Equilibrium states for a class of skew products

Carvalho, Maria; Pérez, Sebastián A.

doi:10.1017/etds.2019.32

Cited by 5 publications

(5 citation statements)

References 39 publications

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“…For potentials with a small variational condition, Rios and Siqueira [24] obtained uniqueness of equilibrium states for partially hyperbolic horseshoes. For the same class of potentials that we will study in this work, Carvalho and Pérez [10] obtained similar results about equilibrium states for a class of skew product. Recently, Climenhaga, Pesin and Zelerowicz [13] studied uniqueness of equilibrium states for certain transitive partially hyperbolic diffeomorphisms.…”

Section: Introductionsupporting

confidence: 67%

Equilibrium states for maps isotopic to Anosov

2021

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In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on T 4 , with two-dimensional center foliation. To do so we propose to study the disintegration of measures along one-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context.

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Section: Introductionsupporting

confidence: 67%

Equilibrium states for maps isotopic to Anosov

2021

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“…[25]). For Shub's examples the uniqueness of the measure of maximal entropy was obtained in [35] (a generalization for equilibrium states may be read in [16]). Nevertheless, without additional assumptions this measure may not describe the distribution of the periodic points and the topological entropy may be different from the periodic one.…”

Section: Resultsmentioning

confidence: 99%

Periodic points and measures for a class of skew products

Carvalho¹,

Pérez²

2019

Preprint

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We consider the open set constructed by M. Shub in [42] of partially hyperbolic skew products on the space T 2 ×T 2 whose non-wandering set is not stable. We show that there exists an open set U of such diffeomorphisms such that if F S ∈ U then its measure of maximal entropy is unique, hyperbolic and, generically, describes the distribution of periodic points. Moreover, the non-wandering set of such an F S ∈ U contains closed invariant subsets carrying entropy arbitrarily close to the topological entropy of F S and within which the dynamics is conjugate to a subshift of finite type. Under an additional assumption on the base dynamics, we verify that F S preserves a unique SRB measure, which is physical, whose basin has full Lebesgue measure and coincides with the measure of maximal entropy. We also prove that there exists a residual subset R of U such that if F S ∈ R then the topological and periodic entropies of F S are equal, F S is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding.

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“…It is also worth mentioning [15], which includes the same construction although the setting is somewhat different. We also mention [13], where the authors prove existence and uniqueness of equilibrium states for the Shub example on T 4 . As in our case, they consider potentials which are well projected along certain central foliation to the two-torus where the dynamic is Anosov.…”

Section: Corollary Bmentioning

confidence: 99%

“…We state that an f -invariant measure μ maximizes the entropy if its metric entropy realizes the supremum in (13): that is,…”

Section: Topological Pressure and Equilibrium Statesmentioning

confidence: 99%

Measures maximizing the entropy for Kan endomorphisms

2021

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In 1994, Kan provided the first example of maps with intermingled basins. The Kan example corresponds to a partially hyperbolic endomorphism defined on a surface, with the boundary exhibiting two intermingled hyperbolic physical measures. Both measures are supported on the boundary, and they also maximize the topological entropy. In this work, we prove the existence of a third hyperbolic measure supported in the interior of the cylinder that maximizes the entropy. We also prove this statement for a larger class of invariant measures of large class maps including perturbations of the Kan example.

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Equilibrium states for a class of skew products

Cited by 5 publications

References 39 publications

Equilibrium states for maps isotopic to Anosov

Equilibrium states for maps isotopic to Anosov

Periodic points and measures for a class of skew products

Measures maximizing the entropy for Kan endomorphisms

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