Conservativity of random Markov fibred systems

Abstract In this paper we extend results concerning conservativity and the existence of σ-finite measures to random transformations which admit a countable relative Markov partition. We consider random systems which are locally fibre-preserving and which admit a countable, relative Markov partition. If the system is relative irreducible and satisfies a relative distortion property we deduce that the system is either totally dissipative or conservative and ergodic. For conservative systems, we provide sufficient conditions for the existence of absolutely continuous σ-finite invariant measures.


Introduction
The class of random transformations considered in this note is best described by the following example. Let (X, B, m) and ( , F, P) be Lebesgue spaces, and let θ be an invertible, probability-preserving transformation of . A map τ from to the set of non-singular transformations of X defines a skew product transformation over the base ( , F, P, θ ) by T : X × → X × , (x, ω) → (τ (ω)(x), θ (ω)).
For a measurable set A ∈ B × F, the return time is defined by φ A (x, ω) := min{n ≥ 1 | T n (x, ω) ∈ A} and its induced transformation is As a special case, consider A = X × B for some B ∈ F and observe that φ A only depends on the second coordinate. Since θ is conservative, T A is well defined and is a skew product over the base θ B . However, in case A = B × for some B ∈ B the situation is different. If T A is well defined, that is φ A < ∞ almost everywhere, then the induced map is no longer a skew product. Defining τ B (ω)(x) := τ (θ φ A (x,ω)−1 (ω)) • · · · • τ (θ ω) • τ (ω)(x), the induced transformation has the form T A : A → A, (x, ω) → (τ B (ω)(x), θ φ A (x,ω) (ω)).
The definition of the class of generalized random transformations is motivated by this observation. Namely, a generalized random transformation consists of a transformation T of the measure space (Y, B, m), a probability-preserving dynamical system ( , F, P, θ ), a surjective and measurable map π : Y → and a measurable map ψ : Y → Z such that for almost every y ∈ Y , π • T (y) = θ ψ(y) • π(y). Moreover, we require that there exists a disintegration of m such that each fibre is a standard measure space. As in the case of random transformations, is called the base space and θ the base transformation. Note that each random dynamical system [4,13], random bundle transformation [14] and Markov chain in random environment [8,15] is a generalized random transformation, and that the class of generalized random transformations is stable under inducing to subsets (see §4). Finally, we remark that there are many situations in which generalized random transformations may occur, e.g. by inducing as in the above example or for the random coding of the geodesic flow (see §5. 3).
In §2, we introduce the basic notions for generalized random Markov fibred systems, including the relative version of a Markov partition and of a metric distortion property, given by the absolute continuity of the fibre measures. If the base space and the base transformation are trivial the system reduces to a Markov fibred system as defined in [2]. Here, we give a sufficient condition for the existence of an absolutely continuous T -invariant probability measure (see Proposition 2.1).
In §3, we investigate several notions of relative irreducibility and aperiodicity and discuss how they relate to each other.
In §4, a relative distortion property is formulated, which generalizes the Schweiger condition to the random situation and thus will be termed relative Schweiger property. The class of random systems considered here is stable under inducing. From this we obtain the main results of this paper (Theorem 4.1) which generalizes the corresponding theorem in [2].
If a relatively irreducible generalized random Markov fibred system satisfies the relative Schweiger condition, then it is either totally dissipative or conservative and ergodic.
Finally, if (Y, T ) is conservative and ergodic, we give sufficient conditions for the existence of a σ -finite T -invariant equivalent measure µ (Theorem 4.2). In particular, if (Y, T ) is a Markov chain with stationary transition probabilities (see §5.2) this gives an answer to an open problem presented by Orey [15,Problem 1.3.1].
Section 5 contains applications to the theory of random Markov shifts, skew products, coding of the geodesic flow and random hyperbolic rational maps on the Riemann sphere.

Random Markov fibred systems
We now give the details of the underlying measure of a generalized random transformation.

69
Definition. Let (Y, B, m) be a measure space, ( , F, P) be a probability space and let π : Y → be a surjective and measurable map such that the following holds. (a) For almost every (a.e.) ω ∈ , the space Y ω := π −1 ({ω}) is a Polish space and There exists a family of measures {m ω } ω∈ such that m ω is a Borel measure on (Y ω , B ω ) for a.e. ω, and such that dm = dm ω d P. Furthermore, let T : Y → Y be a non-singular and measurable transformation, θ : → an invertible, bi-measurable and probability-preserving transformation and ψ : Y → Z be measurable. If, for a.e. y ∈ Y , π • T (y) = θ ψ(y) • π(y), then the system Y = ((Y, B, m, T ), ( , F, P, θ ), π, ψ) is called a generalized random transformation.
We will refer to a measure m on Y as a relative Borel measure if there exists a family of measures {m ω } ω∈ for which the properties (a) and (b) in the above definition are satisfied. Moreover, we will refer to Y ω as the fibre over ω and to m ω as the fibre measure. For ω ∈ and n = 0, 1, 2, . . ., we denote by T n ω : Y ω → T n (Y ω ) the restriction of T n to Y ω . Here, we will use the convention T n Note that the existence of {m ω } ω∈ is in most cases provided by the existence of conditional measures. If for example (Y, B, m) and ( , F, P) are standard probability spaces, that is Polish spaces with Borel probability measures, then by the disintegration theorem there exists a family of probability measures {m ω } with properties (a) and (b) (e.g. [1]). Definition. The system ((Y, B, m, T ), ( , F, P, θ ), π, ψ, α) is called a random Markov fibred system if ((Y, T ), ( , θ ), π, ψ) is a generalized random transformation, and if there exists a countable measurable partition α with the following properties for a.e. ω ∈ . (a) The map ψ is constant (= ψ(a)) on every set a ∈ α, i.e. π • T | a = θ ψ(a) • π | a . (b) α is a relative generator, that is n>0 {a ∩ Y ω | a ∈ α n } generates B ω where, for n ∈ N ∪ {0}, (c) α is a relative Markov partition, that is for all a ∈ α, bi-measurable and together with its inverse non-singular.
We begin with a characterization of T -invariant measures for random Markov fibred systems which project to the base probability measure under π . LEMMA 2.1. Let µ be a relative Borel measure on (Y, B). Then µ is T -invariant if and only if for a.e. ω ∈ , Proof. For n ∈ Z, let b n := {y ∈ Y | y ∈ a, ψ(a) = n}. By θ-invariance of P, we determine that, for f ∈ L ∞ (µ), Conservativity of random Markov fibred systems 71 LEMMA 2.2. Let ((Y, T ), ( , θ ), π, ψ, α) be a random Markov fibred system with m(Y ) < ∞. We then have, for a.e. ω ∈ , Proof. First note that D ·,n is measurable, since the sum extends only over countably many non-zero summands. Observe that By θ-invariance of P we conclude that converges in L 1 (P) by Birkhoff's ergodic theorem. In particular, condition (d) in the following proposition is satisfied if ψ ≡ 1.

M. Denker et al
Using equation (1) and assumption (b), for A ∈ B and a.e. ω ∈ this gives Combining Lemma 2.2 with assumption (c), we determine that for A ∈ B and a.e. ω ∈ a∈α n We therefore have for m ω -a.e. x ∈ Y ω and P-a.e. ω ∈ that We use this estimate to deduce that there exist convergent subsequences of Observe that M D when D → ∞ by assumption (d). Moreover, g n |M D , h n |M D ∈ L ∞ (m |M D ) are uniformly bounded by the above estimate, from which there exist weak * convergent subsequences.
Passing through a countable sequence of D → ∞ shows that there exist a monotonically increasing sequence (n k ) and g, h ∈ L ∞ (m) with g = lim k→∞ g n k and h = lim k→∞ h n k . Let µ ω := gm w and ν ω := hm w . Clearly, the above estimate also implies that dµ ω /dν ω ∈ and It follows that µ ω (A) = a∈α µ a * ω (v a,ω (A)). By Lemma 2.1, the measure µ is T -invariant.
Note that the invariance of µ (see equation (2)) implies that µ ω (a) ≤ µ aω (T ω a) for a ∈ α. This then implies that If in addition to the assumptions in Proposition 2.1, we have that for a.e. ω ∈ the family {T ω (a) ∩ Y ω | a ∈ α} is finite, then there exist two families of positive constants {l ω } ω∈ and {u ω } ω∈ such that for µ ω -a.e. y ∈ Y ω , Proof. As it was shown in the proof of Proposition 2 the non-zero values of dν ω /dm ω are bounded from below. Combining this with dµ ω /dν ω ∈ [C −2 ω , C 2 ω ] yields the assertion. 2 For the case that the base transformation θ is a factor of T , that is ψ ≡ 1, these results have the following immediate implications. Using Lemma 2.1, we determine that the measure µ is T -invariant if and only if dµ θω = dµ ω • T −1 ω for a.e. ω ∈ . This implies that the function ω → µ ω (Y ) is invariant under θ. Under the additional assumptions that µ ω (Y ) < ∞ for a.e. ω ∈ (e.g. under the assumptions of Proposition 2.1) and that θ is ergodic, we then have that {µ ω } is a family of probability measures.

Relative versions of irreducibility and aperiodicity
In this section several generalizations of the notions of irreducibility and aperiodicity are discussed. Let α := a ∈ n≥1 α n m(a) > 0 .
First recall that for a deterministic Markov fibred system (Y, B, m, T, α) the transformation T is called irreducible if, for all a, b ∈ α there exists n ∈ N such that m(a ∩ T −n (b)) > 0. Moreover, T is called aperiodic if there exists N ∈ N such that m(a ∩ T −n (b)) > 0 for each n > N (e.g. [1]). As a first approach we obtain the following two weaker notions.

M. Denker et al
Definition. The random Markov fibred system Y = ((Y, T ), ( , θ ), π, ψ, α) is called weakly relatively irreducible if for a.e. ω ∈ and for all a, b ∈ α with m ω (a) > 0 there is n(ω) ∈ N such that m ω (a ∩ T −n(ω) A weakly relatively irreducible system is called weakly relatively aperiodic, if for m-a.e. y ∈ a there exists n(y) ∈ N such that relation (4) holds for all n > n(y) with m π(T n (y)) (b) > 0.
Note that the relative irreducibility imposes the following conditions on the random fibred system. For a, b ∈ α, Therefore, there exists k ∈ N such that P({ω | n(ω) = k}) > 0. Hence m(a ∩ T −n (b)) > 0 which implies the above non-relative version of irreducibility. Using the same arguments, it can easily be seen that the system is (non-relatively) irreducible if and only if it is weakly relatively irreducible. On the other hand, a system is weakly relatively aperiodic if it is aperiodic, but the converse is not true in general. This can be seen by the following example. Let the random system Y be the cross product of an aperiodic fibred system and ( , θ ), where = {0, 1} with θ (0) = 1 and θ (1) = 0. For ψ = 1, the system is clearly relatively aperiodic but not aperiodic.
Furthermore, consider the following stronger versions which, as will be shown later, imply that the base transformation is ergodic.
Definition. The random Markov fibred system Y = ((Y, T ), ( , θ ), π, ψ, α) is called relatively irreducible if for a.e. ω ∈ and for all a, b ∈ α with m ω (a) > 0 and A ⊂ π(b), Note that, if the base is a point, these notions reduce to the classical ones. In order to relate relative irreducibility (respectively, aperiodicity) to its weak version the following condition on ψ turns out to be useful.
(1) A relatively irreducible system is weakly relatively irreducible, and a relatively aperiodic system is weakly relatively aperiodic. (2) If the system is relatively irreducible then θ is ergodic.
(3) If the system is orbit covering, weakly relatively aperiodic and θ is ergodic then the system is relatively aperiodic.
In order to prove assertion (3), let a, b ∈ n≥1 α n , A ⊂ π(b), A ∈ F, m ω (b) > 0 for a.e. ω ∈ A. First note that, by the weak relative aperiodicity of the system, for y ∈ a and n ∈ N with π(T n (y)) ∈ A, we have that m π(y) (a ∩ T −n π(y) (b)) > 0. Moreover, since T is orbit covering and θ is ergodic, for a.e. y ∈ a there exists n ∈ N such that π(T n (y)) ∈ A. This proves the assertion.

Infinite invariant measures and the relative Schweiger condition
As a motivation for the following definition, consider the following. For b ∈ α and ω ∈ with m ω (b) > 0 assume that there exists a constant C bω such that, for all a ∈ α ∩ α n , n ∈ N with m a * ω (a ∩ v a,ω )(b) > 0, C a∩T −n (b),bω < C.
By a refinement α of α with respect to the base , one can now assume without loss of generality that the above property holds for a.e. ω ∈ π(b). For ease of exposition, we now introduce the relative Schweiger condition with respect to the latter assumption.
Definition. The random Markov fibred system Y is said to possess the relative Schweiger property if there exists a measurable function C : ω → R + , ω → C ω and a family R(C) ⊂ α such that the following holds.
(c) For almost all ω ∈ , b∈R(C) b = Y ω mod m ω . LEMMA 4.1. Suppose Y is a random Markov fibred system having the relative Schweiger property with respect to R(C). For a ∈ R(C) and a.e. ω ∈ π(a) we have for all n ∈ N, b ∈ α n b * ω , and B ∈ B such that B ⊂ a and m ω (B) > 0, Proof. If b ∩ T −n (a) = ∅ then by the Schweiger property c := b ∩ T −n (a) is an element of R(C). For ω ∈ with a ∈ α m ω and m b * ω (b ∩ T −n b * ω (a)) > 0, we obtain by equation (1)

The lower estimate is shown analogously. 2
Recall that B ∈ B is called a wandering set if {T −n (B)} n∈N is a family of disjoint sets. Moreover, the measurable union of the wandering sets is called the dissipative part of Y with respect to T for which we will write D(T ). The set C(T ) := D(T ) c is referred to as the conservative part of Y . PROPOSITION 4.1. Suppose Y is a random Markov fibred system having the relative Schweiger property with respect to R(C). For a ∈ R(C), there exist disjoint sets C a , D a ∈ F with π(a) = C a ∪ D a such that a ∩ π −1 (C a ) ⊂ C(T ) and a ∩ π −1 (D a ) ⊂ D(T ).
Then B D,q is also a wandering set, and for a.e. ω ∈ π(B D,q ) we obtain the following estimate by applying Lemma 4.1 with respect to the refined partition where a is replaced by a ∩ π −1 • π(B D,q ) and a ∩ π −1 • π(B c D,q ): Now let E ∈ B be a subset of C(T ) ∩ a ∩ π −1 • π(B D,q ). Then, by Halmos' recurrence theorem (e.g. [1]), we have for each F ⊂ E, F ∈ B that ∞ n=0 1 T n (F) (y) = ∞ for a.e. y ∈ F. For a.e. ω ∈ π(F) we obtain the contradiction • π(B D,q ))) < ∞.
Hence a ∩ π −1 • π(B D,q ) ⊂ D(T ). Since n∈N B n,1/n = B, it follows that a ∩ π −1 • π(B) ⊂ D(T ). Hence C a := π(a ∩ C(T )) and D a := π(a ∩ D(T )) are disjoint which proves the assertion. 2 We now obtain the following generalization of Theorem 2.5 in [2]. T ), ( , θ ), π, ψ, α) is a relatively irreducible random Markov fibred system with the relative Schweiger property with respect to R(C), then T is either conservative or totally dissipative. Moreover, if T is conservative, then T is ergodic.
Proof. Let us first establish that the system is either conservative or totally dissipative. By the previous proposition, we have where the union is taken over all a ∈ n≥1 α n such that a ∈ R(C). Since Y is relatively irreducible we have for all a, b ∈ R(C, ω) and a.e. x ∈ a ∩ π −1 (C a ) that there exists n ∈ N such that θ n (π(x)) ∈ D b . Since θ is ergodic, the assertion follows. Now let T be conservative. We show it is also ergodic. Suppose that B ∈ B, T −1 B = B, and m(B) > 0. Then there exists n ∈ N and b ∈ α n such that m(B ∩ T n b) > 0 and b ∈ R(C). Moreover, for all q > 0, there exists M q ∈ F such that m bω (B|T n b) > q for all ω ∈ M q . Since T is assumed to be conservative, b ∩ π −1 (M q ) is a set of full returns. Therefore, for m-a.e.
x ∈ there exists a sequence (n k ) k∈N , n k ∞ such that T n k x ∈ b ∩ π −1 (M q ) for all k ∈ N. For n ∈ N, let a n (x) ∈ α n be given by x ∈ a n (x). By the relative Schweiger property, we have that a n k (x) ∩ T −n k (b) ∈ R(C). Combining estimate (1) with the invariance of B for sufficiently large k, m π(x) (B|a n k (x) ∩ v a n k ,a n k xπ( By the martingale convergence theorem, for m-a.e. x ∈ Y ω , Estimate (5) then shows that 1 B (x) = 1 for m-a.e. x ∈ n∈Z T n (b ∩ π −1 (M q )). Since q > 0 was chosen arbitrarily, if we let M := {ω | m bω (B ∩ T n b) > 0} it follows that n∈Z T n (b ∩ π −1 (M)) ⊂ B mod m.
Using similar arguments and estimate (1), we obtain that for m-a.e.
Since T is irreducible, it follows that The assertion follows by the same arguments as the first part of the proof. 2 In order to prove the existence of an invariant σ -finite measure, we have to use the fact that the class of random systems considered here is closed under inducing to a set of full returns. This can be seen by the following argument. For a dynamical system S : X → X and B ⊂ X , denote by ϕ S B : X → N, x → min{n ≥ 1 | S n (y) ∈ B} the return time to B ⊂ X with respect to (X, S). Furthermore, for a random dynamical system Y := ((Y, B, m, T ), ( , F, P, θ ), π, ψ), A ∈ B and y ∈ Y , let ψ(T i (y)) .
Then the system ((A, T A ), (π(A), θ π(A) ), π | A , ψ A ) is a random dynamical system as defined in §1. Moreover, if Y is a random Markov fibred system with respect to the partition α and b ∈ α, then the induced system ((b, T b ), (π(b), θ π(b) ), π | b , ψ b ) is again a random Markov fibred system with respect to the partition β, where Suppose Y is a conservative and ergodic random Markov fibred system with the relative Schweiger property. Moreover, assume that there exists b ∈ R(C) such that property (d) of Proposition 2.1 holds for the induced transformation T b . Then there exists a σ -finite invariant relative Borel measure µ ∼ m.
In case of a random fibred system, condition (d) is satisfied for a cylinder b ∈ R(C) if, for a.e. ω ∈ π(B), This follows by a simple calculation using equation (3).
Proof. As noted before, T b is also a random Markov fibred system with respect to partition β. By the relative Schweiger property, we have that C a,ω < C ω for all a ∈ n β n and ω ∈ π(b). Therefore, by Proposition 2.1 and Corollary 2.1 there is a finite invariant measure equivalent to m| b and hence a σ -finite invariant measure µ ∼ m.
Example 2. Let τ be a measurable map taking values in the space of open and expanding maps of a compact metric space M. We assume that there exists a probability measure λ on the Borel σ -algebra of M which is non-singular with respect to each map τ (ω) for P-a.e. ω ∈ . Then m = P × λ is non-singular with respect to T (ω, m) = (θ (ω), τ (ω)(m)). Since each map is open and expanding there exists ω > 1 and d ω > 0 such that for x ∈ M and dist(τ (ω)(x), y) < d ω there is z ∈ M, dist(x, z) < d ω with τ (ω)(z) = y and dist(τ (ω)(x), y) > ω dist(x, z). By [10], there exists a relative Markov partition if there exist constants > 1 and d > 0 which bound ω and d ω from below. Hence, this is a random Markov fibred system possessing the relative Schweiger property with R(C ω , ω) = α ω . We suspect that the condition open and expanding can be replaced by open and expansive in the relative sense. This would lead to a random Markov fibred system with a countable partition and the relative Schweiger property only holds for a proper subclass of cylinders, which generates the σ -field.
Example 3. For each ω ∈ let f ω be a hyperbolic polynomial [6], so that T : × C → × C, T (ω, z) = (θ (ω), f ω (z)) is a skew product. Restricting to its Julia set we obtain a random dynamical system. If the maps are uniformly hyperbolic, then for Hölder continuous potentials a Gibbs family exists [9] in particular it defines a random Markov fibred system by [10]. Since it satisfies the relative Schweiger property for all sets and is irreducible, it is conservative and ergodic by Theorem 4.1 and there exists an equivalent invariant probability measure by Theorem 4.2 (in fact, Proposition 2.1 suffices).

Random Markov chains.
We now give an application of Proposition 2.1 to random Markov chains. That is, for a countable index set I with the discrete topology, let I N be the space of sequences in I , B be the Borel σ -algebra with respect to the product topology and let σ : I N → I N , (i 1 i 2 . . .) → (i 2 . . .) be the shift. For a standard probability space ( , F, P) and a probability-preserving invertible transformation θ of , let , θ (ω)). In order to randomize the system, we now define a system of random Markov measures as follows. Denote by V the set of probability vectors and by M the set of transition matrices, that is the hyperbolic metric d (e.g. [5]). We now fix 6 geodesics γ 1 , γ 2 , . . . , γ 6 of Euclidean radius √ 3/2 which are determined as follows. With the indices taken modulo 6, for k = 1, 2, 3 we have that γ 2k−1 and γ 2k intersect inside D with interior angle 2π/3, and that γ 2k and γ 2k+1 intersect at the ideal boundary ∂D (and hence are tangential to each other at the point of intersection). Furthermore, if we assume that γ 1 , γ 2 , . . . , γ 6 are ordered counter-clockwise, these properties uniquely determine the geodesics up to rotation around the origin (see Figure 1).
In order to define a Fuchsian group G, for each k = 1, 2, . . . , 6, let g k be the uniquely determined, orientating preserving, hyperbolic isometry such that we have g k (γ k ) = γ k+3 , and that γ k is contained in the isometric circle of g k , or equivalently |g k (x)| = 1 for all x ∈ γ k . Note that this definition immediately implies that g k = g −1 k+3 for k = 1, 2, . . . , 6. By Poincaré's theorem (see [11]) we hence obtain a Fuchsian group G which is generated by {g 1 , g 2 , g 3 }, and for which the component of D \ {γ 1 , γ 2 , . . . , γ 6 } which contains the origin is a fundamental polygon (see Figure 1). Moreover, by the shape of the fundamental polygon it follows immediately that D/G is a punctured torus.
Proof. By construction, T (a) is α-measurable for each a ∈ α. In complete analogy to the deterministic case, it follows by standard means in the theory of hyperbolic geometry that α is relative generator. These observations immediately imply the first assertion. In order to show that T is relatively aperiodic, note that by construction we have for k = 1, 2, . . . , 6 and l = 1, 2, 3, The system is therefore relatively aperiodic. 2 The aim is to use our results to deduce that the above system is conservative and ergodic with respect to an infinite measure which is equivalent to m. Let A := 3 k=1 b 2k−1,2k × . Note that by the fact that the deterministic Bowen-Series map is conservative [7,17] for a.e. y ∈ A c there exists n(y) ∈ N such that T n (y) ∈ A. In particular, the induced system Y = ((A, T A ), ( , θ ), π A , ψ A , α A ) as introduced in §4 is well defined. Also note that by the choice of A, we clearly have that ψ A (y) = 1 for all y ∈ A. Hence, Y is a skew product. We will proceed with the following result which relies on the application of Proposition 2.1 to the system Y.
As a consequence of Stadlbauer [17,Lemma 4.3], we determine that there exists C > 1 such that for the first and second derivatives D and D 2 in the first coordinate, we have D 2 (T n A ) (D(T n A )) 2 < C for all n ∈ N. By standard arguments (e.g. [1, p. 147]), it follows that C a,ω > C for all a ∈ α A and a.e. ω ∈ . Hence, by Proposition 2.1 and Corollary 2.1, we obtain an invariant finite measure µ on A and a family {l ω } ω∈ such that 0 < l ω < dµ ω /dm ω < C 2 /η for a.e. ω ∈ .