Measures of maximal entropy for surface diffeomorphisms

Buzzi, Jérôme; Crovisier, Sylvain; Sarig, Omri

doi:10.48550/arxiv.1811.02240

Cited by 7 publications

(24 citation statements)

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“…Moreover, if the diffeomorphism is transitive 5 then there is only one m.m.e. (see [BCS18]). Since for a generic large parameter the standard map has elliptic islands (see [Du94]), it is not transitive for these parameters.…”

Section: Measures Of Maximal Entropy and Main Resultsmentioning

confidence: 96%

“…The study of existence and finiteness of ergodic measures of maximal entropy has a long history, which we will not try to state here. We refer the reader to Section 1.9 in [BCS18] for many references on the history of this problem.…”

Section: Measures Of Maximal Entropy and Main Resultsmentioning

confidence: 99%

“…for g n , for each n ∈ N. Then any accumulation for the weak * -topology of the sequence (µ n ) n∈N is a m.m.e. for g (see for instance Section 5.1 in [BCS18]). Let P(T 2 ) be the set of probability measures of T 2 endowed with the weak * -topology.…”

Section: Theorem Bmentioning

confidence: 99%

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Uniqueness of the measure of maximal entropy for the standard map

Obata

2020

Preprint

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In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e. We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the measure on the manifold. For a generic large parameter, we prove that the support of the m.m.e. has Hausdorff dimension 2. We also obtain the C 2 -robustness of several of these properties. Contents 1 Introduction 2 Preliminaries 3 Estimates on invariant manifolds for measures with large exponents 4 Homoclinic relation for measures with large exponents 5 Proof of Theorem A 6 Growth and equidistribution of periodic points: proof of Theorem B 7 Further properties of the m.m.e.: proof of Theorems C and D Bibliography * D.O. was supported by the projects ANR BEKAM : ANR-15-CE40-0001 and ERC project NUHGD.

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Section: Measures Of Maximal Entropy and Main Resultsmentioning

confidence: 96%

Section: Measures Of Maximal Entropy and Main Resultsmentioning

confidence: 99%

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Uniqueness of the measure of maximal entropy for the standard map

Obata

2020

Preprint

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“…To see this, we need to review the steps in the proof of [4, Theorem B'], which is a combination of [4, Theorem B] and Katok's approximation theorem [32] and its extensions in [4,Theorem. 3.3] and [22,Theorem 2.12].…”

Section: 2mentioning

confidence: 99%

Homoclinic tangencies leading to robust heterodimensional cycles

Barrientos¹,

Díaz²,

Pérez³

2021

Preprint

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We consider C r (r 1) diffeomorphisms f defined on manifolds of dimension 3 with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism f can be C r approximated by diffeomorphisms with C 1 robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka's examples of diffeomorphisms with C 1 robust homoclinic tangencies also display C 1 robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain C 1 robust cycles after C 1 perturbations.

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“…If moreover the system is topologically mixing, then the SRB measure when it exists is Bernoulli [16]. By the spectral decomposition of C ∞ surface diffeomorphisms [16] there are at most finitely many ergodic SRB measures with entropy and thus maximal exponent larger than a given positive constant. Therefore, in the Main Theorem, the set Λ = {χ(µ i ), i ∈ I} is either finite or a sequence decreasing to zero.…”

Section: Introductionmentioning

confidence: 99%