Maximal Entropy Measures for Non-Accessible Topological Skew Products

Ali Tahzibi; Richard Cubas

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Maximal Entropy Measures for Non-Accessible Topological Skew Products

Ali Tahzibi, Richard Cubas

Dynamical Systems · Submitted 4 Apr 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism.

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In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism. We prove that if the whole manifold is the unique minimal invariant set saturated by unstable foliation, then either there exists a unique measure of maximal entropy which is non-hyperbolic or there are exactly two hyperbolic ergodic measures of maximal entropy.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism.	contribution 0.79	0.93	0.72	asserted 1.00
s1	We prove that if the whole manifold is the unique minimal invariant set saturated by unstable foliation, then either there exists a unique measure of maximal entropy which is non-hyperbolic or there are exactly two hyperbolic ergodic measures of maximal entropy.	result 0.99	0.94	0.64	asserted 1.00

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Original abstract

In this paper we establish a dichotomy for the ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms with one-dimensional compact center leaves which are virtually skew products over (transitive) Anosov homeomorphism. We prove that if the whole manifold is the unique minimal invariant set saturated by unstable foliation, then either there exists a unique measure of maximal entropy which is non-hyperbolic or there are exactly two hyperbolic ergodic measures of maximal entropy.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 8 sentences without a confident label

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The proof this claim is analogous to the proof of $s$-invariance in Proposition 4.2 from [ref] .

A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows · The $cu$-conditionals · 0.87
We will relate the iterations of different $cu$-functions by using the invariance under $s$-holonomy and especially the following result (See proof of Theorem C in [ref] ).

Absolute continuity, Lyapunov exponents and rigidity I : geodesic flows · The $cu$-conditionals · 0.88

Cites as background 3 papers

Following Margulis, the autors in [ref] introduced the notion of Margulis measures system ${m_x}_x∈M$ whose main feature is the existence of a constant $D>0$ such that for every $x∈M$ we have $f_*m_x=D^-1.m_f(x)$.

A dichotomy for measures of maximal entropy near time-one maps of transitive Anosov flows · Introduction · 0.87
Recently, new approaches were developed for the study of maximal measures of non-uniformly hyperbolic maps and partially hyperbolic diffeomorphisms [ref] .

Measures of maximal entropy for surface diffeomorphisms · Introduction · 0.99
For some recent progress in the more general setting of equilibrium states, see [ref] .

Equilibrium measures for some partially hyperbolic systems · Introduction · 0.98

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