Sinai’s Work on Markov Partitions and SRB Measures

Pesin, Yakov

doi:10.1007/978-3-319-99028-6_11

Cited by 5 publications

(3 citation statements)

References 58 publications

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“…Notice that a rectangle is not always connected, and might even have an infinite number of connected component, even if Ω is itself connected. This technicality is noticed in [Pe19], at the first paragraph of subsection 3.3, and is an obstruction to the existence of finite Markov partition with connected elements.…”

Section: Markov Partitionsmentioning

confidence: 99%

Fourier decay of equilibrium states for bunched attractors

Leclerc

2024

Nonlinearity

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Let M be a closed manifold, and let f : M → M be a C 2 + α Axiom A diffeomorphism. Suppose that f has an attractor Ω with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on Ω. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.

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Section: Markov Partitionsmentioning

confidence: 99%

Fourier decay of equilibrium states for bunched attractors

Leclerc

2024

Nonlinearity

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“…Another generalization of uniformly hyperbolic dynamical systems is the theory of nonuniformly hyperbolic dynamical systems ( Pesin theory ); a detailed account can be found in [14, 15]; see also [7, 32, 69].…”

Section: Attractors With Nonuniformly Hyperbolic Structurementioning

confidence: 99%

“…We consider partially hyperbolic attractors [13, 15, 24, 46, 68, 69, 75], hyperbolic attractors with singularities [32, 66] and attractors with nonuniformly hyperbolic structure [14, 15], endowed with Gibbs

u

‐measures [32, 59, 69, 70, 89]. We construct Laplacians and diffusions that act (respectively, move) leafwise in the unstable directions and are self‐adjoint (respectively, symmetric) with respect to the given Gibbs

u

‐measure.…”

Section: Introductionmentioning

confidence: 99%

Self‐adjoint Laplacians and symmetric diffusions on hyperbolic attractors

Alikhanloo

Hinz

2023

Journal of London Math Soc

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We construct self‐adjoint Laplacians and symmetric Markov semigroups on hyperbolic attractors, endowed with Gibbs u$u$‐measures. If the measure has full support, we can also conclude the existence of an associated symmetric diffusion process. In the special case of partially hyperbolic diffeomorphisms induced by geodesic flows on negatively curved manifolds the Laplacians we consider are self‐adjoint extensions of well‐known classical leafwise Laplacians. We observe a quasi‐invariance property of energy densities in the u$u$‐conformal case and the existence of nonconstant functions of zero energy.

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Markov Approximations and Statistical Properties of Billiards

Szász

2019

The Abel Prize

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However, for hyperbolic dynamical systems which are either singular or whose hyperbolicity is nonuniform, the construction of a Markov partition, which in these cases is necessarily countable, is a rather delicate issue even when such a construction exists. An additional problem is the use of a countable Markov partition for proving probabilistic statements. For a wide class of hyperbolic systems, L. S. Young, in 1998, constructed so called Markov towers, which she could apply successfully to establish nice, for instance, exponential correlation decay, and, moreover, as a consequence, a central limit theorem. The aim of this survey is twofold. First we show how the Markov tower construction is applicable for obtaining finer stochastic properties, like a local limit theorem of probability theory. Here the fundamental method is the study of the spectrum of the Fourier transform of the Perron-Frobenius operator. These ideas and results are applicable to all systems Young has been considering. Second, we survey the problem of recurrence of the planar Lorentz process. As an application of the results from the first part, we obtain a dynamical proof of recurrence for the finite horizon case. Here basically different proofs were given by K. Schmidt, in 1998, and J.-P. Conze, in 1999. As another application we can also treat the infinite horizon case, where already the global limit theorem is absolutely novel. It is not a central one, the scaling is √ n log n in contrast to the classical √ n one. Beyond thus giving a rigorous proof for earlier heuristic ideas of P. Bleher, which used three delicate and hard hypotheses, we can also a) verify the local version of this limit theorem for the free flight function and b) prove the recurrence of the planar Lorentz process in the infinite horizon case.

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Sinai’s Work on Markov Partitions and SRB Measures

Cited by 5 publications

References 58 publications

Fourier decay of equilibrium states for bunched attractors

Fourier decay of equilibrium states for bunched attractors

Self‐adjoint Laplacians and symmetric diffusions on hyperbolic attractors

Markov Approximations and Statistical Properties of Billiards

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