Entropies of commuting transformations on Hilbert spaces

By establishing Multiplicative Ergodic Theorem for commutative transformations on a separable infinite dimensional Hilbert space, in this paper, we investigate Pesin's entropy formula and SRB measures of a finitely generated random transformations on such space via its commuting generators. Moreover, as an application, we give a formula of Friedland's entropy for certain $C^{2}$ $\mathbb{N}^2$-actions.

entropy formula and SRB measures for random transformations generated by finitely commutative transformations in infinite dimensional Hilbert spaces via its generators, which can be viewed as a generalization of the work in [7,8,24,25] to the infinite dimension spaces. However, the techniques and strategies are completely different due to the feature of infinite dimensional smooth dynamics. For more recent progress of SRB measures in infinite dimensional spaces, we refer to the elegant survey [23].
To obtain the relations of metric entropy of the random transformation and the Lyapunov exponents of its generators, the basic strategy is to estimate the random exponential expanding rate in a deterministic subspace by exponential expanding rates of generators in this subspace. Intuitively, random exponential expanding rate should be the weighted combination of exponential expanding rates of generators, and the weights depend on the probability law of choosing the generators for each iteration.
So we first establish Multiplicative Ergodic Theorem (Theorem A) for commutative transformations on a separable infinite dimensional Hilbert space, which is a higher rank group actions version of [15] and infinite dimensional version of [7]. By our assumptions, the deterministic subspace is the common expanding subspace of each generators. Then by comparing the dynamics of the random transformation with the dynamics of its generators, we reformulate Ruelle's entropy inequality (Theorem B), the Pesin's entropy formula and SRB measures (Theorem C) via the generators. Moreover, as an application, we give a formula of Friedland's entropy (Theorem D) for certain C 2 N 2 -actions. This paper is organized as follows. In Section 2, basic notions such as finitely generated random transformations, Lyapunov exponents, metric entropy and Friedland's entropy will be introduced. Then we will state the main results (Theorem A-Theorem D). Section 3 is devoted to the proofs of the main results.

. Preliminaries and Statement of Main Results
Let X be a separable infinite dimensional Hilbert space with inner product < ·, · >, norm · , distance function d and σ-algebra B of Borel sets.

Deterministic Infinite Dimensional Dynamical Systems
We begin with the notion of C 1 map. Let L(X, Y ) denote the collection of bounded linear operators from Banach space X to Y . Let U be a non-empty open subset of X.
A measurable map g : U → Y is said to be C 1 if there exists {d x g : X → Y } x∈U of L(X, Y ) such that i) for each x ∈ U, lim y→x g(x) − g(y) − d x g(x − y) x − y = 0; ii) the map x → d x g is continuous from U to L(X, Y ). The map g is said to be C 2 if its derivative d (·) g is also C 1 from U to L(X, Y ). For any bounded subset A of X, denoted by α(A) the smallest nonnegative real number r such that A can be covered by finite many Borel balls of X with radius at most r. (It is called the Kuratowski measure of non-compactness of the set A.) Define also the index of compactness of a map g : X → X as being the number g α := inf{k > 0 : α(g(A)) ≤ kα(A) for any bounded set A of X}. (2.1) In case g is a continuous linear operator, we have g α = α(g(B X )), where B X is the open unit ball of X. Let h be another continuous linear operator of X, then we have Then for any C 1 map g : X → X and g-invariant compact set K ⊂ X, where µ is any g-invariant measure.
For T ∈ L(X, X), define By a detailed exploration of the asymptotic behaviors of {V p (d x g n )} p∈N , Lian-Lu [15] proved the following theorem concerning the existence of Lyapunov exponents, we only present the part which is adequate for our purposes. We need the following assumptions to get Multiplicative Ergodic Theorem.

H.
(i) g is C 1 Fréchet differentiable and injective; (ii) the derivative of g at x ∈ X, denoted d x g, is also injective; (iii) there exists a g-invariant compact set K ⊂ X.
and at most finitely many real numbers with λ r(x,g) (x, g) > λ α for which the following properties hold. For any x ∈ Γ g , there is a splitting (b) the distribution E α (x, g) is closed and finite-codimensional, satisfies d x gE α (x, g) ⊂ E α (gx, g) and (e) writing π j (x, g) for the projection of X onto E j (x, g) via the splitting at x, we have lim n→∞ 1 n log |π j (g n x, g)| = 0 a.s.
In order to get SRB measures, it is necessary to put some restrictions on g. Under above setting, it is true by seeing Theorem 2.1 that l α (g) < 0 implies the existence of Lyapunov exponents λ 1 (x, g) > λ 2 (x, g) > · · · with multiplicities m 1 (x, g), m 2 (x, g), · · · at µ-a.e. x, which can be infinitely many but only admits finitely many positive ones. and l α (g) < 0. µ is supported on K with h µ (g) < ∞, where K is a compact invariant set. If µ is an SRB measure, then The converse is also true if (g, µ) has no zero Lyapunov exponents and the set K has finite box-counting dimension.  extremely useful in the next section. We need the following assumptions on generators which will be needed throughout the paper.

H0.
(i) f 1 , f 2 are C 1 Fréchet differentiable and injective; (ii) the derivatives of f 1 and f 2 are also injective; (iii) there exists a compact set K ⊂ X such that f 1 (K) = K and f 2 (K) = K.

Random Transformations with Finite Commuting Generators
Again, without loss of generality, we consider random transformations generated by Let Ω = F N = ∞ 0 F be the infinite product of F, endowed with the product topology and the product Borel σ-algebra A, and let θ be the left shift operator on Ω which is defined by (θω) n = ω n+1 for ω = (ω n ) ∈ Ω. Given ω = (ω n ) ∈ Ω, we write There is a natural skew product transformation F : Ω × X −→ Ω × X over (Ω, θ) which is defined by F (ω, x) = (θω, f ω (x)). For any probability measure ν on F, we can define a probability measure P ν = ν N on Ω which is invariant with respect to θ. By the induced finitely generated (i.i.d.) random transformation f over (Ω, A, P ν , θ) we mean the system generated by the randomly composition of f i , i = 1, 2 in the law of ν. We are interested in dynamical behaviors of these actions for P ν -a.e. ω or on the average on ω. It is clear that f (n, ω) is injective and strongly measurable for any n ∈ N, (in This gives the existence of the limits x ∈ Γ 0 . For the Lyapunov exponents for random transformations, we only present the part which is adequate for our purposes.
For any x ∈ Γ f , there are at most finitely many real for which the following properties hold. For P ν -a.e ω ∈ Ω, there is a splitting is a C 1,1 immersed Hilbert manifold of X, the so called unstable manifold at (ω, x).
A measurable partition η of Ω × X is subordinate to W u manifolds of (f, µ), if for P ν × µ-a.e. (ω, x), denote by η(ω, x) the element of η that contains (ω, x), then and η ω (x) contains an open neighborhood of x in W u (ω, x), this neighborhood being taken in the submanifold topology of W u (ω, x). A Borel probability measure µ is said to have absolutely continuous conditional measures on W u -manifolds of (f, µ), if for any measurable partition η subordinate to W u -manifolds of the system, one is the Lebesgue measure on W u (ω, x) induced by its inherited Riemannian metric as a submanifold of X. We call such measure an SRB measure. Similarly, we denote by Now we give more assumptions on the generators.
We are in a situation to state the main results of this paper.
Theorem B. Let f be a finitely generated random transformation of an infinite di- For Pesin's entropy formula, we need more smooth condition of the maps, thus we replace (H0) with the following conditions: H2.
(i) f 1 , f 2 are C 2 Fréchet differentiable and injective; (ii) the derivatives of f 1 and f 2 are also injective; (iii) there exists a compact set K ⊂ X such that f 1 (K) = K and f 2 (K) = K.
Theorem C. Let f be a finitely generated random transformation of an infinite di- satisfies (H1-H2) and h µ (f ) < +∞. If µ is an SRB measure, then If the following assumption (H3) on the generators are made, we will get Pesin's entropy formula and look more closely at SRB measures. The main purpose in making such assumption lies in the fact that we lose control of the random transformation when the stable and unstable directions of the generators mixes together with an infinite dimensional freedom. A trivial motivative example is the random transformations generated by hyperbolic torus automorphisms f 1 = 2 1 It is easy to see that Corollary 2.1 and Corollary 2.2 fail without (H3). More precisely, let λ α = 0 in Theorem A and denote by H3.
Corollary 2.1. Let f be a finitely generated random transformation of an infinite dimensional Hilbert space X over (Ω, A, P ν , θ). Suppose µ ∈ M with suppµ ⊂ K and holds if µ is an SRB measure.
Corollary 2.2. Let f be a finitely generated C 2 random transformation of an infinite dimensional Hilbert space X over (Ω, A, P ν , θ). Suppose µ ∈ M with suppµ ⊂ K and To date, to the best of our knowledge, there has been little discussion of relation of SRB measures of finitely generated smooth random transformation and the SRB measures of its generators. This paper only severs as a first attempt towards this direction, and the results are still far from satisfaction. The assumption (H3) in this setting seems artificial and redundant, but we can not remove it for technical reasons.
We believe that if the generators have common SRB measures, then they could be SRB measures of the random transformation, and if we add some mild conditions (for example condition H3) on the generators the converse could hold true. We leave them as further questions.
Further Questions. (a) Does equality (2.7) imply that µ is an SRB measure by adding assumption that (f i , µ) has no zero Lyapunov exponents and the set K has finite box-counting dimension?
(b) If µ is an SRB measure of every generators, then is µ an SRB measure of f ?
(c) If µ is an SRB measure of f , and assumption (H3) is satisfied, then is µ an SRB measure of every generators?

. Friedland's entropy of N 2 -actions
Friedland's entropy of N k -actions was introduced by Friedland [5] via the topological entropy of the shift map on the induced orbit space. More precisely, let f : This is a closed subset of the compact space n∈Z K and so is again compact. A natural metricd on K f is defined byd Thus we have associated an N-action σ f with the N 2 -action f. where sd(σ f , n, ε, K f ) is the largest cardinality of any (σ f , n, ε)-separated sets of K f .  [5,6,4]).
However, applying the entropy formula (2.7) for finitely generated random transformation, we give some formulas and bounds of Friedland's entropy for smooth N 2 -actions in a infinite dimensional Hilbert space.
Theorem D. Let f : N 2 −→ C 2 (X, X) be a C 2 N 2 -action on an infinite dimensional Hilbert space X. Suppose µ ∈ M f with suppµ ⊂ K and (f i , µ) satisfies (H1-H3) above and h µ (f ) < +∞, where f is a random transformation generated by {f 1 , f 2 } over (Ω, A, P ν , θ). If P ν × µ is a measure with maximal entropy of F and µ is an SRB measure, then Furthermore, if µ ∈ M e f and µ({x ∈ X : f 1 (x) = f 2 (x)}) = 0, then we get the following formula of Friedland's entropy (2.11) Remark 2.1. In Theorem D, we require that the invariant measure of F is in the form of P ν × µ, we can see [17, section 3.4 ] for the existence of such a measure for certain systems.

Proof of Theorem A
Recall that Γ f i ⊂ X is a full measure set such that Lemma 3.1. For all i, j = 1, 2, i = j, we have Proof. By symmetry, we only prove the case for i = 1, j = 2. There exists C > 0 such that for any Similarly, there exists C > 1 such that for any Thus Proof of Theorem A. For any point x ∈ Γ f 1 , let be the decomposition for f 1 . By Corollary 3.1, and , {d x f n 2 } is a cocycle on K with respect to f 2 . Now we use Multiplicative Ergodic Theorem (Theorem 2.1) for E k (x, f 1 ) and E α (x, f 1 ) to get subsets Γ k ⊂ Γ f 1 and Γ α ⊂ Γ f 1 , such that µ(Γ α ) = µ(Γ k ) = 1 for any µ ∈ M. Then for any x ∈ Γ k (resp. x ∈ Γ α ), after relabeling the subscript, if necessary, E k,j 2 (x) and E k,r(x,f 2 )+1 (resp. E r(x,f 1 )+1,j 2 (x) and E α (x)) have desired properties. We take Γ = ∩ We now show (2.4) by claiming that for any ǫ > 0, s 1 , s 2 ∈ Z + , 1 ≤ j i ≤ r(x, f i ) and i = 1, 2, the set satisfies µ(A ǫ ) = 0 for all µ ∈ M. Suppose it is not true. Then there exists a µ ∈ M with µ(A ǫ ) > 0. Choose C > 0 such that the sets , n ∈ Z + } have measures larger than 1 2 µ(A ǫ ) . Then µ(A 1 ∩ A 2 ) > 0. By Poincaré Recurrence Theorem we can take x ∈ A 1 ∩ A 2 such that there exists a sufficient large integer Since which contradicts the fact x ∈ A 1 . Similar claim for the set is also true. Then (2.4) follows by these two claims. Using the same idea, with some modification, we can prove (2.5).

Proof of Theorem
Proof. For any j = 1, 2, let n(ω, f j ) = n−1 m=0 χ f j f θ m ω , by Birkhoff Ergodic Theorem lim n−→∞ 1 n n(ω, f j )dP ν = ν(f j ), where χ f j is the character function of f j . For any x ∈ Γ f , ω ∈ Ω, Let n → ∞, and take integral with respect to P ν , by Random Subadditive Ergodic Let n → ∞, and take integral with respect to P ν , by Random Subadditive Ergodic Theorem [9, Theorem 2.2], we get the desired result.
By a similar argument, we have the following corollary.
Since X is a separable Hilbert space, we can characterize V p (·) in (2.3) by exterior power. Denote by X ∧p the p-th exterior power space of X, i.e., the collection of completely antisymmetric elements of the Hilbert space of tensor product of p copies of X. Let {ξ i } ∞ i=1 be a countable orthonormal base of X. Then is a basis of X ∧p . Define an inner product ·, · on X ∧p by letting ξ i 1 ∧ · · · ∧ ξ ip , ξ j 1 ∧ · · · ∧ ξ jp = 1, if (i 1 , · · · , i p ) = (j 1 , · · · , j p ); 0, otherwise.
This inner product is independent of the choice of the orthonormal basis {ξ i } ∞ i=1 . Denote by | · | the norm on X ∧p induced by this inner product. For any vectors ξ 1 , · · · , ξ p of X, |ξ 1 ∧ · · · ∧ ξ p | is just the volume of the parallelotope formed by these vectors and its square is the classical Gram determinant of the vectors. So we have Given a C 1 map T : U → X for some open subset U of X, define for x ∈ U, p ∈ N, It is true by Lemma 3.5 that As a consequence, we have for f as in Theorem 2.3, if X is a separable Hilbert space, then for P ν -a.e ω ∈ Ω, x ∈ Γ f and p ≤ Σ Similarly, let Next, let E u (x) := r(x,f 1 ) r(x,f 2 ) r(x,f 2 ) r(x,f 1 ) Lemma 3.6. Let f be a finitely generated random transformation of an infinite dimensional Banach space X over (Ω, A, P ν , θ). Suppose µ ∈ M and (f i , µ) satisfies (H0-H1), then

Proof. By Lemma 3.4 and Corollary 3.2, it is clear that
combining with Birkhoff ergodic theory, we have Similarly, To prove Theorem B, we need to establish the relation between local covering numbers of tangent maps of f and Lyapunov exponents of generators. For A ⊂ X, ǫ > 0, define r(A, ǫ, d) = inf{n ≥ 1 : there exist (x 1 , · · · , x n ) ∈ X n and (ǫ 1 , · · · , ǫ n ) ∈ R + n For T ∈ L(X), ǫ > 0, let where B X denotes the unit ball in X. Let β > 0. For ω ∈ Ω and x ∈ K, define whenever the limit exists. By [12, Proposition 3.4], the limit exists P ν × µ-a.s.
For the other inequality, let γ be such that max{−λ r(x,f 1 ) , −λ r(x,f 2 ) } < γ < β. Let n be large such that r(x,f 2 ) where S(A, δ) is the maximal number of subcollection of A such that any two points of it has distance at least δ. Now for 1 ≤ j 1 ≤ r(x, f 1 ), 1 ≤ j 2 ≤ r(x, f 2 ), From this we deduce that Since γ < β is arbitrary, (3.2) is also proved. We are done.
Proof of Theorem B. The main strategy of the proofs are making necessary modifications by comparing the dynamics of f and the generators. We indicate that Lemma 3.7 is the key step to establish the relation between Lyapunov exponents of f and those of its generators f 1 and f 2 . So we will concentrate upon the necessary modifications and omit most of the parallel arguments, for which we refer the reader to [12]. By ergodic decomposition theorem [16, Theorem 1.1], we restrict ourselves to the case µ ∈ M e f .
Then by Lemma 3.7 and similar argument in [12, Lemma 3.5], for P ν × µ-a.e. (ω, x), Let P be a finite measurable partition of Ω × X. For k ∈ N, define a function where χ P is the characteristic function of the set P . We have the following relation between f k and A(f k , P) [12, Proposition 3.7] : We begin with the selection of a sequence of "good" sets A m k,l . Let 0 < β < − 1 5 max{l α (f 1 ), l α (f 2 )} be fixed. Let { P m } m∈N be as above so that (3.4) holds. Let ∆ = r(x,f 1 ) For m ∈ N, consider It is clear that P ν × µ(D m ) tends to 1 as m tends to infinity. For k ∈ N, let Then P ν × µ(D m k ) increases to P ν × µ(D m ) as k increases to infinity. For k ∈ N, let A k be as in (3.3). For l ∈ N, define for x, y ∈ K, x − y ≤ ǫ, ǫ < ǫ 0 /l}.
Since f (k, ω) is C 1 in a neighbourhood of K and K is compact, we see that P ν (A k,l ) increases to P ν (A k ) as l goes to infinity and hence lim k→∞ lim l→∞ Ω\A k,l log + sup Fix A k,l and for m ∈ N, define By [12,Lemma 3.8], where B ω A,n (x, ǫ) := {y ∈ K : The proof is thus finished by [12, Proposition 3.9], since there exists N 0 ∈ N such that for P ν × µ-a.e. (ω, x), the inequality holds for any k, l, m ≥ N 0 .

Proof of Theorem C
Proof of Theorem C. Again we will concentrate upon the necessary modifications and omit most of the parallel arguments, for which we refer the reader to [13].
By Lemma 3.6 and Theorem A, we obtain the desired inequality.
Proof of Corollary 2.1. By Lemma 3.2, Lemma 3.6 and assumption (H3), r(x,f 2 ) Thus, Therefore, combining with Theorem C, we get the desired result.
When µ is ergodic, (3.5) becomes Since P ν × µ is a measure with maximal entropy of F , we can apply the variational principle of F as follows where the supremum is taken over all P ν ′ = ν ′N is the product measure of some Borel probability measure ν ′ on F with ν ′ i = ν ′ (f i ), and in the last line we use the variational principle for the topological pressure P (θ, η J ) of J with respect to θ. From [22,Chapter 9 ], we get that, Therefore, by (3.8) and (3.9), h(σ f ) ≤ log 2 i=1 exp( λ j k (x,f i )>0 λ j k (x, f i )m j k (x, f i )) . Moreover, by [22,Theorem 9.16], J has a unique equilibrium state which is the product measure defined by the measure on F which gives the element f i , i = 1, 2, measure .
So any measure ν defined by above ν i satisfies that the product measure P ν × µ is a measure with maximal entropy of F .