Critical covering maps without absolutely continuous invariant probability measure

Simon Lloyd; Edson Vargas

doi:10.3934/dcds.2019101

Article Contents

2019, Volume 39, Issue 5: 2393-2412. Doi: 10.3934/dcds.2019101

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Critical covering maps without absolutely continuous invariant probability measure

Simon Lloyd^1, , and
Edson Vargas^2,

1.
Department of Mathematical Sciences, Xi'an Jiaotong–Liverpool University, 111 Ren'ai Road, Suzhou 215123, China
2.
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo CEP 05508-090, Brazil

^* Corresponding author: Simon Lloyd
^* Corresponding author: Simon Lloyd

Received: October 2017

Revised: October 2018

Published: January 2019

The first author is supported by Fundação de Amparo à Pesquisa do Estado de São Paulo grant numbers 2011/01482-3 and 2017/10106-1. The second author is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico grant number 310749/2015-8

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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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Mathematics Subject Classification: 37E05, 37C40.

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Figure 1. The graph of the critical covering map $f_1$

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Figure 2. The first entry map to the interval $V_{2n-1}^+$, showing the critical branch $\phi_n$ and some of the branches $\sigma_{k,\ell_k}$, with $k \geq n$

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Figure 3. Comparing graphs of the functions $\mathcal{T}_1$ (left) and $\mathcal{T}_2$ (right)

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Figure 4. The graph of $\mathcal{T}$

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