Invariant measures for typical continuous maps on manifolds

Catsigeras, Eleonora; Troubetzkoy, Serge

doi:10.1088/1361-6544/ab28b1

Cited by 6 publications

(13 citation statements)

References 22 publications

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“…It has been known for some time that a generic homeomorphism of a compact manifold of dimension at least 2 has infinite topological entropy. Catsigeras and Troubetzkoy [10] recently proved that a generic homeomorphism of a compact manifold of dimension at least 2 admits an ergodic measure having infinite entropy.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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A Borel Zd dynamical system (X,S) is ‘almost Borel universal’ if any free Borel Zd dynamical system (Y,T) of strictly lower entropy is isomorphic to a Borel subsystem of (X,S), after removing a null set. We obtain and exploit a new sufficient condition for a topological Zd dynamical system to be almost Borel universal. We use our main result to deduce various conclusions and answer a number of questions. Along with additional results, we prove that a ‘generic’ homeomorphism of a compact manifold of topological dimension at least two can model any ergodic transformation, that non‐uniform specification implies almost Borel universality, and that 3‐colorings in Zd and dimers in Z2 are almost Borel universal.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Chandgotia

Meyerovitch²

2021

Proc. London Math. Soc.

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“…More generally, Catsigeras has shown in Theorem 2 in [8] that for a typical map f ∈ C(M ), where M is a compact manifold of dimension m, there exists a sequence of ergodic measures µ n , with h µn (f ) = ∞, such that µ n → µ and h µ (f ) = 0; therefore, its entropy function is not upper-semicontinuous. Furthermore, for such f ∈ C(M ), it follows from Corollary 15 in [9] that the set M p (f ) is dense in M e (f ); hence, by Theorem 1.1, E e 0 (f ) is a dense G δ subset of M e (f ), although by Theorem 1 in [9], E e 0 (f ) is a meager subset of M(f ).…”

Section: As a Consequence Of Corollarymentioning

confidence: 99%

“…this is a consequence of Theorems 16 and 24 in [7] and Theorems 1 and 2 in [9]), so it is not true, for such functions, that E 0 (f ) is a dense G δ subset of M(f ).…”

Section: As a Consequence Of Corollarymentioning

confidence: 99%

On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces

Carvalho¹,

Condori²

2020

Preprint

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In this work, we show that if f is a uniformly continuous map defined over a Polish metric space, then the set of f -invariant measures with zero metric entropy is a G δ set (in the weak topology). In particular, this set is generic if the set of f -periodic measures is dense in the set of f -invariant measures. This settles a conjecture posed by Sigmund in [30], which states that the metric entropy of an invariant measure of a topological dynamical system that satisfies the periodic specification property is typically zero.We also show that if X is compact and if f is an expansive or a Lipschitz map with a dense set of periodic measures, typically the lower correlation entropy for q ∈ (0, 1) is equal to zero. Moreover, we show that if X is a compact metric space and if f is an expanding map with a dense set of periodic measures, then the set of invariant measures with packing dimension, upper rate of recurrence and upper quantitative waiting time indicator equal to zero is residual.Finally, we present an alternative proof of the fact that the set of expansive measures is a G δσ set in the set of probability measures M(X), if X is a Polish metric space and if f is uniformly continuous (this result was originally proved by Lee, Morales and Shin in [19] for compact metric spaces).

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“…Remark 2.8. Rather than the generic behaviour of h ∈ H(X, µ) (or h ∈ C(X, µ)) for a given µ, one might ask for the generic behaviour of µ ∈ M(X) h , that is, of an h-invariant Borel probability measure µ, for a given homeomorphism (or continuous map) h. It is shown in [19] that for a generic continuous h on a C 1 compact, connected manifold, ergodic measures are nowhere dense in M(X) h . On the other hand, for an irreducible diffeomorphism on a compact manifold without boundary, ergodicity is generic [44,Theorem 7.1].…”

Section: Theorem 27 ([9]mentioning

confidence: 99%

Chaotic tracial dynamics

Jacelon¹

2022

Preprint

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The action on the trace space induced by a generic automorphism of a suitable finite classifiable C * -algebra is shown to be chaotic and weakly mixing. Model C * -algebras are constructed to observe the Central Limit Theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe KK-contractible stably projectionless C * -algebras as crossed products.

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Invariant measures for typical continuous maps on manifolds

Cited by 6 publications

References 22 publications

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

Borel subsystems and ergodic universality for compact Zd‐systems via specification and beyond

On the generic behavior of the metric entropy, and related quantities, of uniformly continuous maps over Polish metric spaces

Chaotic tracial dynamics

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