# American Institute of Mathematical Sciences

May  2019, 39(5): 2393-2412. doi: 10.3934/dcds.2019101

## Critical covering maps without absolutely continuous invariant probability measure

 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

Received  October 2017 Revised  October 2018 Published  January 2019

Fund Project: 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.

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.

Citation: Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101
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##### References:
The graph of the critical covering map $f_1$
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 $k \geq n$
Comparing graphs of the functions $\mathcal{T}_1$ (left) and $\mathcal{T}_2$ (right)
The graph of $\mathcal{T}$
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