The relative tail pressure with subadditive potentials is introduced via open random covers for continuous bundle random transformations. A variational principle is established and the defined pressure turns out to be invariant under the principal extension.
relative topological tail pressure; subadditive potentials; random covers; variational principle
The topological pressure with additive potential was introduced by Ruelle and Walters [1,2]. The topological pressure for subadditive sequence of continuous potentials was introduced by Falconer  on mixing repellers. Cao et al.  considered it on compact systems and established its corresponding variational principle. Ledrappier and Walters  introduced a relative version of topological pressure in the field of the relativized ergodic theory. Bogenschütz  defined the topological pressure for random transformations in the stationary case. Kifer  proposed the notion of topological pressure for continuous bundle random transformations and established its variational principle. The topological pressure play a fundamental role in statistical mechanics, dimension theory [8,9,10,11,12,13] and in the study of complex properties of a random dynamical system [14,15,16,17,18].
The topological pressure with zero potential reduces to the classical topological entropy. For the purpose of measuring the local complexity of compact dynamical systems at arbitrary small scales, Misiurewicz introduced topological tail entropy  for continuous transformations. Downarowicz [20,21] established a maximal entropy principle for the topological tail entropy for homeomorphism. In terms of the essential partitions, Burguet  proved the principle for continuous transformations. Ledrappier  investigated the defect of upper semi-continuity of the metric entropy on the square of compact systems, presented a maximal entropy principle relating it with the topological tail entropy, and showed that topological tail entropy is invariant under the principal extension. The relative version of the tail entropy for continuous bundle random transformations was introduced by Kifer and Weiss  and they deduced the consistence of the two entropy notions defined by open covers and spanning subsets.
Ma et al. set up a relative tail maximal entropy principle  and a tail variational principle  for continuous random transformations by introducing the relative tail entropy and pressure via open random covers, both of the two quantities are proved to be conserved under principal extensions. The notions defined there, via random covers, can enable ones to investigate different fibers under a natural but more complex cover way. For the nonadditive potentials case, the nonadditive thermodynamic formalisms are the powerful tools for the theory of multifractal analysis [12,27,28,29]. A natural question arises whether the tail variational principle still holds for the relative tail pressure associated with a sequence of subadditive random continuous potentials and whether this kind of tail pressure can be maintained by the action of principal extensions.
In this paper we introduce the relative tail pressure with subadditive potentials for continuous bundle random transformations via open random sets. The notion is a little different from that developed before [24,30] for the same cover of the fibers, which could make us consider various covers on different fibers. We investigate the product continuous bundle random dynamical system (RDS) generated by a given continuous bundle RDS and another continuous bundle RDS over the same probability space. A variational inequality is obtained for the relative tail pressure with subadditive potentials, which shows that the pressure of given continuous bundle RDS is an upper bound of the defect of the upper semi-continuity of the relative entropy and Lyapunov exponent of subadditive potentials in the product continuous bundle RDS. For the self-product of the given continuous bundle RDS, we establish a variational principle for the defined pressure by constructing a maximal invariant measure for the product continuous bundle RDS to ensure that the relative tail pressure may be attained. As for the trivial measure space, the relative tail pressure with the zero potential is just the topological tail entropy defined in Reference  and the variational principle is the deterministic version of maximal entropy principle deduced by Ledrappier . It turns out that from this variational principle that the relative tail pressure with subadditive potentials is an invariant in the sense of the principal extension. The method we adopt is still in the framework of Misiurewicz’s elegant proof .
Organization of the paper is as follows?We recall some basics of the relativized ergodic theory in Section 2. The relative tail pressure with subadditive potentials is introduced in term of open random covers in Section 3. In Section 4, we give the power rule and a variational inequality for the relative tail pressure in the general product RDSs. In Section 5, we state and prove the variational principle in the self-product RDSs for the relative tail pressure with the subadditive potentials and show that the defined pressure can be conserved under the consideration of the principal extension.
2. Relative Entropy
In this section, we recall some basic notions of the relative measure-theoretic entropies for bundle random transformations [24,26]. For a general theory of random dynamical systems, we refer to [24,32,33].
Let be a probability space which is complete countably generated and be a -preserving transformation of this space. Let X be a compact metric space and be its Borel -algebra. Let be subset of which is measurable under the product -algebra and assume that the fibers are compact subsets of X. A continuous bundle random dynamical system (RDS) T over is generated by the mappings so that the map is measurable and the map is continuous for -almost all (a.a.) . The family is called a random transformation and each maps the fiber to . The map defined by is called the skew product transformation. Notice that , where for and .
Let be the space of probability measures on with the marginal on and . Denote by the space of all -invariant measures in .
Let and be a sub--algebra of which is restricted on and satisfies . Let be a finite or countable measurable partition of , the relative entropy of is defined as
where is the conditional entropy of given -algebra and .
The relative entropy of is defined by the formula
where the supremum is taken over all finite or countable measurable partitions of with finite conditional entropy . The defect of upper semi-continuity of the relative entropy ) is defined on as
3. Relative Tail Pressure with Subadditive Potentials
Let be a sequence of random continuous functions on in (see Reference  for the detail). is called subadditive if for any and ,
For any -invariant measure , denote
The existence of the limit follows from the well-known subadditive argument. is called the Lyapunov exponent of with respect to . Denote by for any , then .
The map is called a (closed) random set if Q is measurable, where denotes the space of the (closed) subsets of X. The map is called an open random set if its complement is closed. Let be a finite or countable family of random sets and denote . is called a random cover of if for all . is called an open random cover if all random set U in are open. Let . We will denote by , the set of random covers, open random covers, respectively. A random cover is said to be finer than another random cover , written as , if each element of is a subset of some element of .
For each and any non-empty set , denote
where belongs to the set of all random subcover of . For , let
For each , a standard argument shows that the sequence is subadditive.
By replacing the function in Lemma 3.1 of Reference  with , one can easily get the following Lemma, which provides the basic measurable property needed. In fact, for any measurable function g on , this result also holds.
The map from Ω to is measurable for each and .
-a.s. exists, which follows from the classical subadditive ergodic theorem (see Reference [33,35]). Let
Notice that increase in , a limit (finite or infinite) exists over the directed set ,
is said to be the relative conditional pressure of with subadditive potentials for random cover . For the trivial , will be simply written as . Since decrease in , another limit exists over ,
is said to be the relative tail pressure of with subadditive potentials . Obviously .
4. Variational Inequality for Relative Tail Pressure
In this section we consider the relationship between the relative tail pressure, Lyapunov exponent with subadditive potentials and the relative entropy over the measurable subset of the product space .
We first give the power rules for the relative conditional pressure and relative tail pressure with subadditive potentials in the original continuous bundle RDS.
Let Θ be a skew product transformation, Φ be subadditive and . Then for each .
Let . Notice that
Thus and the result holds. □
Let Θ be a skew product transformation and Φ be subadditive. Then for each .
By Proposition 1,
where belongs to the set of all random covers of . Then
Since for each , then Then by taking infimum over all on this inequality. □
Consider another compact space . Denote by a measurable subset of satisfying that the fibers are compact. The continuous bundle RDS S over and the skew product transformation on can be defined similarly as in Section 2.
A continuous bundle RDS T is called a factor of another continuous bundle RDS S if a family of continuous surjective maps exists, which satisfies the map being measurable and . The factor transformation π from to is defined as and the skew product system is said to be a factor of the skew product system .
We now take up the consideration of the measurable subset based on and with the product -algebra . Denote by and set . The continuous bundle RDS over can be defined as usual by the maps , which requires being measurable and being continuous in for -a.a. . The skew product transformation is defined as , which is generated by the two product transformations and .
Let , be the two natural projections with , , respectively. Then and are obviously two factor transformations. Let be the restriction of on and denote , and .
For the given the algebra , the relative entropy of is then defined as
where belongs to the set of all finite or countable measurable partitions of satisfying .
We need the following two important Lemmas. The first Lemma shows the upper semi-continuity of the conditional entropy, which can be found in many references, for instance [5,7]. The second one is Lemma 5 in , which shows the intrinsic connection relating the relative entropy with the relative tail pressure even in the general additive case.
Let be a finite measurable partition of . If with , , where , then
Let , and be two finite measurable partitions of . Then
For any given finite measurable partition of the original RDS, we give an inequality relating the relative conditional pressure with subadditive potentials and the relative entropy of the product RDS with respect to invariant measures.
Let Γ be the skew product transformation on , and Φ be subadditive with . If is a finite measurable partition of , then
Let be a measurable partition of , and . Let be a measurable partition of such that and be the open random cover of generated by (see  for details). Denote by the Lebesgue number of the open cover for each .
Fix . Denote where and . Choose one point in with . For each pair of elements in , implies that and are in the same element of . Hence for each , there exists at most elements of satisfying
For each , an -separated set satisfying the inequality
can be easily constructed in as follows. Choose the first point with the second point with
Choose the mth point such that
The process will cease at some finite step l since is finite. Let . is obviously an -separated set and at most elements of are deleted for each step. The inequality (1) holds.
It follows from Lemma 3 that
Let be an open random cover of satisfying . Since each cannot contain two or above elements in , then
Since and , then
Let be a refine sequence of finite measurable partitions such that , then the inequality
Using , , and in the inequality (2), by the above equalities (3), (4) and the power rules in Proposition 1, one can easily get
which completes the proof. □
The following theorem describes the variational inequality between the relative tail pressure with subadditive potentials, the defect of upper semi-continuity of the relative entropy function and Lyapunov exponent with subadditive potentials with respect to invariant measures.
Let Γ be the skew product transformation on , . For subadditive potentials Φ with , one has .
Let and be finite. Let be a refine finite measurable partition of with for each element . Then by Proposition 3,
for each and . By Lemma 2, the upper semi-continuity of the conditional entropy implies that
By the arbitrariness of the partition , we have . □
5. Variational Principle for Relative Tail Pressure
In this section we investigate the variational principle between the defect of upper semi-continuity of the relative entropy function of the self-product RDSs and the relative tail pressure with subadditive potentials of the original RDS.
Denote by and . Let with be the skew product transformation on . Let be the natural projection from to with , i=1, 2, where .
We will use the following Lemma  in the proof of Proposition 4. It is a random version of the result presented by Cao et al. .
Let be a sequence probability measures in , where and . Suppose that is a subsequence of with in . Then for each ,
Moreover, is an upper bound of the left limit superior.
For any given open random cover of the original RDS, the following construction of a maximal invariant measure sets up a relationship between the relative conditional pressure with subadditive potentials and the relative entropy of the self-product RDSs, which is essential for the argument of the variational principle.
Let Θ be the skew product transformation on , Φ be subadditive and with . Then there exists some satisfying
the support of is on .
Choose some with such that .
Fix and . Let be an element in with and choose one point . Let be the Lebesgue number of the open cover . Let . There exists a maximal -separated subset with in , where . Let
Notice that is the subset of some element of . It follows that
and we have
Let be the probability measures of such that their disintegrations satisfying
with . Denote
It follows from Theorem 1.5.8 in Reference  that the Krylov-Bogolyubov procedure for continuous bundle RDS guarantees that one can choose a subsequence of such that converges towards some .
We now show that satisfies the properties (i) and (ii).
For the first proposition, let . Let be a finite partition of into measurable subsets with and . Denote . Then since . Let .
Let , where and is the natural projection. It is abbreviated as for convenience in the sequel.
Since different elements of belong to different elements of ,
It is not hard to verify that
For each j with , the section can be separated into subsection , ⋯ and no more than other positive integers. Then
Since the entropy function is concave, then by summing over all j, we have
Considering the selected subsequence , by Lemmas 2 and 4, we have
By taking , we have
Choose a refine sequence of with , and a refine sequence with , where each is a finite measurable partition. Using Lemma l.6 in Reference  we have
and the property (i) holds.
For the second proposition, we omit the argument since it is very similar to that of Proposition 4.2 in Reference  and we complete the proof. □
We now show that the relative tail pressure with subadditive potentials of the original RDS could be reached by the defect of the upper semi-continuity of the relative entropy together with the Lyapunov exponent of the subadditive potentials with respect to some invariant measure of the self-product RDS.
Let Θ be a skew product transformation on and Φ be subadditive with . There exists some with its support on and such that .
Let . Denote by . By Proposition 4, for each , we can find some with its support on and satisfying the inequality
By ( Lemma 2.1) the set of the limit points of the sequence of is contained in . Pick some limit point m with for some subsequence of , then
and the sequence of open random covers is refine, one has
Let be any finite measurable partition of , obviously,
then the two partitions and are the same except zero-measure sets. Notice that a.s. for each i, . One has
and . Hence
Since , then . By Theorem 1, the result holds. □
It follows directly from Theorem 1 and Proposition 5 that the desired variational principle holds.
Let Θ be a skew product transformation on and Φ be subadditive with . Then
Let be -algebra generated by the restriction of the product -algebra on and denote .
A skew product transformation Λ is called a principal extension of the skew product transformation Θ if the relative entropy vanishes for any measure m in .
The following theorem shows that the relative tail pressure with subadditive potentials is invariant under principal extensions. The proof is similar to Theorem 4.3 in Reference  and we omit it.
Let be two skew product transformations and Φ be subadditive with . If Λ is a principal extension of Θ, then , where π is the factor transformation between Λ and Θ and .
All authors have contributed in equal amount to the paper.
The research is supported by the National Natural Science Foundation of China (Grant Nos. 11471114, 11671208, 11431012 and 11271191), the National Basic Research Program of China (973 Program) (Grant No. 2013CB834100), and a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.
Conflicts of Interest
The authors declare no conflict of interest.
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