fibers X(℘)ω , ω ∈ Ω if we take the product topology

on the sequence space and the discrete topology on each S(℘)ω which is compatible

with the metric dωr introduced at the beginning of Section 2. Denote by π(℘)(φ) the

fiber (relative) topological pressure of the skew product transformation T : X(℘) →X(℘), T (ω, x) = (θω, Tωx), x ∈ X

(℘)ω (see [18]).

4.1. Theorem. Let (X, T ) be a topologically mixing countable random Markov shiftand φ be a locally fiber Holder continuous function such that the conditions ofTheorem 3.2 hold true. Then

(4.1) ΠG(φ) = supπ(℘)(φ) : ℘ ∈ L`.


Proof. For any a ∈ S(℘)(ω) and b ∈ S(θn℘)(ω) set

Zω,℘n (φ, a, b) = sup

∑[c]ω∈γn,℘

ω ,c0=a,cn=b

eφωn(x[c]ω ), x[c]ω ∈ [c]ω

where γn,℘

ω is the set of n-cylinders in X(℘)ω and x[c]ω is a point in [c]ω. When the

set of cylinders of length n+1 starting at a and ending at b which appears in bracesis empty we set Zω,℘

n (φ, a, b) = 0. Introduce also

Zω,℘n (φ, a, b) =

∑[c]ω∈γn,℘

ω ,c0=a,cn=b

eφωn [c].

By (2.3),

(4.2) Zω,℘n (φ, a, b) ≤ Zω,℘

n (φ, a, b) ≤ Bθnω1 (φ)Zω,℘

n (φ, a, b).

It follows from Proposition 1.6 in [18] that with probability one,

(4.3) π(℘)(φ) = limn→∞

1n

log∑

a∈S(℘)(ω),b∈S(℘)(θnω)

Zω,℘n (φ, a, b).

Set Ω(℘)l = ω ∈ Ω : ℘(ω) ≤ l and assume that l is large enough so that P (Ω(℘)

l ) >

0. Let κ(l)j (ω) be successive times when θκ

(l)j (ω)ω ∈ Ω(℘)

l , i.e. κ(l)0 (ω) = 0 and

κ(l)j+1(ω) = mini > κ

(l)j (ω) : θiω ∈ Ω(℘)

l . Then, of course, the above limit equals

again π(℘)(φ) with probability one if we take it along the subsequence nj = κ(l)j (ω)

in place of n = 1, 2, .... Set ΓB = ω : Bω1 (φ) ≤ B and choose B > 0 large enough

so that P (Ω(℘)l ∩ ΓB) > 0. Let κ

(l,B)j (ω), j = 1, 2, ... be successive times when

θκ(l,B)j (ω)ω ∈ Ω(℘)

l ∩ ΓB which is a subsequence of κ(l)j (ω), j = 1, 2, ..., and so the

above limit equals π(℘)(φ) with probability one if we take it along the subsequencenj = κ

(l,B)j (ω), j = 1, 2, ... in place of n = 1, 2, .... Then there exists a subsequence

ji, i = 1, 2, ... and some a and b such that with positive probability

(4.4) π(℘)(φ) = limi→∞

1

κ(l,B)ji

(ω)log Zω,℘

κ(l,B)ji

(ω)(φ, a, b).

Since

(4.5) Zω,℘n (φ, a, b) ≤ Zω

n (φ, a, b)

we derive from here and Theorem 3.2 that

(4.6) π(℘)(φ) ≤ ΠG(φ).

In order to prove the inequality in the other direction fix ε > 0 and relying onTheorem 3.2 choose k0 = k0(ω) so large that for all k ≥ k0(ω),

(4.7) ΠG(φ) ≤ 1

n(k)a,K(ω)

log Zε

n(k)a,K(ω)

(φ, a) + ε

for Pa,K-a.a. ω ∈ Ωa,K assuming that K is large enough so that P (Ωa,K) > 0.Let Ωa,K = ω ∈ Ωa,K : k0(ω) ≤ K and choose K so large that P (Ωa,K) > 0.Set Ωa,K,B = Ωa,K ∩ ΓB and choose B so large that P (Ωa,K,B) > 0. Now, letn(0)(ω) = 0 and

n(k+1)(ω) = minn > n(k)(ω) : θnω ∈ Ωa,K,B


which is a subsequence of n(k)a,K(ω), k = 1, 2, .... Next, choose a minimal integer

M = M(ω) so that

(4.8)1

n(K)(ω)log Zω

n(K)(ω)(φ, a) ≤ 1n(K)(ω)

log Zω,M(ω)

n(K)(ω)(φ, a) + ε

where

Zω,Mn (φ, a) =

∑eφω

n [c], [c]ω ∈ γnω , c0 = cn = a, [c]ω = [c0, ..., cn],(4.9)

cj ∈ 1, ...,M(ω)− 1 ∀j = 0, 1, ..., n

if the set of n-cylinders appearing in braces is nonempty (which holds true ifwe choose first n ≥ Naa(ω) and then pick up M(ω) large enough) and we setZω,M

n (φ, a) = 0, otherwise.Now, observe that for P -almost all ω ∈ Ωa,K,B there exist ω1, ω2, ..., ωK ∈ Ωa,K,B

and j1, j2, ..., jk ≥ 0 such that ω = θjiωi, i = 1, ...,K and ω 6= θjω if ω 6= ω1, ..., ωK

or j < n(K)(ω), j 6= j1, ..., jK . Namely, j1 = j1(ω) = minj ≥ 0 : θ−jω ∈ Ωa,K,B,ω1 = θ−j1ω and, recursively,

ji+1 = ji+1(ω) = minj > ji : θ−jω ∈ Ωa,K,B, ωi+1 = θ−ji+1ω.

Set ℘(ω) = max1≤j≤K M(ωj) and Θω = θn(1)(ω)ω. Then for any cylinder [c]ω =[c0, c1, ..., cn(K)(ω)]ω, ω ∈ Ωa,K,B with c0 = cn(K) = a we have that [c]ω ∩X

(℘)ω 6= ∅.

Then by (4.2),

(4.10) Zω,℘(ω)

n(K)(ω)(φ, a) ≤ BZω,℘

n(K)(ω)(φ, a)

where Zω,℘(ω)n (φ, a) = Z

ω,℘(ω)n (φ, a, a) and the latter was introduced at the begin-

ning of the proof. Since n(NK)(ω) =∑N−1

j=0 n(K)(ΘjKω) it follows from (2.4) and(3.1) that

(4.11)N−1∏j=0

ZΘjKω,℘n(K)(ΘjKω)

(φ, a) ≤ B2N Zω,℘n(NK)(ω)

(φ, a).

Then taking here log, dividing by n(NK)(ω), letting N → ∞ and taking sup over℘ ∈ L` we obtain from above that

(4.12) ΠG(φ) ≤ sup℘∈L`

π(℘)(φ) + 2ε +2K

log B

taking into account that n(K)(ω) ≥ K for all ω. Since ε and K can be chosenarbitrarily small and large, respectively, we obtain the required inequality whichtogether with (4.6) completes the proof of Theorem 4.1.

The following result is the fiber (relative) variational principle for a class ofrandom infinite topologically mixing Markov shifts considered in this paper.

4.2. Theorem. Let (X, T ) be a topologically mixing countable random Markov shiftand φ be a locally fiber Holder continuous function such that the conditions ofTheorem 3.2 hold true. Then

(4.13) ΠG(φ) = suph(r)

µ (T ) +∫

φdµ : µ ∈MT (X)

where h(r)µ (T ) is the fiber (relative) entropy of T (see [18] and [19]) and MT (X) is

the set of T -invariant probability measures on X with the marginal P on Ω.


Proof. By Theorem 4.1 for any ε > 0 we can choose ℘ ∈ L` so that

(4.14) ΠG(φ) ≤ π(℘)(φ) + ε.

The shift T acting on X(℘) has compact fibers, and so we can apply to it the theorydescribed in [19]. Since T is, clearly, fiber (relative) expansive (see [19], Section 1.3)then h

(r)µ (T ) is upper semicontinuous in µ (see Theorem 1.3.5 in [19]), and so there

exists a T -invariant probability measure µ = µ(℘)φ on X(℘), which can be trivially

extended to the whole X, such that

π(℘)(φ) = h(r)µ (T ) +

∫φdµ.

It follows that

(4.15) ΠG(φ) ≤ suph(r)

µ (T ) +∫

φdµ : µ ∈MT (X).

In order to obtain the inequality in the other direction we will show that

(4.16) ΠG(φ) ≥ h(r)µ (T ) +

∫φdµ

for any µ ∈ MT (X). So, fix µ ∈ MT (X) and for each m ∈ N set [≥ m]ω = x ∈Xω : x0 ≥ m, α

(m)ω = [a]ω, a ∈ S(ω), a < m; [≥ m]ω and B(m)

ω = σ(α(m)ω )

which is the (finite) σ-algebra generated by the partition α(ω)ω of Xω. For each set

Γ ⊂ X let Γω = x ∈ Xω : (ω, x) ∈ Γ and set α(m) = Γ : Γω ∈ α(m)ω which is a

measurable partition of X. Since

B(m)ω ↑ ∪mB(m)

ω ⊂ σ(∪mB(m)ω ) = Bω,

where, recall, Bω is the Borel σ-algebra on Xω with respect to the product topology,we obtain relying on properties of the fiber (relative) entropy and the relativeKolmogorov-Sinai theorem (see Section 2.1 in [15] and [2]) that

(4.17) limm→∞

h(r)µ (T, α(m)) = h(r)

µ (T )

where h(r)µ (T, α(m)) is the relative entropy of T with respect to the partition α(m).

Fix m and set β = α(m) and βω = α(m)ω . For ai ∈ βθiω we write [a0, ..., an]ω =

∩nk=0(T

kω )−1ak (or [a]ω), βn

ω = ∨nk=0(T

kω )−1βθkω and βn = Γ ⊂ X : Γω ∈ βn

ω.Since for a ∈ βω, b ∈ βθnω and [a]ω ⊂ a ∩ (Tn

ω )−1b,

µω([a]ω|a ∩ (Tnω )−1b) =

µω([a]ω)µω(a ∩ (Tn

ω )−1b)

(where µω(·|·) denotes the conditional probability) we obtain by Jensen’s inequalityfor the concave function log that

Hµω(βn

ω) +∫

φωndµω ≤

∑a∈βω,b∈βθnω

µω(a ∩ (Tnω )−1b)(4.18) ∑

[a]ω⊂a∩(T nω )−1b,[a]ω∈βn

ωµω([a]ω|a ∩ (Tn

ω )−1b) log eφωn [a]

µω([a]ω)

≤∑

a∈βω,b∈βθnωµω(a ∩ (Tn

ω )−1b) log∑


ω

eφωn [a]

µω(a∩(T nω )−1b)

=∑


ωµω(a ∩ (Tn

ω )−1b)Πωn(a, b) + Hµω

(βω ∨ (Tn

ω )−1βθnω

)where

Πωn(a, b) = log

∑[a]ω⊂a∩(T n

ω )−1b,[a]ω∈βnω

eφωn [a]


and Hν(ξ) is the usual entropy of a partition ξ with respect to a measure ν. Observealso that

(4.19) Hµω

(βω ∨ (Tn

ω )−1βθnω

)≤ Hµω

(βω) + Hµθnω(βθnω) ≤ 2 log m

where, recall, m is the number of elements of β.Next, if a ∈ βω, b ∈ βθnω, a 6= [≥ m]ω and b 6= [≥ m]θnω then, clearly,

Πωn(a, b) = log Zω

n (φ, a, b),

and so by (3.4) we obtain in this case that Pa,N -a.s.,

(4.20) limj→∞

1

n(j)b,N (ω)

Πω

n(j)b,N (ω)

(a, b) = ΠG(φ).

On the other hand, if a = [≥ m]ω or b = [≥ m]θnω then we claim that P -a.s.,

(4.21) lim supn→∞

1n

Πωn(a, b) ≤

∫ (‖φω‖+ log ‖Lω

φ1‖)dP (ω).

In order to prove (4.21) suppose, first, that b = [≥ m]θnω ∈ βθnω. For every[a]ω ∈ βn

ω choose a point xa ∈ [a]ω ⊂ Xω such that

(4.22) φωn [a]ω ≤ φω

n(xa) + log 2.

Write [a]ω ∈ βn

ω : [a]ω ⊂ a ∩ (Tnω )−1b

= ∪n−1

k=0Γωk ,

where for k = 0, 1, ..., n− 1, Γωk are the pairwise disjoint sets

Γωk =

[a]ω ∈ βn

ω : a = (a, a1, ..., ak, bk+1, ..., bn)

where ai ∈ βθiω, ai 6= [≥ m]θiω and bj = [≥ m]θjω

.

SetSω

k =∑

[a]∈Γωk

eφωn(xa)

and let kn(ω) < n be such that Sωkn(ω) is maximal. Note that

Sωkn(ω) =

∑[a]∈Γω

kn(ω)eφω

kn(ω)(xa)+φωn−kn(ω)(T

kn(ω)ω xa)

≤ eΨn−kn(ω)(θkn(ω)ω)

∑[a]∈Γω

kn(ω)eφω

kn(ω)(xa)

where Ψl(ω) =∑l−1

j=0 ‖φθjω‖. Choose arbitrary points xi ∈ [i]ω. Since the kn(ω)thcoordinate of each xa belongs to 1, ...,m− 1 we have that

Sωkn(ω) ≤ Bω

1 eΨn−kn(ω)(θkn(ω)ω)

∑m−1i=1

(L

ω,kn(ω)φ I[a]ω

)(xi)

≤ Bω1 (m− 1)eΨn−kn(ω)(θ

kn(ω)ω)∏kn(ω)−1

j=0 ‖Lθjωφ 1‖.

Then, using (4.22) we obtain that

Πωn(a, b) ≤ log

∑n−1k=0 2Sω

k ≤ log(2nSωkn(ω))

≤ log(2n) + log Bω1 + log(m− 1) +

∑n−1j=0 ‖φθjω‖+

∑n−1j=0 log ‖Lθjω

φ 1‖.


Dividing both parts of this inequality by n and letting n → ∞ we obtain (4.21)by the ergodic theorem. In the case a = [≥ m]ω and b 6= [≥ m]θnω choose xb ∈ b.Then

Πωn(a, b) ≤ log

(Bω

1 (Lω,nφ I[a]ω )(xb)

)≤

n−1∑j=0

log ‖Lθjωφ 1‖+ log Bω

1

and dividing both parts of this inequality by n and letting n →∞ we derive (4.21)again by the ergodic theorem.

Now given ε > 0, by (4.20) and (4.21) for Pa,N -a.a. ω ∈ Ωa,N and any Γ,Ξ ∈ βthere exists JΓΞ(ω) ∈ N such that if j ≥ JΓΞ(ω), a = Γω ∈ βω and b = Ξθnω ∈ βθnω

then

(4.23) Πω

n(j)b,N (ω)

(a, b) ≤ n(j)b,N (ω)(ΠG(φ) + ε)

provided a 6= [≥ m]ω and b 6= [≥ m]θnω, while

(4.24) Πω

n(j)b,N (ω)

(a, b) ≤ n(j)b,N (ω)(C + ε),

where C =∫

(‖φω‖ + log ‖Lωφ1‖)dP (ω), when a = [≥ m]ω or b = [≥ m]θnω. Then

for any j ≥ J(ω) = maxJΓΞ(ω) : Γ,Ξ ∈ β, n = n(j)b,N (ω) and Pa,N -a.a. ω ∈ Ωa,N ,∑

a∈βω,b∈βθnωµω(a ∩ (Tn

ω )−1b)Πωn(a, b) ≤ nµω

([≥ m]cω(4.25)

∩(Tnω )−1[≥ m]cθnω

)(ΠG(φ) + ε) + nµω

([≥ m]ω ∪ (Tn

ω )−1[≥ m]θnω

)(C + ε)

≤ n(1− εm(ω))ΠG(φ) + Cnεm(ω) + Cnεm(θnω) + ε

where [≥ m]cω = Xω \ [≥ m]ω and εm(ω) = µω([≥ m]ω). Clearly, P -a.s.,

(4.26) εm(ω) → 0 and εm =∫

εm(ω)dP (ω) → 0 as m →∞.

Set Qm = ω : µω([≥ m]ω) ≤√

εm then by Chebyshev’s inequality

P (Qm) ≥ 1−√

εm.

Hence, if m is large enough then

P (Qm ∩ Ωa,N ) > 0 and P (Qm ∩ Ωb,N ) > 0.

In particular, we can define n(0)b,N,m(ω) = 0 and inductively

n(k+1)b,N,m(ω) = minn > n

(k)b,N,m(ω) : θnω ∈ Qm ∩ Ωb,N

which produces a subsequence of n(k)b,N (ω), k = 1, 2, ....

It is well known and follows from the subadditive ergodic theorem (see Section2.1 in [15], [2], [18] and [19]) that P -a.s.,

(4.27) limn→∞

1n

Hµω (βnω) = h(r)

µ (T, α(m)).

By the ergodic theorem we have also that P -a.s.,

(4.28) limn→∞

1n

∫φω

ndµω = limn→∞

1n

n−1∑k=0

∫φθkωdµθkω =

∫φdµ.


Hence, dividing the left and the right hand sides of (4.18) by n(j)b,N,m(ω), letting

j → ∞ and taking into account (4.18)–(4.21), (4.25), (4.27) and (4.28) we obtainthat

(4.29) h(r)µ (T, α(m)) +

∫φdµ ≤ ΠG(φ) + 2C

√εm + ε.

Letting, first, m →∞ and then ε → 0 we obtain (4.16) by (4.17), (4.26) and (4.29)completing the proof of Theorem 4.2.

5. Relative Ruelle-Perron-Frobenius theorem

Let (X, T ) be a topologically mixing countable random Markov shift and φ be alocally fiber Holder continuous function. We say that φ is positive recurrent if thereexists a positive random variable λ : Ω → R with the following properties. Thereexist a ∈ S, a measurable set Ωr

a ⊂ Ωa of positive measure and positive measurablefunctions Ma : Ωr

a → R , Na : Ωra → N such that for all ω ∈ Ω′ and n ≥ Na(ω) with

ω ∈ Ωra ∩ θnΩr

a,

(5.1) M−1a (ω) ≤ Zθ−nω

n (φ, a)λ(θ−1ω) · · ·λ(θ−nω)

≤ Ma(ω).

As shown in the following proposition, this definition does not depend on the choiceof a ∈ S, that is for each b ∈ S there exists a subset of Ωb such that the esti-mate (5.1) holds with respect to measurable functions Mb,Nb. Observe that thisthen implies that the notion of positive recurrence also is independent of a fur-ther subdivision of the cylinders with respect to base. For ease of notation, setΛn(ω) := λ(ω)λ(θω) · · ·λ(θn−1ω) for n ∈ N and ω ∈ Ω.

5.1. Proposition. Let φ be positive recurrent and b ∈ S. Then there exists aset Ωr

b ⊂ Ωb of positive measure, such that for each c ∈ S there exist positivemeasurable functions Mbc : Ωc → R and Nbc : Ωc → R such that for a.e. ω ∈ Ωc

and n > Nbc(ω) with ω ∈ Ωc ∩ θn(Ωrb),

(5.2) M−1bc (ω) ≤ Zθ−nω

n (φ, b, c)/Λn(θ−nω) ≤ Mbc(ω).

Proof. Let a ∈ S be the cylinder given by the definition of positive recurrence.Then there exist B > 0 such that Ω′a := ω ∈ Ωr

a : Bω1 ≤ B is of positive measure.

Set τba(ω) := minn ∈ N : n ≥ Nba(ω), τnω ∈ Ω′a, and choose B,Z > 0 andN ∈ N such that

Ω′ba := ω ∈ Ωb : τba(ω) ≤ N,Zωτba(ω)(φ, b, a)/Λτba(ω)(ω) ≥ Z,Bω

1 ≤ B

is of positive measure. Now fix ω ∈ Ωc, and choose l ∈ N with θ−lω ∈ Ω′a andZθ−lω(φ, a, c) > 0. For each n ≥ l + Na(θ−lω) + N with θ−nω ∈ Ω′ba, set k =τba(θ−nω). A straightforward adaption of the proof of Lemma 3.1 then gives

Zθ−nωk (φ, b, a)Zθ−n+kω

n−k−l (φ, a, a)Zθ−lωl (φ, a, c) ≤ BBθ−lω

1 Bω1 Zθ−nω

n (φ, b, c).

In particular, we obtain the following lower bound.

Zθ−nωn (φ, b, c)/Λn(θ−nω) ≥ ZMa(θ−lω)Zθ−lω

l (φ, a, c)BBθ−lω

1 Bω1 Λl(θ−lω)

:= (M ′bc(ω))−1.


Hence the left inequality of (5.2) holds with respect to Ωrb := Ω′ba, M ′

bc(ω) andNbcω := l + Na(θ−lω) + N . In order to show the remaining inequality, first notethat by the above arguments, there exists Z > 0 such that the set

Ωa := ω ∈ Ω′a : Zθ−nωn (φ, a, b)/Λn(θ−nω) > Z

has positive measure. As above, for ω ∈ Ωc, n ∈ N with θ−nω ∈ Ωrb and n ≥ Nbc(ω),

we obtain by Lemma 3.1 that

Zθ−k−nωk (φ, a, b)Zθ−nω

n (φ, b, c)Zωl (φ, c, a) ≤ BBω

1 Bθlω1 Zθ−k−nω

k+n+l (φ, a, a),

where l ∈ N is chosen such that T lω([c]ω) ⊃ [a]θlω, and k ∈ N such that θ−k−nω ∈

Ωa. Hence

Zθ−nωn (φ, b, c) ≤ BBω

1 Bθlω1 Ma(θlω)Λl(ω)

ZZωl (φ, c, a)

:= M ′′bc(ω),

and the assertion follows with respect to Mbc := maxM ′bc,M

′′bc.

Note that by the above result, for b, c ∈ S, the set Ωrb does not depend on c ∈ S.

Moreover, for ω ∈ Ωc, x ∈ [c]ω and n ≥ Nbc(ω) with θ−nω ∈ Ωrb , observe that

(5.3) (Bω1 )−1

(Lθ−nω,n

φ I[b]θ−nω

)(x) ≤ Zθ−nω

n (φ, b, c) ≤ Bω1

(Lθ−nω,n

φ I[b]θ−nω

)(x).

Hence by Proposition 5.1, for M(ω) := Bω1 Mbc(ω),

(5.4) (Mbc(ω))−1

(ω) ≤(Lθ−nω,n

φ I[b]θ−nω

)(x)/Λn(θ−nω) ≤ Mbc(ω).

In particular, if b = c, then (5.1) is equivalent to the existence of a positive randomvariable M and a set Ω′ ⊂ Ωb of positive measure, such that (5.4) holds for allx ∈ [b]ω, ω ∈ Ω′ and n sufficiently large with θ−nω ∈ Ω′. If ΠG(φ) is well defined,we immediately obtain the following.

5.2. Proposition. Let (X, T ) be a topologically mixing countable random Markovshift with ergodic base transformation and φ be a positive recurrent, locally fiberHolder continuous function such that the conditions of Theorem 3.2 hold true andΠG(φ) < ∞. Then ∫

log λ(ω)dP (ω) = ΠG(φ).

Proof. From (5.1) we obtain, for ω ∈ Ωra and n ≥ Na(θnω) with θnω ∈ Ωa that

M−1a (θnω) ≤ Zω

n (φ, a)/Λn(ω) ≤ Ma(θnω).

As in the proof of Proposition 5.1 we can choose a subsequence of n(j)a,N (ω) of arrivals

to Ma ≤ M∩Na ≤ N for big enough M and N so that P (Ma ≤ M∩Na ≤N ∩ Ωa,N ) > 0. Then taking log and dividing by n we obtain the assertion byTheorem 3.2 and the ergodic theorem.

Furthermore, it turns out that positive recurrence is a sufficient condition for arelative version of the Ruelle-Perron-Frobenius Theorem.

5.3. Theorem. Let (X, T ) be a topologically mixing countable random Markov shiftwith ergodic base transformation and φ a positively recurrent, locally fiber Holdercontinuous function. Then there exist unique (up to scalar multiplication) measur-able families νω : ω ∈ Ω, hω : ω ∈ Ω with the following properties.


(1) For a.e. ω ∈ Ω, νω is a σ-finite measure on Xω, the functions hω, log hω

are locally fiber Holder continuous, and

(Lωφ)∗νθω = λ(ω)νω, Lω

φhω = λ(ω)hθω, νω(hω) = 1.

(2) If fω : ω ∈ Ω′ is a family with fω ∈ L1(νω) for a.e. ω ∈ Ω, then

limn→∞

∥∥∥∥∥ Lω,nφ fω

λ(ω) · · ·λ(θn−1ω)− hθnω

∫fωdνω

∥∥∥∥∥L1(νθnω)

= 0.

Proof. The proof of Theorem 5.3 will proceed in several steps. First assume that φis positively recurrent. In the next section, we prove using a tightness argument thatthe family νω exists (Theorem 6.1). In Section 7 we then construct hω as thelimes inferior of the iterates of Lω

φ (Proposition 7.3) and prove the convergence in(2) by showing that the system is relatively exact (Proposition 7.4). The uniquenessof νω is then shown using well known arguments from the theory of conformalmeasures (Theorem 7.5). In particular, as a consequence of the convergence in(2), it follows that hω is the unique locally fiber Holder continuous function withLω

φhω = λ(ω)hθω and νω(hω) = 1.

6. Random conformal measures

We now proceed with the construction of a random Gibbs measure for a positivelyrecurrent. Namely, as a consequence of Propositions 6.3 and 6.7 below, we obtainthe following result.

6.1. Theorem. For a topologically mixing countable Markov shift with ergodic basetransformation, and φ a positively recurrent, locally fiber Holder continuous poten-tial, there exists a σ-finite random measure νω : ω ∈ ΩM, which is finite oncylinders, such that, for a.e. ω ∈ Ω,

(6.1) (Lωφ)∗(νθω

) = (λ(ω))νω.

Note that by well known arguments, for a random measure mω : ω ∈ Ω, theproperty in (6.1) is equivalent to

mθω(Tω(Aω)) = λ(ω)∫

IAωe−φω

dνω

for a.e. ω ∈ Ω and any Aω ∈ Bω. In this situation, we will refer to mω : ω ∈ Ωas a random conformal measure.

6.1. Construction principle. For the construction of the random measure, wewill follow ideas in [26] and [9]. For a cylinder a ∈ N, ω ∈ Ω with a ≤ l(ω) andn ∈ N, let

Eωn := c : c = (c0, c1, ..., cn−1), c0 = a, [ca]ω ∈ γn

ω (n ∈ N, ω ∈ Ω),

and note that Zωn (φ, a, a) =

∑c∈Eω

neφω

n [c]. For Ω′ ⊂ Ωa and ω ∈ Ωa, let Jωa (Ω′) be

the subset of N given by

(6.2) Jωa (Ω′) := n ∈ N : θnω ∈ Ω′.

By the mixing property, it follows that Zωn (φ, a, a) > 0 for all n > Na,a(ω), n ∈

Jωa (Ωa). For a measurable family ξ = (ω, ξω) ∈ [a]ω : ω ∈ Ωa and each ω ∈ Ωa,


n ∈ Jωa (Ωa), set

Eωn (ξ) := (cξθnω) : c ∈ Eω

n, and

Zωn :=

∑x∈Eω

n (ξ)

eφωn(x)/Λn(ω) = Lω,n

φ−log λ(I[a]ω )(ξθnω).

In order to construct the random measure for a positively recurrent potential φ, wefirst restrict our considerations to the measurable set

(6.3) ΩB,M ⊂ ω ∈ Ωra : Bω

1 < B, Ma(ω) < M,where ΩB,M and B,M > 1 are chosen such that P (ΩB,M ) > 0. Hence, for a.e.ω ∈ ΩB,M and all n ∈ Jω

a (ΩB,M ), we obtain by (5.3)

(6.4) Zωn = [(BM)−1, BM ].

The construction principle will make use of the random power series given by

PωΩB,M

(x) :=∑

n∈Jωa (ΩB,M )

xnZωn , and PΩB,M

(x) :=∫

ΩB,M

PωΩB,M

(x)dP,

for ω ∈ ΩB,M . We now collect several immediate implications of positive recurrence.

6.2. Lemma. For a positively recurrent, topologically mixing countable Markov shiftthe following holds for a.e. ω ∈ ΩB,M .

(1) For x > 0, x ∈ R, we have

PωΩB,M

(x)

= ∞ : x ≥ 1< ∞ : x < 1.

(2) For all s ∈ (0, 1),

PωΩB,M

(s)/PΩB,M(s) ∈ [(BM)−2/P (ΩB,M ), (BM)2/P (ΩB,M )]

PθkωΩB,M

(s)/PωΩB,M

(s) ∈ [(BM)−2, (BM)2]

(3) There exists ρ ∈ L∞(P ), and an increasing sequence (sn : n ∈ N), sn ∈(0, 1) for all k ∈ N with limn sn = 1 such that for all f ∈ L1(P ),

limn→∞

∫fPω

ΩB,M(sn)/PΩB,M

(sn)dP =∫

fρdP.

Proof. Combining the conservativity of θ and (6.4), that for a.e. ω ∈ ΩB,M ,

limn→∞,n∈Jω

a (ΩB,M )(Zω

n )1/n = 1.

Hence the radii of convergence of PωΩB,M

and PΩB,Mare equal to one, and Pω

ΩB,M(1),

PΩB,M(1) = ∞ might be deduced from

PωΩB,M

(x) =∑n∈N

IΩ′ θn(ω)xnZωn .

In order to show the existence of (sn), note that the family of functions fx : x ∈(0, 1) given by

fx : ΩB,M → R, ω 7→ PωΩB,M

(x)/PΩB,M(x),

is uniformly bounded, namely fx(ω) ∈ [(BM)−2/P (ΩB,M ), (BM)2/P (ΩB,M )] a.e.Hence fx is (L∞(P ))∗ sequentially compact, from which the existence of (sn) asin assertion (3) follows.


We now construct the advertised random Gibbs measure as the weak limit of thefollowing family of random probability measures using Crauel’s relative Prohorovtheorem. Let

LC1 (P ) = f : X → R : f |Xω

∈ C(Xω),∫‖f|Xω

‖∞dP (ω) < ∞.

We say (see [5]) that a family µιω : ω ∈ Ω : ι ∈ J of random measures

µιω : ω ∈ Ω is relatively tight if for all ε > 0 there is a set K ⊂ Y such

that K ∩ Yω is compact and such that∫

µιω(K)dP (ω) ≥ 1 − ε. It then follows

that a relatively tight family is sequentially compact by Crauel’s random Prohorovtheorem. The convergence of a sequence (µn

ω) towards µω means that for allf ∈ LC

1 (P ),

limn→∞

∫fdµn

ωdP (ω) =∫

fdµωdP (ω).

The family of random measures considered here is defined as follows. For a cylindera ∈ N, s < 1 and B,M > 1 with P (ΩB,M ) > 0, let

ms,ω :=1

PωΩB,M

(x)

∑n∈Jω

a (ΩB,M )

∑y∈Eω

n (ξ)

eφωn(y)(Λn(ω))−1snδy,

where δy denotes the unit mass at the point y ∈ Xω, and note that for every s < 1the measures ms,ω and the measure given by dms = dms,ωdP (ω) are probabilitymeasures. With (sn) referring to the sequence given by Lemma 6.2, we obtain thefollowing.

6.3. Proposition. Let (X, T ) be a topologically mixing countable Markov shift withergodic base transformation, and φ be a positive recurrent, locally fiber Holder con-tinuous potential. If the family

msn,ω : ω ∈ ΩB,M : n ∈ N

is relatively tight, then there exists a random probability measure νω : ω ∈ ΩB,Mwith νω([a]ω) = 1 for a.e. ω ∈ ΩB,M and which satisfies the following conformal-ity property. For a.e. ω ∈ ΩB,M , k ∈ Jω

a (ΩB,M ), c = (c0, . . . ck−1) ∈ γkω with

T k([c]ω) ⊃ [a]θkω, and each bounded, continuous function f : [(ca)]ω → R,

Λk(ω)∫

[(ca)]ω

fdνω =ρ(θkω)ρ(ω)

∫[a]

θkω

eφk(cx)f(cx)dνθkω(x).

Proof. Choose A ∈ Bω, A ⊂ [a]ω and k ∈ Jωa (ΩB,M ) such that T k : A → T k(A) is

invertible, and T k(A) ⊂ [a]θkω. By injectivity it follows that, for s < 1,

ms,θk(ω)(Tk(A)) =

1Pθkω

ΩB,M(s)

∑n∈Jθkω

a (ΩB,M )

∑y∈T k(A)∩Eθkω

n (ξ)

exp(φθkωn (y))

Λn(θkω)sn

=1

PθkωΩB,M

(s)

∑n∈Jθkω

a (ΩB,M )

∑y∈A∩T−k(Eθkω

n (ξ))

exp(φθkωn (T ky))

Λn(θkω)sn

=1

PθkωΩB,M

(s)

∑n∈Jθkω

a (ΩB,M )

∑y∈A∩Eω

n+k(ξ)

exp(φωn+k(y))

Λn+k(ω)sn+k ·

(exp(φω

k (y))Λk(ω)

sk

)−1


In particular this gives∣∣∣∣∣PθkωΩB,M

(s)

PωΩB,M

(s)ms,θk(ω)(Tk(A))−

∫A

(exp(φω

k (y))Λk(ω) sk

)−1

dms,ω

∣∣∣∣∣(6.5)

≤ 1Pω

ΩB,M(s)

∑n∈Jθkω

a (ΩB,M ),

n≤k

∑y∈Eω

k (ξ)

exp(φωn(y))

Λn(ω)

(exp(φω

k (y))Λk(ω)

)−1

sn−k.

Using (1) of Lemma 6.2, it follows that (6.5) tends to 0 as s tends to 1 from below.By the tightness assumption, there exists a subsequence (snk

) of (sn), a randomprobability measure ν = νω : ω ∈ ΩB,M and a set Ω0 ∈ F , Ω0 ⊂ ΩB,M withP (Ω0) = P (ΩB,M ) which is invariant under the first return map of θ to ΩB,M suchthat for ω ∈ Ω0, the Borel measure νω is the weak limit of (msnl

,ω : l ∈ N).Hence for k ∈ N and a measurable family Aω : ω ∈ ΩB,M such that eitherT kAω → [a]θkω is injective and θkω ∈ ΩB,M or Aω = ∅, the following holds.∫ ∣∣∣∣ρ(θkω)

ρ(ω)νθkω(T kAω)− Λk(ω)

∫A

e−Φωk dνω

∣∣∣∣P (dω)

≤∫ ∣∣∣∣∣ρ(θkω)

ρ(ω)νθkω(T kAω)−

PθkωΩB,M

(snl)

PωΩB,M

(snl)νθkω(T kAω)

∣∣∣∣∣P (dω)

+∫ ∣∣∣∣∣Pθkω

ΩB,M(snl

)

PωΩB,M

(snl)νθkω(T kAω)−

PθkωΩB,M

(snl)

PωΩB,M

(snl)msnl

,θkω(T kAω)

∣∣∣∣∣P (dω)

+∫ ∣∣∣∣∣Pθkω

ΩB,M(snl

)

PωΩB,M

(snl)msnl

,θkω(T kAω)− Λk(ω)∫

A

e−Φωk dmsnl

,ω

∣∣∣∣∣P (dω)

+∫ ∣∣∣∣Λk(ω)

∫A

e−Φωk dmsnl

,ω − Λk(ω)∫

A

e−Φωk dνω

∣∣∣∣P (dω).

Observe that for l →∞, the first summand tends to zero since ρ is a weak-* limit,the second and the forth by weak convergence of msl,ω, and the third by (6.5).Hence

ρ(θkω)ρ(ω)

νθkω(T k(A)) = Λk(ω)∫

A

e−φωk dνω

for a.e. ω ∈ ΩB,M . If in addition, A is a cylinder set, then there exists c, [c]ω ∈ γkω

with A ⊂ [c]ω. It then follows that

(6.6)ρ(θkω)ρ(ω)

∫[a]

θkω

IA(cx)dνθkω(x) = Λk(ω)∫

[ca]ω

IAe−φωk dνω.

Note that by the Markov structure, the functions 1A and 1T k(A) are continuous.As it is well known, the equality in (6.6) can be extended to the set of boundedcontinuous functions, namely for each bounded continuous function g : [c]ω → Rwe have

ρ(θkω)ρ(ω)

∫[a]

θkω

g(cx)dνθkω(x) = Λk(ω)∫

[ca]ω

ge−φωk dνω.

The assertion now follows with g = f · eφωk .


We remark that the above result gives together with the assumption of relativetightness, that there exists a random Gibbs measure for the first return map to(ω, x) : x ∈ [a]ω, ω ∈ ΩB,M. This random measure will now be used to constructa globally defined random Gibbs measure, where we adapt the method in the proofof the Main Theorem in [10].

6.4. Proposition. Let (X, T ) be a topologically mixing countable Markov shift withergodic base transformation, and φ be a positive recurrent, locally fiber Holder con-tinuous potential. Furthermore, assume that there exists a cylinder [a], a ∈ N, andB,M > 1 such that the family of measures

msn,ω : ω ∈ ΩB,M : n ∈ Nis relatively tight. Then there exists a σ-finite random measure νω : ω ∈ ΩM,which is finite on cylinders, such that for a.e. ω ∈ Ω,

(Lωφ)∗(νθω

) = λ(ω)νω

Moreover, for a continuous and bounded function fω : Xω → R which is supportedon finitely many cylinders, there exists ω′ ∈ ΩB,M with

(6.7)∫

fωdνω = liml→∞

ρ(ω′)Pω′

ΩB,M(snl

)

∑m∈Jω

a (ΩB,M )

smnl

Lω,mφ−log λ(fω)(ξθmω),

where (snl) is the subsequence for which msnl

,ω converges.

Proof. Let νaω : ω ∈ ΩB,M refer to the measure given by Proposition 6.3, and

for ω ∈ ΩB,M , let νaω := ρ(ω)νa

ω where ρ is given by Lemma 6.2. By conformalityof νa

ω, we have for a.e. ω ∈ ΩB,M , k ∈ Jωa (ΩB,M ), c = (c0, . . . ck−1) ∈ γk−1

ω withT k([c]ω) ⊃ [a]θkω, and a bounded, continuous function f : [a]θkω → R, that

(6.8)∫

[a]θkω

fdνaθkω = Λk(ω)

∫[(ca)]ω

e−φωk f T kdνa

ω.

We will now use this property as the defining relation for the construction of therandom measure νω : ω ∈ Ω. Fix some cylinder b ∈ N and ω ∈ Ω. Thenby the Markov structure and the mixing property of T , there exists k ∈ N andc = (c0, . . . ck−1) such that θ−kω ∈ ΩB,M , [cb]θ−kω ⊂ [a]θ−kω and [cb]θ−kω ∈ γk

θ−kω.The measure νω restricted to [b]ω is now determined by

(6.9)∫

[b]ω

gdνω = Λk(θ−kω)∫

[(cb)]θ−kω

e−φθ−kωk g T kdνa

θ−kω,

for each bounded, continuous function g : [b]ω → R. We hence obtain a randommeasure νω which is finite on cylinders, and for which the restrictions to [a]ω,ω ∈ ΩB,M coincide with νa

ω : ω ∈ ΩB,M by (6.8). Moreover, for a cylinder b =(b0, . . . bl−1) with [ba]ω ∈ γl

ω, [ba]ω ⊂ [b]ω, θlω ∈ ΩB,M and a bounded, continuousfunction g : [ba]ω → R it follows by definition of νω and (6.8) that

Λl(ω)∫

[ba]ω

gdνω = Λl(ω)Λk(θ−kω)∫

[(cba)]θ−kω

e−φθ−kωk g T kdνa

θ−kω

=∫

[a]θlω

eφωl (bx)g(bx)dνa

θlω(x).

We hence have that the random measure νω is also conformal in the sense of(6.8), and in particular, νω is well defined, i.e. the definition by (6.9) does not


depend on the choice of c. By applying the above identity to the cylinder T ([cb]ω),we obtain that

Λl(ω)∫

[ba]ω

gdνω =∫

[a]θlω

eφωl (bx)g(bx)dνa

θlω(x)

= Λl−1(θω)∫

T ([cb])

eφω

(bx)g(bx)dνθω(x).

We hence have for a.e. ω ∈ Ω and each bounded continuous function f : Xω → Rsupported on finitely many cylinders, that∫

Lωφ(f)dνθω =

∫ ∑c∈γω

1

IT ([c]ω)eφ(cx)f(cx)dνθω(x)

= λ(ω)∑c∈γω

1

∫[c]ω

fdνω = λ(ω)∫

fdνω.

In order to obtain the the second assertion, consider b ∈ S, ω ∈ ΩB,M , k ∈ N,[cb]ω ∈ γk

ω and a bounded continuous function g : Xθkω → R which is equal to 0 forx /∈ [b]θkω. We then have∫

gdνω = liml→∞

ρ(ω)Pω

ΩB,M(snl

)

∑m∈Jω

a (ΩB,M )

smnl

∑y∈Eω

k (ξ)

exp(φωm(y))

Λm(ω)Λk(ω)

exp(φωk (y))g T k(y)

= liml→∞

ρ(ω)Pω

ΩB,M(snl

)

∑m∈Jθkω

a (ΩB,M )

smnl

∑y∈[b]

θkω,T m(y)=ξ

θm+kω)

exp(φθkωm (y))

Λm(θkω)g(y)

= liml→∞

ρ(ω)Pω

ΩB,M(snl

)

∑m∈Jθkω

a (ΩB,M )

Lθkω,mφ−log λ(g)(ξθm+kω)).

For ω ∈ Ω, assume that X ′ω is the union of finitely many cylinders of length 1.

Then by topological mixing of (X, T ) and ergodicity of θ, there exists n ∈ N withθ−nω ∈ ΩB,M and Tn

θ−nω([a]θ−nω) ⊃ X ′ω. Now choose for each b ∈ S with [b]ω ⊂ X ′

ω

an element cb ∈ γnθ−nω with [cb]θ−nω ⊂ [a]θ−nω and Tn

θ−nω([cb]θ−nω) = [b]ω. Thisproves the assertion.

6.2. Relative tightness. In order to apply Proposition 6.3 it is left to show that,for a given cylinder [a], a ∈ N and suitable B,M > 1 and ξ = ξω : ξω ∈ [a]ω, ω ∈ΩB,M, the family msn,ω is relatively tight. In order to proceed we introduce thep-th return time ηp

A : X → N ∪ ∞, that is for A ⊂ X and p ∈ N,

ηpA(y) :=

minn ∈ N : SnIA(Ty) = p: S∞IA(Ty) ≥ p∞ : S∞IA(Ty) < p

where Snf :=∑n−1

k=0 1AT k for n ∈ N∪∞. Furthermore, for a measurable subsetΩ′ ⊂ Ω, and l, p ∈ N, l ≥ p, and Xa(Ω′) := (ω, x) : ω ∈ Ω′, x ∈ [a]ω, set

Eω,nξ,Ω′(l, p) = Eω,n(l, p) := x ∈ Eω

n (ξ) : ηpXa(Ω′)(ω, x) = l, and

Zω,pn,l :=

∑x∈Eω,n

ξ,Ω′ (l,p)

eφn(x)/Λn(ω).

6.5. Lemma (Decomposition Lemma). Let (X, T ) be a topologically mixing count-able Markov shift with ergodic base transformation, and φ be a locally fiber Holder


continuous potential. For p ∈ N and Ω′ ∈ F , Ω′ ⊂ Ωa, there exists a measurablefamily ξ such that

(1) Eωn (ξ) =

⋃l≥pE

ω,nξ,Ω′(l, p), where ∪ denotes the disjoint union.

(2) Zω,pn,l = (Bθlω

1 )±1Zω,pl,l Zθlω

n−l.

Proof. Since (X, T ) is topologically mixing and θ is conservative, there exists ameasurable family ξ with ηp

Xa(Ω′)(ξω) < ∞ for all p ∈ N and a.e. ω ∈ Ω′. Thisproves the first assertion. For the proof of the second assertion, note that, forl = p, · · ·n, there is a bijection

Eω,nξ,Ω′(l, p) → Eω,l

ξ,Ω′(l, p)× Eθlωn−l(ξ)

defined as follows. For y ∈ Eln(ω, p) there are unique elements cy ∈ γl

ω, by ∈ γn−lθlω

,such that y = cybyξθnω, and byξθnω ∈ [a]θlω. In particular,

y 7→ (i1(y), i2(y)) := (cyξθlω, byξθnω)

defines an injective map. Since the inverse map can be constructed by concatenationof the corresponding inverse branches, it follows that the map is bijective. Nextobserve that by Lemma 2.1 we have that

exp(φωl (y))

exp(φωl (i1(y)))

= (Bθlω1 )±1, for y ∈ Eω,n

ξ,Ω′ .

It follows that

Zω,pn,l =

∑y∈Eω,n(l,p)

exp(φωl (y))

Λl(ω)exp(φθlω

n−l)(Tl(y))

Λn−l(θl(ω))

= (Bθlω1 )±1

∑y∈Eω,l(l,p)

exp(φωl (y))

Λl(ω)

∑z∈Eθlω

n−l(ξ)

exp(φθlωn−l(z))

Λn−l(θlω)

= (Bθlω1 )±1Zω,p

n,l Zθlωn−l.

6.6. Corollary. Assume that φ is positively recurrent, and that for Ω′ in Lemma6.5, there exists B,M > 1 such that supω∈Ω′ B

ω1 < B, and supω∈Ω′ Ma(ω) < M .

Then there exists a measurable family ξ such that for almost every ω ∈ Ω′ we have∞∑

l=p+1

Zll (ω, p) ≤ M2B.

Proof. Fix p ∈ N and note that Eω,l(l, p) = ∅ unless θl(ω) ∈ Ω′ (by definition ofthe stopping times). Hence for any n > p

Zωn =

n∑l=p

Zω,pn,l +

∞∑l=n+1

Zω,pn,l

= B±1n∑

l=p

Zω,pl,l Z

θlωn−l +

∞∑l=n+1

Zω,pn,l


By supω∈Ω′ Ma(ω) < M , it follows that

M±1 = B±1M±1n∑

l=p

Zω,pl,l +

∞∑l=n+1

Zω,pn,l .

Since the additional term on the right hand side is positive,n∑

l=p+1

Zω,pl,l ≤ M2B for all n ∈ N.

It is left to present the argument for relative tightness. Choose a measurablefamily ξω : ω ∈ ΩB,M according to Lemma 6.5, and choose εp > 0 (p ∈ N) suchthat

∑∞p=1 εp = ε. In order to obtain a set K with compact fibers and satisfying,

for s < 1, ∫ΩB,M

ms,ω(K)P (dω) ≥ (1− 2ε)(1− 3ε)P (ΩB,M ),

we first construct a suitable subset Ω of ΩB,M as follows.

(1) By the recurrence property of ξω, there exists N ∈ N such that forΩN := ω ∈ ΩB,M : η1

Xa(ΩB,M )(ξω) < N,

P (ΩN ) ≥ (1− ε)P (ΩB,M ).

(2) By Corollary 6.6, for each p ∈ N and ω ∈ ΩB,M there exists νp(ω) > Nsuch that

∞∑l=νp(ω)

Zω,pl,l ≤ εpB

−1M−2.

It then follows by induction (setting Ω(1)1 = ΩB,M ) that there exists νp ∈ N

and measurable sets Ω(p)1 ⊂ Ω(p−1)

1 ⊂ ΩB,M such that supω∈Ω(p) νp(ω) ≤ νp

and

P (Ω(p)1 ) ≥

(1−

p∑k=0

εk

)P (ΩB,M ).

For Ω1 :=⋂

p≥1 Ω(p)1 , we hence have that

P (Ω1 ∩ ΩN ) ≥ (1− 2ε) P (ΩB,M ).

(3) Set N1 = 1 and Ω(1)2 = ΩB,M . By induction one can find for each p ∈ N,

n ≥ 1, and ω ∈ Ω(p)2 integers Np(ω, n) > Np−1 with∑

x∈Eωn :∃k≤νp,

xk≥Np(ω,n)

eφωn(x)

Λn(ω)≤ εpM

−1.

Choose Np > Np−1 and Ω(p)2 ⊂ Ω(p)

2 ⊂ ΩB,M such that

P (Ω(p)2 ) ≥

(1−

p∑k=0

εk

)P (ΩB,M )


and for ω ∈ Ω(p)2 and n ∈ N, Np(ω, n) ≤ Np. For Ω2 :=

⋂∞p=0 Ω(p)

2 , wehence have that

P (Ω2 ∩ Ω1 ∩ ΩN ) ≥ (1− 3ε) P (ΩB,M ).

We are now in position to define the following set K ⊂ X for which it will turn outthat the fibers are compact. For Ω := Ω2 ∩ Ω1 ∩ ΩN , let

K :=

((xk), ω) ∈ X : ω ∈ Ω, xk ≤ Np ∀ k ≤ νp, p ∈ N

∩∞⋂

p=1

((xk), ω) ∈ X : ω ∈ Ω, T l

ω((xk)) ∈ [a]θlω, θlω ∈ ΩB,M

for at least p indices l ≤ νp

.

6.7. Proposition. The random measure ms,ω : s > 0 is P -relatively tight.

Proof. Let ε > 0 be given. We first show that K has compact fibers. For ω ∈ Ω,note that ((xk), ω) ∈ K ∩Xω if and only if xνp−1+1, ..., xνp

are bounded by Np andat least p coordinates up to νp are equal to a. Thus

Xω ∩K =((xk), ω) ∈ Xω : xk ≤ Qk, k ∈ N∩ ((xk), ω) ∈ Yω : xj = a at least p times, j ≤ νp, p ∈ N

is a compact set as an intersection of a compact and a closed set, where Qk = Np

for νp−1 < k ≤ νp. For fixed n ∈ N, let x = ((xk), ω) ∈ Eωn ∩Kc with ω ∈ Ω. Then

one of the two possibilities may occur

(i) ηpXa(ΩB,M )(y) > νp for some p ≥ 1 or

(ii) for some p ≥ 1 there exists k ∈ N such that xk ≥ Np.

Let F(j)n ⊂ Eω

n denote the set of points with property (j) for j = i or j = ii. Thenby Lemma 6.5

∑x∈F

(i)n

eφωn(x)

Λn(ω)≤

∞∑p=1

∑l≥νp

Zω,pn,l ≤

∞∑p=1

B∑l≥νp

Zω,pl,l Z

θlωn−l

≤∞∑

p=1

BM∑l≥νp

Zω,pl,l ≤ BM

BM2

∑p≥1

εp

≤ εM−1 ≤ εZωn .

Likewise we can estimate

∑x∈F

(ii)n

eφωn(x)

Λn(ω)≤

∞∑p=1

∑x∈Eω

n :∃k≤νp,

xk≥Np

eφωn(x)

Λn(ω)≤

∞∑p=1

εpM−1 ≤ εZn(ω).


If follows from these two estimates that for ω ∈ Ω,

ms,ω(Kc) =1

PωΩB,M

(s)

∞∑n=1

∑x∈Eω

n∩Kc

eφωn(x)

Λn(ω)sn

≤ 1Pω

ΩB,M(s)

∞∑n=1

∑x∈F n

(i)

eφωn(x)

Λn(ω)+∑

y∈F n(ii)

eφωn(x)

Λn(ω)

sn

≤ 2ε

PωΩB,M

(s)

∞∑n=1

snZωn = 2ε.

Integrating over Ω then gives that∫

ms,ω(K)dP (ω) ≥ (1− 2ε)(1− 3ε).

7. Relative exactness, convergence and uniqueness

We proceed with the construction of a family of eigenfunctions of the relativeRuelle operator, which is obtained as the limit of the iterates of the above relativeoperator applied to an arbitrary positive, relatively Holder function. The key ar-gument here is to show that the random system is relatively exact as introducedin [12] with respect to the relatively conformal random measure given by Theorem6.1. Recall that, adapted to our situation, the random Markov shift is relativelyexact with respect to the relatively nonsingular random measure mω wheneverfor a.e. ω ∈ Ω, the relative terminal σ-algebra

Fω :=∞⋂

n=0

T−nω (Bθnω)

is trivial, that is for each A ∈ Fω either mω(A) = 0 or mω(Xω \ A) = 0. Further-more, we refer to the family Tω : ω ∈ Ω as the random transfer operator withrespect to m, if for a.e. ω ∈ Ω, the operator Tω : L1(mω) → L1(θω) is defined by

(7.1)∫

Tω(fω) · gθωdmθω =∫

fω · gθω Tωdmω

for all fω ∈ L1(mω), gθω ∈ L∞(θω).

7.1. Proposition. Let (X, T ) be a countable random Markov shift, and let m be anonsingular, σ-finite random measure on X. Then (X, T ) is relatively exact withrespect to m if and only if for a.e. ω ∈ Ω and each f ∈ L1(mω) with

∫fdmω = 0,

(7.2) limn→∞

‖Tnω (f)‖L1(mθnω) = 0.

If in addition, there exists a random function h = hω : ω ∈ Ω with hω ∈ L1(mω),hω > 0 and Tωhω = hθω for a.e. ω ∈ Ω, then (X, T ) is relatively exact if and onlyif for a.e. ω ∈ Ω and each f with f ∈ L1(mω),

limn→∞

∥∥∥Tnω (f)− hθnω

∫fdmω

∥∥∥L1(mθnω)

= 0.

Proof. Assume that T is relatively exact, and let gn := sign(Tnω (f)). We then have

for each n ∈ N that

‖Tnω (f)‖L1(mθnω) =

∫Tn

ω (f)gndmθnω =∫

fgn Tnω dmω.


Note that gn Tnω ∈ L∞(mω), and that gn Tn

ω is T−nω (Bθnω) measurable. Since

T−nω (Bθnω) ⊂

⋂nk=0 T−k

ω (Bθkω), it follows that each weak-∗ limit of (gnTnω : n ∈ N)

is Fω-measurable, and hence either equal to the constant function 1 or -1. By thefact that (g Tn

ω ) is weak-∗ compact, and that∫

fdmω = 0, it follows that the limitin (7.3) is equal to 0.

If T is not relatively exact, then there exist ω ∈ Ω and A ∈ Fω such thatmω(A),mω(Xω \ A) > 0. By definition of Fω there exist n ∈ N and An ∈ Bθnω

with A = T−nω An and Tn

ω A = An. Let f ∈ L1(mω) with∫

fdmω = 0 and f |A > 0.It then follows that Tn

ω (f)|T nω A > 0. Hence

‖Tnω (f)‖L1(mθnω) ≥

∫T n

ω A

Tnω (f)dmθnω =

∫A

fdmω > 0.

In order to prove the second assertion, denote by Tω,m and Tω,µ the random transferoperators with respect to the random measures m and µ, where dµω := hωdm. By(7.1) it follows for all f with f · hω ∈ L1(mω) that

Tnω,m(hωf) = hθnωTn

ω,µ(f).

In particular, this gives that∥∥∥Tnω (f)− hθnω

∫fdmω

∥∥∥L1(mθnω)

=∥∥∥Tn

ω,m

(f − hω

∫fdmω

)∥∥∥L1(mθnω)

=∥∥∥hθnω

(Tn

µ

(f

hω−∫

fdmω

))∥∥∥L1(mθnω)

=∥∥∥Tn

µ

(f

hω−∫

fhω

dµω

)∥∥∥L1(µθnω)

.

The assertion then follows by the first part applied to Tω,µ.

Note that for random Markov shifts, the relative transfer operator may be writtenas follows. For n ∈ N and c ∈ γn−1

ω , let

τθnω,c : Tnω ([c]ω) → [c]ω, (θnω, x) 7→ (ω, cx)

refer to the inverse branch associated with c. We then have, for ω ∈ Ω, f ∈ L1(mω)and n ∈ N,

Tnω (f) =

∑c∈γn−1

ω

dmωτθnω,c

dmθnωf τθnω,c.

In particular, if m is equal to the random conformal measure ν = νω associ-ated with a positively recurrent, locally fiber Holder continuous potential given byTheorem 6.1, the relative Ruelle operator Lω

φ−log λ acts as the relative transferoperator. Hence the strategy of proof of the remaining assertions of Theorem 5.3 isthe following. We first construct a random eigenfunction, and then conclude that(X, T ) is conservative and ergodic using a result in [8]. Then, by showing that thesystem is relatively exact, the convergence result of Theorem 5.3 is deduced fromthe second part of the above proposition. Now let a and ΩB,M defined as in Section6.1.

7.2. Lemma. For P -a.a. ω, the functions given by

fωk := Lθ−kω,k

φ−log λ(I[a]θ−kω

)

are uniformly locally Holder continuous, and uniformly bounded from above andbelow on cylinders for k sufficiently large with θ−k(ω) ∈ ΩB,M .


Proof. Let a, b ∈ S and x ∈ [b]ω. Then by Proposition 5.1 and (5.4),

(7.3) (BMab(ω))−1 ≤ fωk (x) ≤ BMab(ω)

for all k > Nab(ω) with ω ∈ Ωb ∩ θkΩB,M . For x, y ∈ [b]ω, we have∣∣(Lθ−kω,kφ I[a]

θ−kω

)(x)−

(Lθ−kω,k

φ I[a]θ−kω

)(y)∣∣(7.4)

≤∑

r∈Rk

θ−kω(b)

∣∣eφθ−kωk (rx)I[a]

θ−kω(rx)− eφθ−kω

k (ry)I[a]θ−kω

(ry)∣∣

≤∑

r∈Rk

θ−kω(b),r0=a eφθ−kω

k (ry)∣∣1− exp

(φθ−kω

k (rx)− φθ−kωk (ry)

)∣∣where Rk

θ−kω(b) is the set of r = (r0, r1, ..., rk−1) such that [ra] ∈ γkθ−kω. By local

fiber Holder continuity and the first inequality in (2.2) for each r ∈ Rkθ−kω(b),∣∣φθ−kω

k (rx)− φθ−kωk (ry)

∣∣ ≤ V θ−kωk+ιω(x,y)(φk)(7.5)

≤∑k−1

l=0 V θl−kωk+ιω(x,y)−l(φ) ≤

∑∞j=1 κ(θ−jω)rj+ιω(x,y) ≤ K(ω)dω(x, y)

where K(ω) =∑∞

j=1 κ(θ−jω)rj < ∞ P -a.s. since∫

log κdP < ∞, and so κ(θ−jω)may only grow subexponentially. By (7.3)–(7.5),

|fωk (x)− fω

k (y)| ≤ 2K(ω)dω(x, y)fωk (x) ≤ 2K(ω)BMab(ω)dω(x, y)

provided k > Nab(ω) and dω(x, y) ≤ (K(ω))−1 since |ec − 1| ≤ 2|c| for c ≤ 1.

7.3. Proposition. There exists a strictly positive, locally fiber Holder continuousfunction hω such that for a.e. ω ∈ Ω,

Lωφ(hω) = λ(ω)hθω.

In particular,∫

hωdνωdP < ∞, and (X, T ) is conservative and ergodic with respectto the finite, invariant random measure µ given by dµ = hωdνωdP .

Proof. Let Jωa (ΩB,M ) be defined as in (6.2), and, for fω

k as defined in Lemma7.2,

(7.6) hω := lim infk→∞,θ−kω∈ΩB,M

fωk .

Note that by construction, hω : ω ∈ Ω is measurable, and

(7.7) Lωφ(hω) ≤ λ(ω)hθω.

By Lemma 7.2, hω is relatively locally Holder continuous and bounded fromabove and below on cylinders. In particular, for a.e. ω ∈ ΩB,M and n ∈ N such thatθ−nω ∈ ΩB,M , we obtain by combining (5.1) and (5.3) that fω

k (x) ∈ [(BM)−1, BM ]for all x ∈ [a]ω and k ∈ N with θ−kω ∈ ΩB,M .

Now let X ′ω ⊂ Xω be the union of finitely many cylinders. Hence there exists

ω′ ∈ ΩB,M such that (6.7) is applicable to hωIX′ω. Using (7.7) and (6.4), we obtain


for a.e. ω ∈ ΩB,M that∫hωIX′

ωdνω ≤ lim

l→∞

ρ(ω′)Pω′

ΩB,M (snl)

∑m∈Jω

a (ΩB,M )

smnl

Lω,mφ−log λ(hω)(ξθmω)

≤ liml→∞

ρ(ω′)Pω′

ΩB,M (snl)

∑m∈Jω

a (ΩB,M )

smnl

hθmω(ξθmω)

≤ BM liml→∞

ρ(ω′)Pω′

ΩB,M (snl)

∑m∈Jω

a (ΩB,M )

smnl≤ (BM)3/P (ΩB,M ).

Hence, for ω ∈ Ω and k ∈ Jωa (ΩB,M ), it follows by (7.7) that∫

hωdνω =∫

Lω,kφ−log λhωdνθkω ≤

∫hθkωdνθkω ≤ (BM)3/P (ΩB,M ).

Since θ is conservative and ergodic, the map g given by g : Ω → R, ω 7→∫

hωdνω

is in L1(P ). Furthermore, observe that g(ω) ≤ g(θω), whence g(ω) = g(θω) byergodicity of θ, which then gives

Lωφ(hω) = λ(ω)hθω

νθω-almost everywhere. Since νω is positive on cylinders, the support of νθω isdense in Xθω, which gives by continuity of hω that (7.7) holds for all x ∈ Xθω.

In order to show the invariance of µ, first note that for functions gω1 : Xω → R

and gθω2 : Xθω → R, we have

Lφω(gω

1 · gθω2 Tω) = Lφ

ω(gω1 ) · gθω

2 .

Since hω ∈ L1(νω) for a.e. ω ∈ Ω we hence have

(7.8)∫

hωgθω Tωdνω = (λω)−1

∫Lω

φ(hωgθω Tω)dνθω =∫

hθω · gθωdνω

for each gθω with gθωhθω ∈ L1(νθω). Hence T is conservative with respect to µω.Moreover, as a consequence of Holder continuity and Lemma 2.1, it follows that,for each non-empty cylinder [b]ω with a ∈ Sn, ω ∈ Ω,

dνθnω Tnω /dνω(x)

dνθnω Tnω /dνω(y)

=eφn(y)

eφn(x)≤ Bθnω

1 .

In particular, the random Markov fibered system has the bounded distortion prop-erty and hence is ergodic by Theorem 4.4 in [8].

In order to identify the eigenfunctions as the limit of iterates of the Ruelle op-erator, we now prove that the random Markov fibered system is relatively exact,following ideas in [1, Theorem 3.2].

7.4. Proposition. The random Markov fibered system ((X, ν, T ), (Ω, P, θ), π) isrelatively exact.

Proof. Fix an element in the relative terminal σ-algebra of positive measure, that isthere exists a set B ∈ B(Ω) of positive measure, a sequence of measurable familiesAθnω

n : ω ∈ B with Aθnωn ∈ Bθnω such that for the family given by

Aω =∞⋂

n=1

T−nω (Aθnω

n ) (ω ∈ B),


P (ω ∈ B : νω(Aω) > 0) > 0. As in the proof of [1, Theorem 3.2], we obtain that

Tnω (Aω) = Aθnω

n .

Since Aω is non-trivial with respect to dνωdP , there exists b ∈ S with P (ω ∈ B :νω(Aω ∩ [b]ω) > 0) > 0. For C > 0, let

Ω0 := ω ∈ B : νω(Aω ∩ [b]ω) > 0, Cω ≤ C,and choose C such that P (Ω0) > 0. With ηk referring to the k-th return time to[b]Ω0 := (ω, x) : ω ∈ Ω0, x ∈ [b]ω, define

T ∗ : X → [b]Ω0 , (ω, x) 7→ (θη1(ω,x)ω, T η1(ω,x)ω (x)),

and note that T ∗ is defined a.e. since T is conservative and ergodic (see Proposition7.3). For x ∈ Xω, ω ∈ Ω0 and k ∈ N, let [a1(x), . . . aηk(ω,x)+1(x)]ω be the uniquecylinder of length ηk(ω, x) + 1 which contains x, and note that aηk(ω,x)+1 = b. Bybounded distortion and [8, eq. (1)], we obtain

νθηk(ω,x)ω(Aθηk(ω,x)ωηk(ω,x) |[b]θηk(ω,x)ω) ≥ C−2

θηk(ω,x)ωνω(Aω|[a1(x), . . . aηk(ω,x)+1(x)]ω).

Using the martingale convergence theorem, we obtain that, for a.e. x ∈ Xω,

limk→∞

νω(Aω|[a1(x), . . . aηk(ω,x)+1(x)]ω) = IAω ,

and consequently, for a.e. y ∈ Aω,

(7.9) lim infk→∞

νθηk(ω,y)ω(Aθηk(ω,y)ωηk(ω,y) |[b]θηk(ω,y)ω) ≥ C−2.

Since T is conservative and ergodic, note that, for a.e. x ∈ Xω there exists l(ω,x) ∈ Nwith

T l(ω,x)(w, x) ∈ Aθl(ω,x)ω ∩ [b]Ω0 .

Hence, for a.e. x ∈ Xω, we obtain that

lim infk→∞

νθnk+lω(Aθnk+lωnk

|[b]θnk+lω)) ≥ C−2,

where l := l(ω,x), nk := ηk(T l(ω, x)). We proceed with showing that

(7.10) lim infk→∞

νθnk+lω(Aθnk+lωnk+l ∩Aθnk+lω

nk|[b]θnk+lω) > 0,

It then follows by bounded distortion, that

lim infk→∞

νω(Aω|[a1(x), . . . aηk(ω,x)+1(x)]ω) > 0,

and hence (ω, x) ∈ Aω. To show claim (7.10), note that it follows from the topolog-ical mixing property, that for all k sufficiently large, there exists (ω, y) ∈ Aω withal+nk+1(y) = b. By bounded distortion, we obtain that

νω(τθl+nk ω,(a1(y)...al+nk(y))(A

θl+nk ωnk

)|[a1(y) . . . al+nk+1(y)]) ≥ C−4,

and, for all k sufficiently large, by (7.9)

νω(Aω|[a1(y) . . . al+nk+1(y)]) ≥ 1− C−4/2.

This then gives that

νω(Aω ∩ τθl+nk ω,(a1(y)...al+nk(y))(A

θl+nk ωnk

)|[a1(y) . . . al+nk+1(y))]) ≥ C−4/2,

By bounded distortion, we hence obtain

νθl+nk ω(Aθnk+lωnk+l ∩Aθnk+lω

nk|[b]θnk+lω) ≥ C−6/2


for all k sufficiently large.

In particular, note that relative exactness implies the uniqueness of the familyhω given by Proposition 7.3, that is hω is the unique family of functions withLω

φ(hω) = λ(ω)hθω and∫

hωdνω = 1 for a.e. ω ∈ Ω. Finally, we also obtain theuniqueness of the random measure νω.

7.5. Theorem. For a topologically mixing countable Markov shift with ergodic basetransformation, a positive recurrent, locally fiber Holder continuous potential φ,the random measure νω constructed in Theorem 6.1 is the unique ρ/φ-conformalrandom measure (up to multiplication by a constant).

Proof. Let µ = µω be an arbitrary random ρ/φ-conformal measure, and chooseb ∈ N and Ω′ ⊂ Ωb, Ω′ ∈ B(Ω) such that there exists C > 0 with Bω

1 < C andµω([b])/νω([b]) = C±1 for all ω ∈ Ω′. Since ν is conservative and ergodic, notethat for a.e. ω ∈ Ω and each cylinder [c] in Xω can be decomposed according tothe return time to

⋃ω∈Ω′ [b]ω. That is, there exists an at most countable index set

I, and sequences (nk ∈ N : k ∈ I), (c(k) ∈ γnk+1ω : k ∈ I such that θnkω ∈ Ω′,

Tnkω ([c(k)]ω) = [b]θnk ω, [c(k)] ∩ [c(k′)] for k 6= k′, and

∑k νω(([c(k)]ω) = νω([c]ω).

Furthermore, by combining (6.9) and Lemma 2.1, we obtain for each k ∈ N, that

µθnk ω([b]θnk ω) = Λnk(ω)

∫[(c(k)b)]ω

e−φωnk dµω

= C±1Λnk(ω)µω([c(k)b]ω).

Hence µω([c(k)b]ω) = C±2νω([c(k)b]ω), which then gives νω([c]ω) ≤ C2µω([c]ω). Byergodic decomposition, we assume without loss of generality that the measure givenby dµωdP is ergodic. As an immediate consequence, we obtain that µω and νω areequivalent a.s., and hence the invariant measures given by hωdνωdP and hωdµωdP(see Proposition 7.3) are equal (up to multiplication by a constant).

References

[1] J. Aaronson, M. Denker, M. Urbanski, Ergodic theory for Markov fibred systems and

parabolic rational maps. Trans. Amer. Math. Soc. 337 (1993), 495–548.

[2] T. Bogenschutz, Entropy, pressure and a variational principle for random dynamical

systems, Random & Comput. Dyn. 1 (1992), 99–116.

[3] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect.

Notes in Math. 470, Springer–Verlag, New York, 1975.

[4] T. Bogenschutz and V.M. Gundlach, Ruelle’s transfer operator for random subshifts of

finite type, Ergod. Th.& Dynam. Sys., 15 (1995), 413–447.

[5] H. Crauel, Random Probability Measures on Polish Spaces, Taylor & Francis, London,

2002.

[6] I.P. Cornfeld, S.V. Fomin and Ya.G. Sinai, Ergodic Theory, Springer Verlag, Berlin, 1982.

[7] M. Denker, M. Gordin, S.-M., Heinemann, On the relative variational principle for fibre

expanding maps, Ergodic Theory Dynam. Systems 22 (2002), 757–782.

[8] M. Denker, Y. Kifer, M. Stadlbauer. Conservativity for random Markov fibred systems,Preprint (2006), to appear in Ergod. Th.& Dynam. Sys.

[9] M. Denker, M. Urbanski, On the existence of conformal measures, Trans. Amer. Math.Soc. 328 (1991), no. 2, 563–587.

[10] M. Denker, M. Yuri. A note on the construction of nonsingular Gibbs measures, Coll.Mathematicum 84/85 (2000), 377–383.


[11] Y. Derriennic, Un theoreme ergodique presque sous-additif, Ann. Probab. 11 (1983),669–677.

[12] Y. Guivarc’h, Proprietes ergodiques, en mesure infinie, de certains systemes dynamiques

fibres, Ergod. Th.& Dynam. Sys. 9 (1989), no. 3, 433–453.

[13] P. Hanus, R.D. Mauldin and M. Urbanski, Thermodynamic formalism and multifractal

analysis of conformal infinite iterated function systems, Acta Math. Hungar. 96 (2002),27–98.

[14] Yu. Kifer, Equilibrium states for random expanding transformations, Rand. & Comput.

Dynam., 1 (1992), 1–31.

[15] Yu. Kifer, Ergodic theory of random transformations, Birkhauser, Boston, 1986.

[16] Yu. Kifer, Fractal dimensions and random transformations, Trans. Amer. Math. Soc.,

348 (1996), 2003–2038.

[17] Yu. Kifer, Perron-Frobenius theorem, large deviations, and random perturbations in ran-

dom environments, Math. Z., 222 (1996), 677–698.

[18] Yu. Kifer, On the topological pressure for random bundle transformations, Amer. Math.Soc. Transl. (2) 202 (2001), 197–214.

[19] Yu. Kifer and P.-D.Liu, Random dynamics, in: Handbook of dynamical systems, vol. 1B

(B.Hasselblatt and A.Katok, eds.), p.p.379–499, Elsevier, Amsterdam, 2006.

[20] K. Khanin and Yu. Kifer, Thermodynamic formalism for random transformations and

statistical mechanics, Amer. Math. Soc. Transl. Ser. 2 171 (1996), 107–140.

[21] R.D. Mauldin and M. Urbanski, Dimensions and measures in infinite iterated function

systems, Proc. London Math. Soc., 73 (1996), 105–154.

[22] R.D. Mauldin and M. Urbanski, Gibbs states on the symbolic space over an infinitealphabet, Israel J. Math. 125 (2001), 93–130.

[23] R.D. Mauldin and M. Urbanski, Graph directed Markov systems, Cambridge Tracts in

Math., 148, Cambridge Univ. Press, Cambridge, 2003.

[24] S. Orey, Markov chains with stochastically stationary transition probabilities, Ann.

Probab. 19 (1991), 907–928.

[25] W. Parry, Intrinsic Markov chains, Trans. Amer. Math. Soc. 112 (1964), 55–66.

[26] S. J. Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976), 241–273.

[27] D. Ruelle, Thermodynamic formalism (Encyclopedia of Mathematics and its Applica-

tions 5), Addison-Wesley, 1978.

[28] O. Sarig, Thermodynamic formalism for countable Markov shifts, Ergod. Th.& Dynam.Sys. 19 (1999), 1565–1593.

[29] O. Sarig, Thermodynamic formalism for null recurrent potentials, Israel J. Math. 121

(2001), 285–311.

[30] O. Sarig, Existence of Gibbs measures for countable Markov shifts, Proc. Amer. Math.

Soc. 131 (2003), 1751–1758.

[31] K. Schurger, Almost subadditive extension of Kingman’s ergodic theorem, Ann. Probab.

19 (1991), 1575–1586.

[32] D. Sullivan, Conformal dynamical systems. In: Geometric dynamics (Rio de Janeiro,1981), 725–752, Lecture Notes in Math. 1007, Springer, Berlin, 1983.

Institut fur Mathematische Stochastik, Universitat Gottingen, Maschmuhlenweg

8-10, 37073 Gottingen, GermanyE-mail address: [email protected]

Einstein Institute of Mathematics, Hebrew University of Jerusalem, Edmond J. SafraCampus, Givat Ram, Jerusalem, 91904, Israel

E-mail address: [email protected]

Institut fur Mathematische Stochastik, Universitat Gottingen, Maschmuhlenweg8-10, 37073 Gottingen, Germany

E-mail address: [email protected]

THERMODYNAMIC FORMALISM FOR RANDOM COUNTABLE · 2017-09-07 · erties of Markov chains with random...

Documents

Transcript of THERMODYNAMIC FORMALISM FOR RANDOM COUNTABLE · 2017-09-07 · erties of Markov chains with random...