We introduce the relative tail pressure to establish a variational principle for continuous bundle random dynamical systems. We also show that the relative tail pressure is conserved by the principal extension.
relative tail pressure; relative entropy; variational principle; principal extension
The notion of topological pressure for the potential was introduced by Ruelle  for expansive dynamical systems. Walters  generalized it to the general case and established the classical variational principle, which states that the topological pressure is the supremum of the measure-theoretic entropy together with the integral of the potential over all invariant measures. In the special case that the potential is zero, it reduces to the variational principle for topological entropy.
The entropy concepts can be localized by defining topological tail entropy to quantify the local complexity of a system at arbitrary small scales . A variational principle for topological tail entropy was established in the case of homeomorphism from subtle results in the theory of entropy structure by Downarowicz . An elementary proof of this variational principle for continuous transformations was obtained by Burguet  in term of essential partitions. Ledrappier  presented a variational principle between the topological tail entropy and the defect of upper semi-continuity of the measure-theoretic entropy on the cartesian square of the dynamical system involved, and proved that the tail entropy is an invariant under any principal extension. Kifer and Weiss  introduced the relative tail entropy for continuous bundle random dynamical systems (RDSs) by using the open covers and spanning subsets and deduced the equivalence between the two notions.
A relative version of the variational principle for topological pressure was given by Ledrappier and Walters  in the framework of the relativized ergodic theory, and it was extended by Bogenschütz  to random transformations acting on one place. Later, Kifer  gave the variational principle for random bundle transformations.
In this paper, we propose a relative variational principle for the relative tail pressure, which is introduced for random bundle transformations by using open random sets. The notion defined here enables us to treat the different open covers for different fibers. We deal with the product RDS generated by a given RDS and any other RDS with the same base. We obtain a variational inequality, which shows that the defect of the upper semi-continuity of the relative measure-theoretic entropy of any invariant measure together with the integral of the random continuous potential in the product RDS cannot exceed the relative tail pressure of the original RDS. In particular, when the two continuous-bundle RDSs coincide, we construct a maximal invariant measure in the product RDS to ensure that the relative tail pressure could be reached, and establish the variational principle. For the trivial probability space and the zero potential, the relative tail pressure is the topological tail entropy defined in  and the variational principle reduces to the version deduced by Ledrappier  in deterministic dynamical systems. As an application of the variational principle we show that the relative tail pressure is conserved by any principal extension.
The paper is organized as follows. In Section 2, we recall some background in the ergodic theory. In Section 3, we introduce the notion of the relative tail pressure with respect to open random covers and give the power rule. Section 4 is devoted to the proof of the variational principle and shows that the relative tail pressure is an invariant under principal extensions.
2. Relative Entropy
Let be a complete countably generated probability space together with a -preserving transformation ϑ and be a compact metric space with the Borel σ-algebra . Let be a measurable subset of with respect to the product σ-algebra and the fibers be compact. 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.) ω. The family is called a random transformation and each maps the fiber to . The map defined by is called the skew product transformation. Observe that , where for and .
Let be the space of probability measures on having the marginal on Ω and set . Denote by the space of all invariant measures in .
Let be a sub-σ-algebra of restricted on , and be a finite or countable partition of into measurable sets. For the conditional entropy of given σ-algebra is defined as:
where is the conditional expectation of with respect to .
Let and let be a sub-σ-algebra of restricted on satisfying . For a given measurable partition of , the conditional entropy is a non-negative sub-additive sequence, where . The relative entropy of Θ with respect to a partition is defined as:
The relative entropy of Θ 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:
Any on disintegrates (see  (Section 10.2)), where is the disintegration of μ with respect to the σ-algebra formed by all sets with . This means that is a probability measure on for -almost all (a.a.) ω and for any measurable set , -a.s. , where and so . The conditional entropy of given the σ-algebra can be written as:
where , is a partition of .
Let be a compact metric space with the Borel σ-algebra and be a measurable, with respect to the product σ-algebra , subset of with the fibers being compact. The continuous bundle RDS S over is generated by the mappings so that the map is measurable and the map is continuous for -almost all (a.a.) ω. The skew product transformation is defined as .
Let be two continuous bundle RDSs over on and , respectively. T is said to be a factor of S, or S is an extension of T, if there exists a family of continuous surjective maps such that the map is measurable and . The map defined by is called the factor or extension transformation from to . The skew product system is called a factor of or is an extension of .
Denote by the restriction of on and set .
A continuous bundle RDS T on is called a principal factor of S on , or that S is a principal extension of T, if for any invariant probability measure m in , the relative entropy of Λ with respect to vanishes, i.e.,
Let T and S be two continuous bundle RDSs over on and , respectively. Let and . It is not hard to see that is a measurable subset of with respect to the product -algebra (as a graph of a measurable multifunction; see  (Proposition III.13)). The continuous bundle RDS over is generated by the family of mappings with . The map is measurable and the map is continuous in for -a.a. ω. The skew product transformation Γ generated by Θ and Λ from to itself is defined as .
Let be the natural projection with , and with . Then, and are two factor transformations from to and , respectively. Denote by the restriction of on and set , and .
The relative entropy of Γ given the -algebra is defined by:
is the relative entropy of Γ with respect to a measurable partition , and the supremum is taken over all finite or countable measurable partitions of with finite conditional entropy .
Let , which is also a measurable subset of with respect to the product -algebra . Let be a skew-product transformation with . The map is measurable and the map is continuous in for -a.a. ω. Let be two copies of , i.e., , and be the natural projection from to with , i = 1, 2. Denote by . The relative entropy of given the -algebra is defined by:
is the relative entropy of with respect to a measurable partition , and the supremum is taken over all finite or countable measurable partitions of with finite conditional entropy .
3. Relative Tail Pressure
A (closed) random set Q is a measurable set valued map , or the graph of Q denoted by the same letter, taking values in the (closed) subsets of compact metric space X. An open random set U is a set valued map whose complement is a closed random set. A measurable set Q is an open (closed) random set if the fiber is an open (closed) subset of in its induced topology from X for -almost all ω (see  (Lemma 2.7)). A random cover of is a finite or countable family of random sets , such that for all , and it will be called an open random cover if all are open random sets. Set , and . Denote by the set of random covers and the set of open random covers. For , is said to be finer than , which we will write if each element of is contained in some element of .
For each measurable in and continuous in function f on , let:
and be the space of such functions f with and identify f and g provided ; then is a Banach space with the norm . Any such f will be called a random continuous function from to .
Let and . Denote by:
For any non-empty set and a random cover , set:
For , let:
For an open random cover , is measurable in ω. The following proof is similar to  (Proposition 1.6).
Let and . The function is measurable.
Fix . Let and . Notice that is the open cover of consisting of sets,
Since each is a random set, then the sets,
are measurable sets of . It follows from Lemma III.39 in  that the function:
is measurable in ω, where if . Since , it follows that (see  (Theorem III.30)) for any collection of strings , , the set:
belongs to . Since is finite, One obtains a finite partition of Ω into measurable sets , where J is a finite family of —strings such that . Thus for each ,
and so this function is measurable in ω.
Since for each ,
Then the function is measurable in ω. ☐
For each ω, the sequence is subadditive. Indeed, if β is a random cover of on and γ is a random cover of on , then is a finite subcover of on , and for each ,
and so is also subadditive.
By the subadditive ergodic theorem (see [14,15]) the following limit:
-a.s. exists and,
which will be called relative topological conditional pressure of Θ of an open random cover given a random cover . If is a trivial random cover, then is called the relative topological pressure of an open random cover (under the action of Θ). Observe that for all .
Notice that is increasing in in the sense of the refinement. There exists a limit (finite or infinite) over the directed set ,
which will be called the relative topological conditional pressure of Θ given a random cover . If is trivial, will be abbreviated as and be called the relative topological pressure of Θ. Since is decreasing in , one can take the limit again:
which is called the relative tail pressure of Θ. It is clear that .
For each open cover of the compact space X, naturally form an open random cover of . In this case, the above definition of relative topological pressure reduces to that given in .
Let T be a continuous bundle RDS on , be a random cover of and . Then for each ,
Let be an open random cover of . Since,
and , then,
By the definition of the relative topological conditional pressureof open random cover given , under the action of , we have:
where the supremum is taken over all open random covers of .
Thus, and the result follows. ☐
The relative tail pressure has the following power rule.
Let T be a continuous bundle RDS on and . Then for each ,
By Proposition 1,
where the infimum is taken over all random covers of . Then,
Since , then,
By taking infimum on the inequality over all random covers of , one gets and the equality holds. ☐
We need the following lemma which shows the basic connection between the relative entropy and relative tail pressure.
Let T be a continuous bundle RDS on and . Suppose that , are two finite measurable partitions of and , then,
where and is the sub-σ-algebra generated by the partition .
A simple calculation (see for instance  (Section 14.2)) shows that,
Let . Notice that μ can disintegrate , and a.s. Then,
4. Variational Principle for Relative Tail Pressure
We now take up the consideration of the relationship between the relative entropy and relative tail pressure on the measurable subset of with respect to the product -algebra .
Let . A partition is called —contains a partition if there exists a partition such that , where the infimum is taken over all ordered partitions obtained from and .
The following lemma comes essentially from the argument of Theorem 4.18 in  and Lemma 4.15 in . We omit the proof.
Given and . There exists , such that if the measurable partition —contains , where is a finite measurable partition with k elements, then
We need the following result, which has appeared already at several places (see for instance [8,10]).
Let be a finite measurable partition of . Given satisfying for each , where ∂ denotes the boundary and , then m is a upper semi-continuity point of the function defined on , i.e.,
Let be the continuous bundle RDSs on and . Suppose that , are two finite measurable partitions of and , then,
Let be the sub-σ-algebra generated by the partition . Since is a sub-σ-algebra of and , then,
Let , then . By Lemma 2, one has,
Let be the continuous bundle RDS on , and . Then for each finite measurable partition of ,
Let be a measurable partition of and .
Recall that can be viewed as a Borel subset of the unit interval . Then is also a probability measure on the compact space with the marginal on . Let and as desired in Lemma 3. Since ν is regular, there exists a compact subset with for each . Denote by . Then is a measurable partition of and By Lemma 3,
Let . Choose with such that implies . Fix . Since is compact, for each , there exists a finite separated subset in , which fails to be separated when any point is added. Recall that .
For each , let . Choose some point with , and an element with , where is the Bowen metric defined as for . Then, . Since each ball of radius meets at most the closure of two members of , then for each , the cardinality of the set cannot exceed . Therefore,
Hence by Lemma 5, one has,
Let be an open random cover with , then each contains at most one element of . Thus,
and by the inequality (1), one has,
Since , then,
then by the inequality (2), one has,
Let be an increasing sequence of finite measurable partitions with , by Lemma 1.6 in , one has,
it is not hard to see that,
where denotes the relative entropy of with respect to the partition ξ.
By the equality (4), (5) and Proposition 1, and applying , , and to the inequality (3), dividing by m and letting m go to infinity, one has:
and we complete the proof. ☐
Now, we can give the variational inequality between defect of upper semi-continuity of the relative entropy function on invariant measures and the relative tail pressure.
Let be the continuous bundle RDS on , and . Then .
Let be a finite random cover of and . Choose a finite measurable partition of with and for each . By Proposition 3 and , for each and ,
Then by Lemma 4,
Since the partition is arbitrary, then . ☐
Next, we are concerned with the variational principle relating the relative entropy of and the relative tail pressure of Θ. Recall that is a measurable subset of with respect to the product algebra and . The skew product transformation is given by . Let be two copies of , i.e., , and be the natural projection from to with , i = 1, 2.
The following important proposition relating the relative tail pressure and the relative entropy is necessary for the proof of the variational principle.
Let T be a continuous bundle RDS on , be an open random cover of and . There exists a probability measure such that,
is supported on the set .
Choose an open random cover of with such that, . Recall that is the collection of all open random covers on , and
Let and . Choose one element with , and a point . Since is an open random cover of , by the compactness of , there exists a Lebesgue number for the open cover and a maximal separated subset in such that,
where denotes the open ball in center at y of radius 1 with respect to the Bowen metric for each , i.e., . Let,
Notice that for each , the open ball is contained in some element of , then must be contained in some element of . This means that,
Consider the probability measures of via their disintegrations:
so that , and let,
By the Krylov–Bogolyubov procedure for continuous RDS (see  (Theorem 1.5.8) or  (Lemma 2.1 (i))), one can choose a subsequence such that convergence to some probability measure is . Next we will verify that the measure satisfies (i) and (ii).
Let . Choose a finite measurable partition of with for each ω and , , in the sense of , where ∂ denotes the boundary. Set . Since , then . Denote by . For each ω, let , where are two copies of the space X and is the natural projection from the product space to the space . We abbreviate it as for convenience.
Since each element of contains at most one element of , one has,
Since for each ,
For , one can cut the segment into disjoint union of segments , ⋯ and less than other natural numbers. Then,
By summing over all j, and considering the concavity of the entropy function , one has,
Replacing the sequence by the above selected subsequence , letting and , by Lemma 4, one has,
By letting , one gets,
Let be an increasing sequence of finite measurable partitions with , by Lemma l.6 in  one has,
which shows that the measure satisfies property (i).
For the other part of this proposition, let . Recall that and notice that for all . Let and , . All of them are the measurable subsets of with the product algebra , and is contained in for some and . It follows from the construction of that,
Therefore, the probability measure satisfies the property (ii) and we complete the proof. ☐
Let T be a continuous bundle RDS on and . There exists a probability measure , which is supported on , and satisfies .
Let be an increasing sequence of open random covers of . Denote by . By Property 4, for each , there exists a probability measure such that
and is supported on . Let m be some limit point of the sequence of , then (see  (Lemma 2.1 (i))) and
On the other hand, notice that the support of m,
where is the subsequence of such that convergence to m in the sense of the narrow topology. Since is a refining sequence of measurable partition on , then,
Thus for every finite measurable partition on ,
This means that and coincide up to sets of measure zero. Observe that -a.s. for all . Then,
and by the definition of the relative entropy. Hence,
By Theorem 1, and we complete the proof. ☐
The following variational principle comes directly from Theorem 1 and Proposition 5.
Let T be a continuous bundle RDS on and . Then,
We are now in a position to prove that the relative tail pressure of a continuous bundle RDS is equal to that of its factor under the principal extension.
Let be two continuous bundle RDSs over on and , respectively. Suppose that S is a principal extension of T via the factor transformation π, then for each , .
Denote by , which is a measurable subset of with respect to the product algebra . Let be the map induced by the factor transformation π as . Then ϕ is a factor transformation from to .
Let and be the natural projection defined as . By the equality 4.18 in , for each , where is the usual measure-theoretical entropy. Let be the natural projection defined as . Then and .
Notice that . One obtains . Since the continuous bundle RDS S is a principal extension of the RDS T via the factor transformation π, by the Abramov-Rokhlin formula (see [20,21]) one has . It follows that , and then . Observe that and , then
Thus by Theorem 2,
For each , there exists some such that Therefore, the other part of the above inequality holds and we complete the proof. ☐
The research is supported by the National Natural Science Foundation of China (Grant Nos. 11471114, 11671208, 11431012 and 11271191) and the National Basic Research Program of China (973 Program) (Grant No. 2013CB834100).
All authors are equally contributed. All authors have read and approved the final manuscript.
Conflicts of Interest
The authors declare no conflict of interest.
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