VARIATIONAL PRINCIPLES OF INVARIANCE PRESSURES ON PARTITIONS

. We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley’s Theorem for the Hausdorﬀ dimension; and give varia- tional principles between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures under some technical assump- tions.


(Communicated by Xiangdong Ye)
Abstract. We investigate the relations between Bowen and packing invariance pressures and measure-theoretical lower and upper invariance pressures for invariant partitions of a controlled invariant set respectively. We mainly show that Bowen and packing invariance pressures can be determined via the local lower and upper invariance pressures of probability measures, which are analogues of Billingsley's Theorem for the Hausdorff dimension; and give variational principles between Bowen and packing invariance pressures and measuretheoretical lower and upper invariance pressures under some technical assumptions.
1. Introduction. Topological feedback entropy was first introduced by Nair et al. [14] by using invariant open covers to characterize the minimal data rate for making a subset of the state space invariant. Later, Colonius and Kawan [5] introduced invariance entropy, which is defined via spanning sets, to describe the exponential growth rate of the minimal number of different control functions sufficient for orbits to stay in a given set when starting in a subset of this set. The fact that these two entropies are equivalent was shown by Colonius, Kawan, and Nair [6]. Recently, Huang and Zhong [10] use the theory of Carathéodory-Pesin structure to obtain a dimension-like characterization for invariance entropy, which is called Bowen invariance entropy. We refer the reader to the monograph written by Kawan [11] for more about invariance entropy.
By choosing conditionally invariant measures and quasi-stationary measures, Colonius [2,3] first introduced four notions of metric invariance entropy in analogy to the topological notion of invariance entropy of deterministic control systems. In [2], Colonius showed that the metric entropy of a given controlled invariant set coincides with the minimal entropy of coder-controllers associated with a quasi-stationary measure rendering that set invariant. Variational principle [17] for topological entropy [1] and measure theoretic entropy [12] in classical dynamical systems states that topological entropy is determined by the supremum of measure theoretic entropies. Feng and Huang gave variational principles between Bowen and packing topological entropy and measure-theoretical lower and upper topological entropies of subsets respectively. Motivated by the works of Huang-Zhong [10], Feng-Huang [9], and Colonius [2,3], Wang, Huang, and Sun [18] introduced packing invariance entropy and gave variational principles between Bowen and packing invariance entropies and measure-theoretical lower and upper invariance entropies in some special situations respectively.
Invariance pressure, as a generalization of invariance entropy, was first introduced by Colonius, Santana, and Cossich [8,4]. Recently, the same authors in [7] obtain some bounds of invariance pressure and get an explicit formula for hyperbolic linear control systems. Zhong and Huang [19] introduced a version of invariance pressure in a way resembling Hausdorff dimension, which is called Bowen invariance pressure. On the other hand, Tang, Cheng, and Zhao [16] extended Feng and Huang's result and gave variational principle between Pesin-Pitskel topological pressure (also called Bowen topological pressure) and measure-theoretic lower pressure.
Invariance feedback pressure via invariant open covers was first introduced by Colonius, Santana, and Cossich [8]. Wang, Huang, and Sun [18] introduced packing invariance entropy for invariant partitions. Encouraged by these works, we in this paper define packing invariance pressure and measure-theoretic lower and upper invariance pressures for invariant partitions and give corresponding variational principles which extend Theorems 6.4 and 7.2 in [18](see Theorems 3.1 and 4.2). The outline of this paper is as follows. In Section 2, we give the definitions and some basic properties of invariance pressures. In Sections 3 and 4, we respectively give variational principles for Bowen and packing invariance pressures for some special cases.
2. Invariance pressures of invariant partitions. In this paper, we mainly consider a discrete-time control system on a metric space X of the following form where F is a map from X × U to X, U is a compact set, and F u (·) ≡ F (·, u) is continuous for every u ∈ U . Given a control sequence ω = (ω 0 , ω 1 , . . .) in U , the solution of (1) can be written as For convenience, we denote system (1) by Σ = (N 0 , X, U, U , φ), where U ⊂ U N0 is a nonempty set such that (i). if ω = (ω 0 , ω 1 , . . .) ∈ U and m ∈ N then (ω m , ω m+1 , . . .) ∈ U ; (ii). if m, n ∈ N and In the rest of this paper, we always assume that Q is a compact subset of X.
Given a ∈ {1, . . . , q}, let ω a denote the control function v(A a ). Put S A = {1, . . . , q}. Given s = (s 0 , s 1 , . . . , If C s (Q) = ∅, then we say that s is an admissible word of length N . Let l(s) denote the length of s and let L j (C) denote the set of all admissible words of length j, Let C(U, R) denote the collection of all continuous functions from U to R. Given f ∈ C(U, R) and n ∈ N, let 2.1. Bowen invariance pressure. Let C = (A , τ, v) be an invariant partition of Q and f ∈ C(U, R). For any K ⊂ Q, α ∈ R, N ∈ N, let where the infimum is taken over all finite or countable covers G ⊂ S(C) of K with l(s) ≥ N for any s ∈ S. Since M C (N, α, K, Q, f ) is monotonically increasing with N , the following limit exists It is routine to check that m C (α, K, Q, f ) as a function of α has a critical point α ∈ [−∞, +∞] denoted by P C (K, Q, f ) such that m C (α, K, Q, f ) = 0, α > α and m C (α, K, Q, f ) = ∞, α < α .
Definition 2.2. The quantity P C (K, Q, f ) is called Bowen invariance pressure of K with respect to Q, C, and f .

2.2.
Lower and upper capacity invariance pressures. Let C = (A , τ, v) be an invariant partition of Q and f ∈ C(U, R). For any K ⊂ Q, α ∈ R, N ∈ N, we define R C (N, α, K, Q, f ) = inf

XING-FU ZHONG
where the infimum is taken over all finite covers G ⊂ L N (C) of K. Set Define the "jump-up" points of r C (α, K, Q, f ) and r C (α, K, Q, f ) as respectively.
Definition 2.3. We call the quantities CP C (K, Q, f ) and CP C (K, Q, f ) the lower and upper capacity invariance pressures of K with respect to Q, C, and f , respectively.

2.3.
Packing invariance pressure. Given s, s ∈ S(C), we say that s is equal to s if l(s) = l(s ) and s i = s i for every 0 ≤ i ≤ l(s) − 1. Given G ⊂ S(C), we say that G is pairwise disjoint if C s (Q) ∩ C s (Q) = ∅ for any s, s ∈ G with s = s . Let C = (A , τ, v) be an invariant partition of Q and f ∈ C(U, R). For any K ⊂ Q, α ∈ R, and N ∈ N, define where the supremum is taken over all finite or countable pairwise disjoint collections G such that C s (Q) ∩ K = ∅ and l(s) ≥ N for every s ∈ G. The quantity P C (N, α, K, Q, f ) decreases as N increases. Therefore the following limit exists: There clearly exists a unique point denoted by P P C (K, Q, f ) such that P C (α, K, Q, f ) = ∞, α < P P C (K, Q, f ); P C (α, K, Q, f ) = 0, α > P P C (K, Q, f ). Definition 2.4. We call the quantity P P C (K, Q, f ) packing invariance pressure of K with respect to Q, C, and f .

Properties of invariance pressures.
Proposition 1. Let Q be controlled invariant and C be an invariant partition of Q. For any f ∈ C(U, R), the following assertions hold: Then Proof. (i) and (v) follow directly from the definitions of invariance pressures. The first inequality in (ii) comes from (3) of Proposition 1.1 in [15] and we shall prove the second one. Given δ > 0, for every i ∈ N, since Let us rearrange the elements in K, say, Hence K is at most countable, and K ⊂ ∪ ∞ i=1 L i . This yields that Since δ is arbitrarily small, we get the desired result. The third and fourth inequalities in (iii) follow from monotonicity. We now show the other equalities in (iii). If s > P(K i , Q, f ) for all i ∈ N, where P denotes either P C or P P C , then P (s, K i , Q, f ) = 0, where P denotes either m C or P C . This implies P (s, ∪ i K i , Q, f ) = 0 by (ii). Hence P(∪ i K i , Q, f ) ≤ s. The opposite inequalities follow from monotonicity. (iv) comes from Theorem 2.4 in [15].
where the infimum is taken over all finite covers G ⊂ L n (C) of K. From Theorem 2.2 in [15], we have Then for any f ∈ C(U, R) we have We shall prove the first inequality. Without loss of generality, we can assume that By taking n → ∞, we have Hence P C (α, K, Q, f ) = +∞ and P P C (K, Q, f ) ≥ α, which implies the desired inequality.
We now prove the second inequality. Suppose This implies that there exists a disjoint collection G ⊂ S(C) such that C s (Q)∩K = ∅, l(s) ≥ N for every s ∈ G, and There must be some Using Theorem 1 in [19], we have Theorem 2.6. Let Q be controlled invariant and f ∈ C(U, R). Then (2) is taken over all finite collections, then Using Proposition 5 in [19], we have . Then for any m ∈ N we have 2.5. Measure-theoretic invariance pressures. Let M(Q) denote the set of all Borel probability measures on Q.
Definition 2.7. Let µ ∈ M(Q). Given a controlled invariant set Q, an invariant partition C of Q, and f ∈ C(U, R), the measure-theoretical lower and upper invariance pressures of µ for Q with respect to C are defined respectively by where

Remark 1.
It is easy to check that h µ,C (x, f ) and h µ,C (x, f ) are measurable by using the fact that [C n (x)] = [C n (y)] for any y ∈ [C n (x)]. Let m = inf u∈U f (u).
For any N ≥ 1, let Then K is contained in the union of the sets in F N . Since F N ⊂ {C s (Q) : s ∈ S(C)} and S(C) is at most countable, F N is at most countable. It is clear that C s (Q) = C s (Q) if s = s with l(s) = l(s ). Hence, by induction, we can pick a finite or countable pairwise disjoint subfamily G ⊂ S(C) such that This implies that It follows that m C (β + r, K, Q, f ) ≤ 1 and P C (K, Q, f ) ≤ β + r.
Since r is arbitrary, we get P C (K, Q, f ) ≤ β.
Similar to the proof for 1, we have which implies that, by Proposition 1 and arbitrariness of r, 3. For any α > β and N ∈ N, let r = α−β It follows that P P C (K, Q, f ) ≤ α. Therefore, we get the desired result. 4. Given α < β, we shall show P P C (K, Q, f ) ≥ α. To see this, we only need to show that P (α, K , Q, f ) = ∞ for any K ⊂ K with µ(K ) > 0. Let r = β−α 2 . Then there exists a strictly increasing sequence {n j (x)} ∞ j=1 such that .
For any N ≥ 1, let Then K is contained in the union of the sets in K N . Similar to the proof of 2, we can find a finite or countable pairwise disjoint subfamily G ⊂ S(C) such that K ⊂ ∪ s∈G C s (Q), and C s (Q) ∈ K N , ∀ s ∈ G.
where the infimum is taken over all finite or countable families of {(s, γ s )} such that 0 < γ s < ∞, s ∈ S(C), l(s) ≥ N for all s, and s γ s χ Cs(Q) ≥ g, where χ B denotes the characteristic function of B, i.e., χ B (x) = 1 if x ∈ B and 0 if x ∈ X \ B. For K ⊂ Q and g = χ K , we set Since the quantity W C (N, α, K, Q, f ) increases as N increases, the following limit exists: Obviously, there exists a critical value denoted by P W C (K, Q, f ) such that W C (α, K, Q, f ) = ∞, α < P W C (K, Q, f ); W C (α, K, Q, f ) = 0, α > P W C (K, Q, f ). Definition 3.2. We call P W C (K, Q, f ) weighted invariance pressure of K with respect to Q, C, and f .
Remark 4. If f = 0 then weighted invariance pressure is the weighted C-P dimension [18].
Proof. Let K ⊂ Q, α ∈ R. Taking f = χ K and γ s = 1 in the definition (4), we see that the second inequality holds. In the following, we prove the first inequality holds when N is large enough. Given ε > 0 and α ∈ R, choose N ∈ N such that n 2 e −nετ ≤ 1 for any n ≥ N . Let {(s, γ s )} s∈G be a family such that G is at most countable, s ∈ S(C), 0 < γ s < ∞, l(s) ≥ N , and s γ s χ Cs (Q) ≥ χ K .

We show below that
For any n ≥ N , let G n = {s ∈ G : l(s) = n}. Without loss of generality, we can assume that C s (Q) ∩ C s (Q) = ∅ for s, s ∈ G n with s = s (otherwise we replace (C s (Q), γ s ) and (C s (Q), γ s ) by (C s (Q), γ s + γ s )). For t > 0, put K n,t = {x ∈ K : s∈Gn γ s χ Cs(Q) (x) > t}, and G n,t = {s ∈ G n : K n,t ∩ C s (Q) = ∅}.
Since K n,t ⊂ ∪ s∈Gn,t C s (Q), we have Noting that which is a contradiction. Hence We finish the proof by letting t → 1.

A dynamical
Proof. By definition, we have c < ∞. We define a function p on the space C(Q) of continuous real-valued functions on Q by Let 1 denote the constant function 1(x) ≡ 1. It is easy to check that (1) p(h + g) ≤ p(h) + p(g) for any h, g ∈ C(Q).
If g ∈ C(Q) with g ≥ 0, then p(−g) = 0 and so L(g) ≥ 0. Hence combining the fact that L(1) = 1, we can use the Riesz representation theorem to find a Borel probability measure µ on Q such that L(g) = Q g d µ for g ∈ C(Q).
Remark 5. Since we use Riesz representation theorem and Urysohn lemma in this proof, we need that Q is compact and Q \ C n (x) is closed. That is why we suppose Q is a controlled invariant compact set and C is a clopen invariant partition.
Proof of Theorem 3.1. We first show that P C (K, Q, f ) ≥ h µ,C (Q, f ) for any µ ∈ M(Q) with µ(K) = 1. Fix l ∈ N. Let Thus µ(K l ) > 0. By Theorem 2.8, we have Letting l → ∞, we have the desired inequality. We now show the converse inequality. We can assume that P C (K, Q, f ) > −∞, otherwise we have nothing to prove. By Proposition 4, we see that for any x ∈ Q and n ≥ N . By a direct computation we obtain Thus h µ,C (Q, f ) ≥ α and we get the desired inequality.

4.
Variational principle for packing invariance pressure. Since e −l(s)τ α+S l(s)τ f (ωs) < b − a, we can discard elements in G one by one until we get a desired collection. Proof. Using an analogous method in the proof of the first part of Theorem 3.1, we can show that . We now prove that the converse inequality holds. To see this, we employ the approach given by Wang, Huang and Sun [18], which is adapted from the method used by Feng and Huang [9].
Without losing generality, we can assume that P P C (K, Q, f ) > −∞. Let −∞ < α < β < P P C (K, Q, f ). We are going to construct inductively a sequence of finite sets (K i ) ∞ i=1 and a sequence of finite measures (µ i ) ∞ i=1 so that K i ⊂ K and µ i is supported on K i for each i. Together with these two sequences, we construct a sequence of integer-valued functions (m i : K i → N) and a sequence of positive numbers (M i ). The construction is divided into three steps: Step 1. Construct K 1 , µ 1 , m 1 (·), and M 1 . Since Then P C (β, K ∩H, Q, f ) = 0 by the separability of Q. Let K = K \H = K ∩(Q\H). We claim that for any open set G ⊂ Q, either K ∩G = ∅ or P C (β, K ∩G, Q, f ) > 0. To see this, assume P C (β, K ∩ G, Q, f ) = 0 for an open set G ⊂ Q. Since K = K ∪ (K ∩ H), we have and P C (β, K ∩ H, Q, f ) = 0, we have It follows that P C (α, K , Q, f ) = ∞.
Using Lemma 4.1 again, we can find a finite set E 2 (x) ⊂ F and an integer-valued function m 2 :
In particular, for any z x ∈ C Mp (x), x ∈ K p , we have, for each x ∈ K p , and , Q, f ) > 0. We construct below each term of them for i = p + 1 in a way similar to Step 2.
It follows from (7) and (3-b) that the elements in {C mp+1 (y)} y∈Kp+1 are pairwise disjoint. Let M p+1 = max{m p+1 (x) : x ∈ K p+1 }. Then for any z x ∈ C Mp+1 (x), x ∈ K p+1 , we have, for each x ∈ K p+1 , Furthermore, for any x ∈ K p+1 , there exists y ∈ K p such that C Mp+1 (x)∩E p+1 (y) = ∅. Then, by (3-a), As in the above steps, we can construct by induction the sequences {K i }, {µ i }, {m i (·)}, and {M i }. We summarize some of their basic properties as follows: (a) For each i, the family F i := {C Mi (c) : x ∈ K i } is disjoint. For any x ∈ K i+1 , there exists y ∈ K i such that C Mi+1 (x) ⊂ C Mi (y).
(b) For each x ∈ K i and z ∈ C Mi (x), Using the above inequalities repeatedly, we have for any j > i, where C := ∞ n=1 (1 + 2 −n ). Letμ be a limit point of {µ i } in the weak-star topology. Let Thenμ is supported on K * and K * ⊂ K.
On the other hand, by (9) .
Fix j > n. Since we have µ j (K * ) > 1. It follows thatμ(K * ) ≥ 1. From (a), we see that K * ⊂ ∪ x∈Ki [C Mi (x)] for any i ∈ N. This implies that By the first part of (b), for each x ∈ K i and z ∈ C Mi (x), µ(C mi(x) (z)) ≤μ(C Mi (x)) ≤ Ce .
Since for each z ∈ K * and i ∈ N, there exists x ∈ K i such that z ∈ C Mi (x). Hencê . 508