Pre-image Variational Principle for Bundle Random Dynamical Systems

The pre-image topological pressure is defined for bundle random dynamical systems. A variational principle for it has also been given.


Introduction
In deterministic dynamical systems, the thermodynamic formalism based on the notions of pressure, of Gibbs and equilibrium states plays a fundamental role in statistic mechanics, ergodic theory and dimension theory [3,13,18,23]. The background for the study of equilibrium states is an appropriate form of the variational principle. Its first version was formulated by Ruelle [2]. In random dynamical systems (RDS), the thermodynamic formalism is also important in the study of chaotic properties of random transformations [5,8,20,25]. The first version of the variational principle for random transformations was given by Ledrappier and Walters in the framework of the relativized ergodic theory [6], and it was extended by Bo-Email addresses: xianfengma@gmail.com (Xianfeng Ma), ecchen@njnu.edu.cn (Ercai Chen) genschütz [17] to random transformations acting on one place. Later Kifer [26] gave the variational principle for random bundle transformations.
In recent years, the pre-image structure of a map has also been studied by many authors [4,11,14,15,21,22,28]. In deterministic dynamical systems, Fiebig [4] studied the relation between the classical topological entropy and the dispersion of pre-images. Cheng and Newhouse [21] introduced the notions of the pre-image entropies and obtained a variational principle which is similar to the standard one. Zeng [7] defined the notion of the pre-image pressure and investigated its relationship with invariant measures. He also established a variational principle for the pre-image pressure, which was a generalization of Cheng's result for the pre-image entropy. In random dynamical systems, Zhu [27] introduced the analogous notions as that in the deterministic case and gave the analogs of many known results for entropies, such as Shannon-McMillan-Breiman Theorem, the Kolmogorov-Sinai Theorem, the Abromov-Rokhlin formula and the variational principle.
In this paper, we present the notion of the pre-image topological pressure and derive the corresponding variational principle for bundle random dynamical systems. In fact, we formulate a random variational principle between the pre-image topological pressure, the pre-image measure-theoretic entropy and some functions of the invariant measure. We also introduce a revised definition of the random pre-image topological entropy without any additional assumptions, while the original notion defined in [27] need a strong measurability condition. All results in [27] still hold for our new notion. For the probability space consisting of a single point, we establish the pre-image variational principle on any compact invariant subset for deterministic dynamical systems, which is a generalization of Zeng's result [7] for the whole space. The method we use is in the framework of Misiurewicz's elegant proof [10]. Kifer's method [24] and Cheng's technique [21] are also adopted in the argument of our theorem. In fact, our proof generalizes Kifer's proof of the standard variational principle for random bundle transformations. This paper is organized as follows. In Section 2, we define the pre-image measure-theoretic entropy as a conditional entropy of the induced skew product transformation and give another fiberwise expression for bundle random dynamical systems. In Section 3, we define the pre-image topological pressure for bundle random dynamical systems and give the power rule for this pressure. In Section 4, we state and prove the pre-image variational principle.

Pre-image measure-theoretic entropy for bundle RDS
Let (Ω, F , P) be a probability space together with an invertible P-preserving transformation ϑ, where F is complete, countably generated and separated points. Let (X, d) be a compact metric space together with the Borel σalgebra B. Let E ⊂ Ω × X be measurable with respect to the product σalgebra F × B and the fibers E ω = {x ∈ X : (ω, x) ∈ E}, ω ∈ Ω be compact. A continuous bundle random dynamical system (RDS) T over (Ω, F , P, ϑ) is generated by map T ω : E ω → E ϑω with iterates T n ω = T ϑ n−1 ω · · · T ϑω T ω , n ≥ 1, and T 0 ω = id, so that the map (ω, x) → T ω x is measurable and the map x → T ω x is continuous for P-almost all (a.a) ω. The map Θ : E → E defined by Θ(ω, x) = (ϑω, T ω x) is called the skew product transformation.
Let P P (E) = {µ ∈ P P (Ω × X) : µ(E) = 1}, where P P (Ω × X) is the space of probability measures on Ω × X with the marginal P on Ω. Any µ ∈ P P (E) on E can be disintegrated as dµ(ω, x) = dµ ω (x) dP(ω) (See [16]), where µ ω are regular conditional probabilities with respect to the σ-algebra F E formed by all sets For µ ∈ P P (E), the conditional entropy of Q given by the σ-algebra of [24]) by The pre-image measure-theoretic entropy h (r) pre, µ (T ) of bundle RDS T with respect to µ is defined by the formula and the supremum is taken over all finite or countable measurable partitions  [17,19,24] for detail).
Compared with Zhu [27], we define the pre-image measure-theoretic entropy on the measurable subset E instead of on the whole space Ω×X. Moreover, if E = Ω × X, then the above definition is just the measure-theoretic pre-image entropy in [27]. In this sense, the definition is a generalization of Zhu's.
In [27], Zhu gave another fiberwise expression for his defined measuretheoretic pre-image entropy, which can be seen as a generalization of Kifer's discussion on the standard measure-theoretic entropy for random bundle transformations [26]. In a similar way, we can give a fiberwise expression for the above definition. For completeness of this paper, we state this proposition and give the proof.
Proof. Note that for any f ∈ L 1 (Ω×X, µ) and where f ω (x) = f (ω, x). Therefore, Hence for any finite measurable partition Q of E, we have where I · (·|·) denotes the standard conditional information function. Thus Dividing by n and letting n → ∞, we obtain the equality (1).
Moreover, if we use Zhu's methods and restrict the whole space Ω × X to the measurable subset E in [27], then all results with respect to the pre-image measure-theoretic entropy defined by Zhu also hold for the above definition. Then, we can use those results directly without giving any proof whenever we consider the pre-image measure-theoretic entropy.

Pre-image topological pressure for bundle RDS
Let X E = {x ∈ X : (ω, x) ∈ E, ω ∈ Ω}. For each x ∈ X E , by the measurability of bundle RDS T , E(x) = {(ω, y) : ω ∈ Ω, y ∈ T −1 ω x} is measurable with respect to the product σ-algebra F × B. For each k ∈ N, let E(x, k) = {(ω, y) : ω ∈ Ω, y ∈ (T k ω ) −1 x}. By the continuity of bundle RDS T and [1, Theorem III.30], it is not hard to see that Chaper III]) with respect to the Borel σ-algebra induced by the Hausdorff topology on the space K(X), and the distance function For each n ∈ N, a family of metrics d ω n on E ω is defined as It is not hard to see that for each k ∈ N and x ∈ X E , the set . Since for each m ∈ N, ǫ > 0 and a real number a the set Due to the compactness, there exists a smallest natural number s n (ω, ǫ) such that card(F ) ≤ s n (ω, ǫ) < ∞ for every (ω, n, ǫ)-separated F . Moreover, there always exists a maximal (ω, n, ǫ)-separated set F in the sense that for every y ∈ E ω with y ∈ F the set F ∪{y} is not (ω, n, ǫ)-separated anymore. In particular, this is also true for any compact subset K of E ω . Let s n (ω, ǫ, K) be the smallest natural number such that card(F ) ≤ s n (ω, ǫ, K) < ∞ for every (ω, n, ǫ)-separated set F of K.
For each function f on E, which is measurable in (ω, x) and continuous in x ∈ E ω , let For any δ > 0, let where the supremum is taken over all (ω, n, ǫ)-separated sets of (T k ω ) −1 x in E ω . Based on the foregoing analysis, clearly, any (ω, n, ǫ)-separated set can be completed to a maximal one. Then, the supremum can be taken only over all maximal (ω, n, ǫ)-separated sets.
In this paper, we always assume that for each k ∈ N and x ∈ X E , (T k ω ) −1 x = ∅. Alternatively, we can also assume that for each ω the mapping T ω is surjective.
The following auxiliary result, which relies on Kifer's work [26] and restricts his result to the family of compact subsets (T k ω ) −1 x of E ω for nonrandom positive number ǫ, provides the basic properties of measurability needed in what follows. We make a little adjustment to Kifer's proof for the purpose of defining the pre-image topological pressure.

Lemma 2.
For each x ∈ X E , k, n ∈ N with k ≥ n, and a nonrandom small positive number ǫ, the function P pre, n, ω (T, f, ǫ, (T k ω ) −1 x) is measurable in ω, and for any δ > 0, there exists a family of maximal (ω, n, ǫ)-separated sets and depending measurably on ω in the sense that G = {(ω, x) : x ∈ G ω } ∈ F × B, which also means that the mapping ω → G ω is measurable with respect to the Borel σ-algebra induced by the Hausdorff topology on the K(X) of compact subsets of X. In particular, the supremum in the definition of P pre, n, ω (T, f, ǫ, (T k ω ) −1 x) can be taken only over families of (ω, n, ǫ)-separated sets, which are measurable in ω.
In view of this assertion, for each f ∈ L 1 E (Ω, C(X)) and any positive number ǫ we can introduce the function Note that though we set x ∈ X E , only those points such that x ∈ E ϑ k ω act in the function P pre (T, f, ǫ).
The limit exists since P pre (T, f, ǫ) is monotone in ǫ and, in fact, lim ǫ→0 above equals sup ǫ>0 . where s n (ω, ǫ, (T k ω ) −1 x) is the largest cardinality of an (ω, n, ǫ)-separated set of (T k ω ) −1 x. Remark 5. Definition 4 is different from the random pre-image topological entropy defined by Zhu [27], which in our terminology can be expressed as The measurability of the function sup x∈X E s n (ω, ǫ, (T k ω ) −1 x) can not been guaranteed in most cases. In Definition 4, based on Lemma 2, s n (ω, ǫ, (T k ω ) −1 x) is always measurable. On the other hand, through a rigorous investigation, it is not hard to see that if we replace Zhu' definition by Definition 4 and make a little change to Zhu's argument, then the variational principle for the pre-image entropy for bundle RDS T still holds, namely, h pre, µ (T ) : µ ∈ M 1 P (E, T )}. In fact, we just need to make some adjustment to the order of the supremum and the logarithm and restrict the whole space Ω × X to the measurable subset E in his proof. Remark 6. If the measure space Ω consists of a single point, i.e., Ω = {ω}, then bundle RDS T reduces to a deterministic dynamical system (E ω , d, T ), where T : E ω → E ω is continuous. Furthermore, if E ω = X, then Definition 3 is just the pre-image topological pressure defined by Zeng [7] for deterministic dynamical systems except for the order of the supremum and the logarithm, and the difference between the two kinds of order does not affect the variational principle and the method of the argument.
For a given m ∈ N + , if we replace ϑ by ϑ m and consider bundle RDS T m defined by (T m ) n ω = T m ϑ (n−1)m ω · · · T m ϑ m ω T m ω , i.e., (T m ) n ω = T mn ω , then the pre-image topological pressure has the following power rule. Proof. Fix m ∈ N. Let n ∈ N, k ≥ n and x ∈ X E . If E is an (ω, n, ǫ)separated set of ((T m ) k ω ) −1 x for T m , then E is also an (ω, mn, ǫ)-separated set of (T mk ω ) −1 x for T . Since (T m ) n ω = T mn ω , so For any ǫ > 0, by the continuity of T m , there exists some small enough δ > 0 such that if d(y, z) ≤ δ, y, z ∈ E ω , then d m ω (y, z) ≤ ǫ. For any positive integer n, there exists some integer l such that mn ≤ l < m(n + 1). It is easy to see that any (ω, l, ǫ)-separated set of (T k ω ) −1 x for T is also an Hence, any (ω, n, δ)separated set of (T k ω ) −1 x for T m is also an (ω, n, δ)-separated set of ((T m ) k ′ ω ) −1 x ′ for T m . Therefore, for k ≥ l, E is an (ω, n, δ) -separated set of (( Since f ∈ L 1 E (Ω, C(X)), so Then by the definition of P pre (T, f, ǫ), we have Hence, mP pre (T, f, ǫ) ≤ P pre (T m , S m f, δ).
If ǫ → 0, then δ → 0. Hence we obtain mP pre (T, f ) ≤ P pre (T m , S m f ) and complete the proof.

Pre-image variational principle for bundle RDS
Proof.
Since diam(β k j ) → 0 as j → ∞, we can choose large enough j such that for for any probability vector (p 1 , · · · , p m ), we have In the sequel, let A be the maximal one satisfying the above inequality and Hence there is y 1 ∈ (T k ω ) −1 u B such that d(x C , y 1 ) < ǫ 1 and then d ω n (x C , y 1 ) < δ. Let E ω A be a maximal (ω, n, δ)-separated set in (T k ω ) −1 u B . Since E ω A is also a spanning set, there is a point z(x C ) ∈ E ω A such that d ω n (y 1 , z(x C )) ≤ δ, then d ω n (x C , z(x C )) ≤ 2δ and hence We now show that if y ∈ E ω A , then card{A ∩ C ∈ Q n (ω, A) : z(x C ) = y} ≤ 2 n .
Suppose z(x C ) = z(x e C ). There exist x C ∈ A ∩ C and x e C ∈ A ∩ C so that d ω n (x C , z(x C )) < 2δ and d ω n (x e C , z(x e C )) < 2δ. Therefore d ω n (x C , x e C ) < 4δ; so T i ω (x C ) and T i ω (x e C ) are in the same element of η, say Q 0 ∪ Q j i , 0 ≤ i < n. Hence there are at most 2 n elements A ∩ C of Q n (ω, A) equal to a fixed member of E ω A . Combining (6) and (7), we get Taking the logarithm of both parts and using the resulting inequality in order to estimate the righthand side of (5), we obtain ≤n log 2 + log P pre,n,ω (T, f, δ, Integrating this with respect to P and letting j → ∞, we have Letting k → ∞, dividing by n and letting n → ∞, by Proposition 1 and the equality (3), we have Using (4), we derive the inequality Since this is true for all finite partitions P of X and all positive δ, we have pre,µ (T ) (See [27,Proposition 4] ), the same arguments as above applied to T n and to S n f yield n(h (r) pre, µ (T ) + f dµ) ≤ log 2 + 1 + P pre (T n , S n f ).
Taking into account Proposition 7, dividing by n and letting n → ∞, we conclude that h (r) pre, µ (T ) + f dµ ≤ P pre (T, f ).
(2) By the equality 3, we can choose a sequence n i → ∞, k i ≥ n i , and points x ϑ k i ω ∈ E ϑ k i ω for each ω ∈ Ω such that For a small nonrandom ǫ > 0, by Lemma 2, we can choose a family of maximal (ω, Next, we define probability measures ν (i) on E via their measurable disintegrations Then, by Lemma 2.1 (i)-(ii) of [26], we can choose a subsequence n i l of {n i } such that lim l→∞ µ (i l ) = µ for some µ ∈ M 1 P (E, T ). Without loss of generality, we still assume that lim i→∞ µ (i) = µ.
Next, we choose a partition P = {P 1 , · · · , P k } of X with diamP ≤ ǫ, which satisfies µ ω (∂P i ) dP(ω) = 0 for all 1 ≤ i ≤ k, where ∂ denotes the boundary. Let P(ω) = {P 1 (ω), · · · , P k (ω)}, P i (ω) = P i ∩ E ω . Since each element of n i −1 l=0 (T l ω ) −1 P(ϑ l ω) contains at most one element of G(ω, n i , ǫ), then by the inequality (8), we have For ω ∈ Ω, let C ω be the subcollection of B − ω consisting of µ ω -null sets. For any σ-algebra A of subsets of E ω , there is an enlarged σ-algebra A Cω defined by A ∈ A Cω if and only if there are sets B, M, N such that A = B ∪ M, B ∈ A, N ∈ C ω and M ⊂ N. The σ-algebra B − Cω is simply the for all l ≥ 1. Similar to the proof of Proposition 1, we conclude that for each k and any finite partition where ω is supported on (T k i ω ) −1 x ϑ k i ω , the canonical system of conditional measures induced by ν Hence for any finite partition γ of E ω , we have . Integrating in (9) with respect to P, then by (10), (11) and S n i f dν (i) = n i f dµ (i) , we obtain the inequality ≥ log P pre, n i , ω (T, f, ǫ, (T k i ω ) −1 x ϑ k i ω ) dP(ω) − 1.
For q, n i ∈ N with 1 < q < n i , let a(s) denote the integer part of (n i − s)q −1 for all 0 ≤ s < q. Then, clearly, for each s, we have where cardS ≤ 2q. Since cardQ = k, by the subadditivity of conditional entropy (See [24, Section 2.1]), we have Summing this inequality over s ∈ {0, 1, · · · , q − 1}, we get where the second inequality, as in Kifer's works [26], relies on the general property of conditional entropy of partition i p i H η i (ξ|A) ≤ H P i p i η i (ξ|A) which holds for any finite partition ξ, σ-algebra A, probability measures η i , and probability vector (p i ), i = 1, . . . , n, in view of the convexity of t log t in the same way as that in the unconditional case (cf. [12, pp.183 and 188 ] ). This together with (12) yields q log P pre, n i , ω (T, f, ǫ, (T k i ω ) −1 x ϑ k i ω ) dP(ω) − q Diving by n i , passing to the lim sup i→∞ and using the inequality 10 in [27], i.e., we get Dividing by q and letting q → ∞, we have P pre (T, f, ǫ) ≤ h (r) pre, µ (T, Q) + f dµ ≤ h (r) pre, µ (T ) + f dµ Let ǫ → 0, then we have P pre (T, f ) ≤ h (r) pre, µ (T ) + f dµ and complete the proof of Theorem 8.
Remark 9. If f = 0, then, without any additional assumption, Theorem 8 can be expressed as h pre, µ (T ) : µ ∈ M 1 P (E, T )}. In [27], the variational principle for the pre-image topological entropy needs a measurability condition which in most cases cannot be satisfied. Thus Theorem 8 can be regarded as a revised version of Zhu's. On the other hand, if Ω consists of only one point, that is, Ω = {ω}, then by Remark 6, Theorem 8 generalizes Zeng's deterministic variational principle on the whole space X [7] to any compact invariant subset E.