Lyapunov optimization for non-generic one-dimensional expanding markov maps

Mao Shinoda; Hiroki Takahasi

Lyapunov optimization for non-generic one-dimensional expanding markov maps

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

For a non-generic, yet dense subset of C¹ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C¹ perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

37A40, 37C40, 37D05, 37D35, 37E05.

37A40, 37C40, 37D05, 37D35, 37E05

Original language	English
Journal	Unknown Journal
Publication status	Published - 2017 May 22

Keywords

Expanding Markov map
Lyapunov optimizing measure
Non-generic property

ASJC Scopus subject areas

General

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abstract = "For a non-generic, yet dense subset of C1 expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some H{\"o}lder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C1 perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.37A40, 37C40, 37D05, 37D35, 37E05.37A40, 37C40, 37D05, 37D35, 37E05",

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AU - Takahasi, Hiroki

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N2 - For a non-generic, yet dense subset of C1 expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C1 perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.37A40, 37C40, 37D05, 37D35, 37E05.37A40, 37C40, 37D05, 37D35, 37E05

AB - For a non-generic, yet dense subset of C1 expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new C1 perturbation lemma which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.37A40, 37C40, 37D05, 37D35, 37E05.37A40, 37C40, 37D05, 37D35, 37E05

KW - Expanding Markov map

KW - Lyapunov optimizing measure

KW - Non-generic property

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