Variational Principle for Relative Tail Pressure

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.


Introduction
The notion of topological pressure for the potential was introduced by Ruelle [1] for expansive dynamical systems.Walters [2] 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 [3].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 [4].An elementary proof of this variational principle for continuous transformations was obtained by Burguet [5] in term of essential partitions.Ledrappier [6] 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 [7] 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 [8] in the framework of the relativized ergodic theory, and it was extended by Bogenschütz [9] to random transformations acting on one place.Later, Kifer [10] 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 [3] and the variational principle reduces to the version deduced by Ledrappier [6] 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.

Relative Entropy
Let (Ω, F , P) be a complete countably generated probability space together with a P-preserving transformation ϑ and (X, B) be a compact metric space with the Borel σ-algebra B. Let E be a measurable subset of Ω × X with respect to the product σ-algebra F × B and the fibers E ω = {x ∈ X : (ω, x) ∈ E } be compact.A continuous bundle random dynamical system (RDS) T over (Ω, F , P, ϑ) is generated by the mappings T ω : E ω → E ϑω 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 family {T ω : ω ∈ Ω} is called a random transformation and each T ω maps the fiber E ω to E ϑω .The map for n ≥ 0 and T 0 ω = id.Let P P (Ω × X) be the space of probability measures on Ω × X having the marginal P on Ω and set P P (E ) = {µ ∈ P P (Ω × X) : µ(E ) = 1}.Denote by I P (E ) the space of all Θ-invariant measures in P P (E ).
Let S be a sub-σ-algebra of F × B restricted on E , and R = {R i } be a finite or countable partition of E into measurable sets.For µ ∈ P P (Ω × X) the conditional entropy of R given σ-algebra S is defined as: is the conditional expectation of 1 R i with respect to S.
Let µ ∈ I P (E ) and let S be a sub-σ-algebra of F × B restricted on E satisfying Θ −1 S ⊂ S. For a given measurable partition R of E , the conditional entropy H µ (R (n) | S) is a non-negative sub-additive sequence, where R (n) = n−1 i = 0 (Θ i ) −1 R. The relative entropy h µ (R | S) of Θ with respect to a partition R 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 R of E with finite conditional entropy H µ (R | S) < ∞.The defect of upper semi-continuity of the relative entropy h µ (Θ | S) is defined on I P (E ) as: otherwise.
Any µ ∈ P P (E ) on E disintegrates dµ(ω, x) = dµ ω (x)dP(ω) (see [11] (Section 10.2)), where ω → µ ω is the disintegration of µ with respect to the σ-algebra F E formed by all sets (F × X) ∩ E with F ∈ F .This means that µ ω is a probability measure on E ω for P-almost all (a.a.) ω and for any measurable set R ∈ E, P-a.s.
The conditional entropy of R given the σ-algebra F E can be written as: Let (Y, C) be a compact metric space with the Borel σ-algebra C and G be a measurable, with respect to the product σ-algebra F × C, subset of Ω × Y with the fibers G ω being compact.The continuous bundle RDS S over (Ω, F , P, ϑ) is generated by the mappings S ω : G ω → G ϑω so that the map (ω, y) → S ω y is measurable and the map y → S ω y is continuous for P-almost all (a.a.) ω.The skew product transformation Λ : G → G is defined as Λ(ω, y) = (ϑω, S ω y).Definition 1.Let T, S be two continuous bundle RDSs over (Ω, F , P, ϑ) on E and G, 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 π ω : G ω → E ω such that the map (ω, y) → π ω y is measurable and π ϑω S ω = T ω π ω .The map π : G → E defined by π(ω, y) = (ω, π ω y) is called the factor or extension transformation from G to E .The skew product system (E , Θ) is called a factor of (G, Λ) or (G, Λ) is an extension of (E , Θ).
Denote by A the restriction of F × B on E and set A G = {π −1 A : A ∈ A}.Definition 2. A continuous bundle RDS T on E is called a principal factor of S on G, or that S is a principal extension of T, if for any Λ−invariant probability measure m in I P (G), the relative entropy of Λ with respect to A G vanishes, i.e., h m (Λ | A G ) = 0.
Let T and S be two continuous bundle RDSs over (Ω, F , P, ϑ) on E and G, respectively.Let H = {(ω, y, x) : y ∈ G ω , x ∈ E ω } and H ω = {(y, x) : (ω, y, x) ∈ H}.It is not hard to see that H is a measurable subset of Ω × Y × X with respect to the product σ-algebra F × C × B (as a graph of a measurable multifunction; see [12] (Proposition III.13)).The continuous bundle RDS S × T over (Ω, F , P, ϑ) is generated by the family of mappings (S × T) ω : H ω → H ϑω with (y, x) → (S ω y, T ω x).The map (ω, y, x) → (S ω y, T ω x) is measurable and the map (y, x) → (S ω y, T ω x) is continuous in (y, x) for P-a.a.ω.The skew product transformation Γ generated by Θ and Λ from H to itself is defined as Γ(ω, y, x) = (ϑω, S ω y, T ω x).
Let π E : H → E be the natural projection with π E (ω, y, x) = (ω, x), and π G : H → G with π G (ω, y, x) = (ω, y).Then, π E and π G are two factor transformations from H to E and G, respectively.Denote by D the restriction of F × C on G and set The relative entropy of Γ given the σ-algebra D H is defined by: where, is the relative entropy of Γ with respect to a measurable partition R, and the supremum is taken over all finite or countable measurable partitions R of H with finite conditional entropy Let E (2) = {(ω, x, y) : x, y ∈ E ω }, which is also a measurable subset of Ω × X 2 with respect to the product σ-algebra F × B 2 .Let Θ (2) : E (2) → E (2) be a skew-product transformation with Θ (2) (ω, x, y) = (ϑω, T ω x, T ω y).The map (ω, x, y) → (T ω x, T ω y) is measurable and the map (x, y) → (T ω x, T ω y) is continuous in (x, y) for P-a.a.ω.Let E 1 , E 2 be two copies of E , i.e., E 1 = E 2 = E , and π E i be the natural projection from E (2)  (2) : A ∈ F × B}.The relative entropy of Θ (2) given the σ-algebra A E (2) is defined by: where, is the relative entropy of Θ (2) with respect to a measurable partition R, and the supremum is taken over all finite or countable measurable partitions R of E (2) with finite conditional entropy

Relative Tail Pressure
or the graph of Q denoted by the same letter, taking values in the (closed) subsets of compact metric space Denote by P(E) the set of random covers and U(E) the set of open random covers.For R, Q ∈ P(E), R is said to be finer than Q, which we will write R Q if each element of R is contained in some element of Q.
For each measurable in (ω, x) and continuous in x ∈ E ω function f on E, let: and L 1 E (Ω, C(X)) be the space of such functions f with f < ∞ and identify f and g provided f − g = 0; then L 1 E (Ω, C(X)) is a Banach space with the norm • .Any such f will be called a random continuous function from E to R.
Let f ∈ L 1 E (Ω, C(X)) and n ∈ N. Denote by: For any non-empty set U ⊂ E and a random cover R ∈ P(E), set: For R, Q ∈ P(E), let: For an open random cover R, The following proof is similar to [10] (Proposition 1.6).
Since each R j is a random set, then the sets, are measurable sets of E. It follows from Lemma III.39 in [12] that the function: [12] (Theorem III.30)) for any collection of n−strings j i = (j i 0 , . . ., j i n−1 ), i = 1, . . ., k, the set: belongs to F .Since l n is finite, One obtains a finite partition of Ω into measurable sets Ω J , where J is a finite family of n-strings such that Ω J = (j 1 ,...,j k )∈J Ω j 1 ,...,j k .Thus for each ω ∈ Ω J , and so this function is measurable in ω.
Since for each t ∈ R, Then the function For each ω, the sequence log which implies: and so ) is also subadditive.By the subadditive ergodic theorem (see [14,15]) the following limit: the sense of the refinement.There exists a limit (finite or infinite) over the directed set U(E), which will be called the relative topological conditional pressure of Θ given a random cover Q.If Q is trivial, π f Θ (Q) will be abbreviated as π Θ ( f ) and be called the relative topological pressure of Θ.Since π f Θ (Q) is decreasing in Q, one can take the limit again: naturally form an open random cover of E. In this case, the above definition of relative topological pressure reduces to that given in [10].
By the definition of the relative topological conditional pressureof open random cover R (m) given Q (m) , under the action of Θ m , we have: Then, where the supremum is taken over all open random covers R of E.
Since R ≺ R (m) , then: and so, Thus, π and the result follows.
The relative tail pressure has the following power rule.
Proposition 2. Let T be a continuous bundle RDS on E and f where the infimum is taken over all random covers of E.
By taking infimum on the inequality over all random covers of E, one gets π * Θ m (S m f ) ≥ mπ * Θ ( f ) and the equality holds.
We need the following lemma which shows the basic connection between the relative entropy and relative tail pressure.Lemma 2. Let T be a continuous bundle RDS on E and µ ∈ P P (E).Suppose that R, Q are two finite measurable partitions of E and f ∈ L 1 E (Ω, C(X)), then, where α(R(ω)) = sup x∈R(ω)∩Q(ω) f (ω, x) and Q is the sub-σ-algebra generated by the partition Q.

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 H of Ω × Y × X with respect to the product σ-algebra F × C × B.
Let µ ∈ P P (E).A partition P is called γ-contains a partition Q if there exists a partition R P such that inf where the infimum is taken over all ordered partitions R * , Q * obtained from R and Q.
The following lemma comes essentially from the argument of Theorem 4.18 in [17] and Lemma 4.15 in [15].We omit the proof.Lemma 3. Given > 0 and k ∈ N.There exists γ = γ( , k) > 0, such that if the measurable partition P γ-contains Q, where Q is a finite measurable partition with k elements, then H µ (Q | P) < .
Proof.Let Q be the sub-σ-algebra generated by the partition Q.Since F H is a sub-σ-algebra of D H and F H = π −1 E F E , then, Let ν = π E µ, then ν ∈ P P (E).By Lemma 2, one has, Proposition 3. Let S × T be the continuous bundle RDS on H, µ ∈ I P (H) and f ∈ L 1 E (Ω, C(X)).Then for each finite measurable partition Q of E, Proof.Let R = {R 1 , . . ., R k } be a measurable partition of E and ν = π E µ.
Recall that (Ω, F, P) can be viewed as a Borel subset of the unit interval [0, 1].Then ν ∈ P P (E) is also a probability measure on the compact space [0, 1] × X with the marginal P on [0, 1].Let > 0 and γ > 0 as desired in Lemma 3. Since ν is regular, there exists a compact subset , and an element y(P(ω)) ∈ E Q (ω) with d n ω (x, y(P(ω))) ≤ δ(ω), where d n ω is the Bowen metric defined as d n ω (x, y) = max 0≤i<n d(T i ω x, T i ω y) for x, y ∈ X.Then, α(P(ω)) ≤ S n f (ω, y(P(ω))) + n .Since each ball of radius δ(ω) meets at most the closure of two members of P(ω), then for each y ∈ E Q (ω), the cardinality of the set {P(ω) ∈ P (n) and so, log ∑ e S n f (ω,y) + n log 2 + n .
Hence by Lemma 5, one has, Let U = {U} be an open random cover with diam(U(ω)) < δ(ω), then each and by the inequality (1), one has, Since, then by the inequality ( 2), one has, [14], one has, (3) Since, where h µ,Γ m (ξ | D H ) denotes the relative entropy of Γ m with respect to the partition ξ.By Lemma 1.4 in [14], for each m ∈ N, where h µ (Γ m | D H ) is the relative entropy of Γ m .By the equality (4), ( 5) and Proposition 1, and applying Γ m , Θ m , Q (m) and S m f 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.Theorem 1.Let S × T be the continuous bundle RDS on H, m ∈ I P (H) and f ∈ Proof.Let Q be a finite random cover of E and ν = π E m. Choose a finite measurable partition R of E with Q ≺ R and ν(∂R) = 0 for each R ∈ R. By Proposition 3 and π E Γ = Θπ E , for each µ ∈ I P (H) and n ∈ N, Then by Lemma 4, lim sup Thus, lim sup Next, we are concerned with the variational principle relating the relative entropy of E (2) and the relative tail pressure of Θ. Recall that The skew product transformation Θ (2) : E (2) → E (2) is given by Θ (2) (ω, x, y) = (ϑω, T ω x, T ω y).Let E 1 , E 2 be two copies of E , i.e., E 1 = E 2 = E , and π E i be the natural projection from E (2) to E i with The following important proposition relating the relative tail pressure and the relative entropy is necessary for the proof of the variational principle.

Proposition 4. Let T be a continuous bundle RDS on E
There exists a probability measure µ Q ∈ I P (E (2) ) such that, Proof.Choose an open random cover P = {P 1 , . . ., P l } of E with P (ω) = {P 1 (ω), . . ., P l (ω)} such that, and a point x ∈ Q(ω).Since P is an open random cover of E, by the compactness of E ω , there exists a Lebesgue number δ(ω) for the open cover {P 1 (ω), . . ., P l (ω)} and a maximal (n, δ)−separated subset where B y (ω, n, δ) denotes the open ball in E ω center at y of radius 1 with respect to the Bowen metric Notice that for each 0 ≤ i ≤ n − 1, the open ball B(T i ω y, δ(ϑ i ω)) is contained in some element of P(ϑ i ω), then B y (ω, n, δ) must be contained in some element of P (n) (ω).This means that, exp S n f (ω, y), and so, Consider the probability measures σ (n) of E (2) via their disintegrations: ω (x, y)dP(ω), and let, By the Krylov-Bogolyubov procedure for continuous RDS (see [18] (Theorem 1.5.8) or [10] (Lemma 2.1 (i))), one can choose a subsequence {n j } such that µ (n j ) convergence to some probability measure is ) ).Next we will verify that the measure µ Q satisfies (i) and (ii).Let ν = π E 2 µ Q .Choose a finite measurable partition R = {R 1 , . . ., R q } of E with diamR i (ω) < δ(ω) for each ω and ν(∂R i ) = 0, 0 ≤ i ≤ q, in the sense of ν(∂R) = ν ω (∂R(ω))dP(ω), where ∂ denotes the boundary.Set ω : B ∈ B}, where X 1 , X 2 are two copies of the space X and π X 1 is the natural projection from the product space X 1 × X 2 to the space X 1 .We abbreviate it as π −1 X 1 B for convenience.Since each element of R (n) (ω) contains at most one element of E n (ω), one has, Then, exp S n f (ω, y).

Since for each
Then, Therefore, For 0 ≤ j < m < n, one can cut the segment (0, n − 1) into disjoint union of [ n m ] − 2 segments (j, j + m − 1), . . ., (j + km, j + (k + 1)m − 1), . . .and less than 3m other natural numbers.Then, By summing over all j, 0 ≤ j < m and considering the concavity of the entropy function H (•) , one has, Then, by inequality (7), Replacing the sequence {n} by the above selected subsequence {n j }, letting j → ∞ and δ → 0, by Lemma 4, one has, By letting m → ∞, one gets, • be an increasing sequence of finite measurable partitions with ∞ i = 1 = A, by Lemma l.6 in [14] one has, , which shows that the measure µ Q satisfies property (i).
For the other part of this proposition, let n ∈ N.
All of them are the measurable subsets of E (2) with the product σ−algebra F × B 2 , and Then, Therefore, the probability measure µ Q satisfies the property (ii) and we complete the proof.
Proposition 5. Let T be a continuous bundle RDS on E and f ∈ Ł 1 E (Ω, C(X)).There exists a probability measure m ∈ I P (E (2) ), which is supported on {(ω, x, x) ∈ E (2) : x ∈ E ω }, and satisfies h * m (Θ (2) | A E By Property 4, for each n ∈ N, there exists a probability measure µ n ∈ I P (E (2) ) such that ω)}.Let m be some limit point of the sequence of µ n , then m ∈ I P (E (2) ) (see [10] On the other hand, notice that the support of m, where {n l } is the subsequence of {n} such that µ n l convergence to m in the sense of the narrow topology.Since {Q n l } is a refining sequence of measurable partition on E, then, suppm = {(ω, x, x) ∈ E (2) : x ∈ E ω }.
The following variational principle comes directly from Theorem 1 and Proposition 5.
For each µ ∈ I P (E (2) ), there exists some m ∈ I P (G (2) ) such that φm = µ.Therefore, the other part of the above inequality holds and we complete the proof.