Critical covering maps without absolutely continuous invariant probability measure

Simon Lloyd; Edson Vargas

doi:10.3934/dcds.2019101

Critical covering maps without absolutely continuous invariant probability measure

Simon Lloyd^*
, Edson Vargas

^*Corresponding author for this work

Universidade de São Paulo

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

Abstract

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than 1. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.

Original language	English
Pages (from-to)	2393-2412
Number of pages	20
Journal	Discrete and Continuous Dynamical Systems
Volume	39
DOIs	https://doi.org/10.3934/dcds.2019101
Publication status	Published - May 2019

Keywords

Arnold family
Ergodicity
Kneading map

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10.3934/dcds.2019101

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abstract = "We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than 1. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.",

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AU - Vargas, Edson

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AB - We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than 1. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.

KW - Arnold family

KW - Ergodicity

KW - Kneading map

UR - https://www.scopus.com/pages/publications/85061341532

U2 - 10.3934/dcds.2019101

DO - 10.3934/dcds.2019101

M3 - Article

AN - SCOPUS:85061341532

SN - 1078-0947

VL - 39

SP - 2393

EP - 2412

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

ER -

Critical covering maps without absolutely continuous invariant probability measure

Abstract

Keywords

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