The metric entropy of random dynamical systems in a Hilbert space: Characterization of invariant measures satisfying Pesin's entropy
formula

Li, Zhiming; Shu, Lin

doi:10.3934/dcds.2013.33.4123

Cited by 12 publications

(15 citation statements)

References 29 publications

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“…In [17], such a partition was constructed by appealing to a lemma due to Mañé [29], and here lies another difference between finite and infinite dimensions: the lemma in [29] uses the finite dimensionality of the ambient manifold. Our next lemma contains a slight strengthening of this result that is adequate for our purposes; see [21] for a similar result. Lemma 6.14 (following Lemma 2 in [29]).…”

Section: Technical Issues Arising From Noninvertibilitymentioning

confidence: 74%

“…The results in Theorems 1 and 2 were proved in [18], [17] in a finite dimensional context, more precisely for diffeomorphisms of compact Riemannian manifolds, and extended to Hilbert spaces in [21]. In all likelihood, the no zero exponents assumption (H4) is not necessary, but in the presence of zero Lyapunov exponents, the proofs are more elaborate and we have elected to treat that case elsewhere.…”

Section: Statement Of Resultsmentioning

confidence: 99%

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Entropy, volume growth and SRB measures for Banach space mappings

Blumenthal

Young

2016

Invent. math.

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We consider C 2 Fréchet differentiable mappings of Banach spaces leaving invariant compactly supported Borel probability measures, and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces. SRB measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds, equivalently the sum of positive Lyapunov exponents of the map. In addition to numerous difficulties incurred by our infinite-dimensional setting, a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions. The results here generalize previously known results for diffeomorphisms of finite dimensional Riemannian manifolds, and are applicable to dynamical systems defined by large classes of dissipative parabolic PDEs.

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Section: Technical Issues Arising From Noninvertibilitymentioning

confidence: 74%

Section: Statement Of Resultsmentioning

confidence: 99%

Entropy, volume growth and SRB measures for Banach space mappings

Blumenthal

Young

2016

Invent. math.

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“…By Theorem 2.5, there is a unique SRB µ i of f n | Λ i for each 1 ≤ i ≤ n, which is mixing. Under the setting of this section, an equivalent statement of Theorem 2.5 is (5.2), for details we refer the reader to [23]. Then, we have that for any 1 ≤ i ≤ n, µ i is the unique measure satisfying…”

Section: 2mentioning

confidence: 98%

“…A consequence of Theorem 2.10 is the following: Let h ν i (f ) be the metric entropy of f with respect to ν i . Then, by applying results from [23] and [24], one obtains the following corollary: Corollary 2.11. For each ergodic attractor (K i , ν i ), letting (λ k , m k ) be its Lyapunov spectrum, one has h ν i (f ) = m k λ + k > 0, and for any ǫ > 0, there exists a horseshoeK i ⊂ K i such that…”

Section: Introductionmentioning

confidence: 97%

SRB measures for a class of partially hyperbolic attractors in Hilbert spaces

Zeng

Liu

2016

Journal of Differential Equations

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In this paper, we study the existence of SRB measures and their properties for infinite dimensional dynamical systems in a Hilbert space. We show several results including (i) if the system has a partially hyperbolic attractor with nontrivial finite dimensional unstable directions, then it has at least one SRB measure; (ii) if the attractor is uniformly hyperbolic and the system is topological mixing and the splitting is Hölder continuous, then there exists a unique SRB measure which is mixing; (iii) if the attractor is uniformly hyperbolic and the system is non-wondering and and the splitting is Hölder continuous, then there exists at most finitely many SRB measures; (iv) for a given hyperbolic measure, there exist at most countably many ergodic components whose basin contains an observable set.2000 Mathematics Subject Classification. Primary: 37D45, 37C40.

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“…These techniques to handle the non-uniform compactness of the random attractors will also be useful for estimates of the dimension of K w or further consideration of the Ledrappier-Young formula in infinite-dimensional spaces (cf. [14]), the study of which will enhance the knowledge of chaotic behavior of infinite-dimensional systems, as was pointed out by Eckmann and Ruelle [7] and Young [36]. But it is still not clear how to formulate and verify the integrabillity conditions (1.2) and (1.3) for stochastic differential equations (see [1] for a discussion of this topic).…”

mentioning

confidence: 99%