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Anosov endomorphisms on the two-torus: regularity of foliations and rigidity

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We provide sufficient conditions for smooth conjugacy between two Anosov endomorphisms on the two-torus. From that, we also explore how the regularity of the stable and unstable foliations implies smooth conjugacy inside a class of endomorphisms including, for instance, the ones with constant Jacobian. As a consequence, we have in this class a characterisation of smooth conjugacy between special Anosov endomorphisms (defined as those having only one unstable direction for each point) and their linearisations.
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Nonlinearity
Nonlinearity 36 (2023) 5334–5357 https://doi.org/10.1088/1361-6544/acf267
Anosov endomorphisms on the two-torus:
regularity of foliations and rigidity
Marisa Cantarino1,3,and Régis Var˜
ao2,4
1School of Mathematical Sciences, Monash University, Clayton, VIC, 3800,
Australia
2Instituto de Matemática, Estatística e Computaç˜
ao Cientíca, Universidade
Estadual de Campinas—UNICAMP, Rua Sérgio Buarque de Holanda, 651, Cidade
Universitária, Campinas, SP, Brazil
E-mail: marisa.cantarino@monash.edu
Received 25 July 2022; revised 18 July 2023
Accepted for publication 21 August 2023
Published 1 September 2023
Recommended by Dr Rafael de la Llave
Abstract
We provide sufcient conditions for smooth conjugacy between two Anosov
endomorphisms on the two-torus. From that, we also explore how the regularity
of the stable and unstable foliations implies smooth conjugacy inside a class
of endomorphisms including, for instance, the ones with constant Jacobian. As
a consequence, we have in this class a characterisation of smooth conjugacy
between special Anosov endomorphisms (dened as those having only one
unstable direction for each point) and their linearisations.
Keywords: smooth dynamics, hyperbolic dynamics, non-invertible dynamics
Mathematics Subject Classication numbers: Primary: 37C15; Secondary:
37D20
1. Introduction
In this work we study rigidity results for Anosov endomorphisms: hyperbolic maps that are
not necessarily invertible. The term rigidity is associated with the idea that the value of an
invariant or a specic property of the system determines its dynamics. In our case, we want
to determine the smooth conjugacy class of the system, and the invariant will be given by its
Lyapunov exponents.
3M C was partially nanced by the Coordenaç˜
ao de Aperfeiçoamento de Pessoal de Nível Superior—Brasil—Grant
88882.333632/2019-01 and the Fundaç˜
ao Carlos Chagas Filho de Amparo `
a Pesquisa of the State of Rio de Janeiro
E-26/202.014/2022.
4R V was partially nanced by CNPq and Fapesp Grants 18/13481-0 and 17/06463-3.
Author to whom any correspondence should be addressed.
1361-6544/23/+24$33.00 © 2023 IOP Publishing Ltd & London Mathematical Society Printed in the UK 5334
... (We remark that if the linear part is a hyperbolic endomorphism, such a map may not exist. See [CVa23].) The map h is called a semiconjugacy from f to A. When h is a homeomorphism, we say that it is a conjugacy between f and A. ...
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