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Non-uniformly hyperbolic endomorphisms

Part of: Ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  01 September 2025

Martin Andersson Open the ORCID record for Martin Andersson [Opens in a new window] ,
Pablo D. Carrasco Open the ORCID record for Pablo D. Carrasco [Opens in a new window]  and
Radu Saghin Open the ORCID record for Radu Saghin [Opens in a new window]
Martin Andersson
Affiliation:
Universidade Federal Fluminense, Departamento de Matemática aplicada, Rua Professor Marcos Waldemar de Freitas Reis, 24210-201, Niterói, RJ, Brazil nilsmartin@id.uff.br
Pablo D. Carrasco
Affiliation:
Universidade Federal de Minas Gerais, ICEx-UFMG, Av. Pres. Antônio Carlos, 6627 - 31270-901, Belo Horizonte, MG, Brazil pdcarrasco@gmail.com
Radu Saghin
Affiliation:
Pontificia Universidad Católica de Valparaíso, Instituto de Matemática, Blanco Viel 596, 2350050, Cerro Barón, Valparaíso, Chile rsaghin@gmail.com
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Abstract

We show the existence of large $\mathcal C^1$ open sets of area-preserving endomorphisms of the two-torus which have no dominated splitting and are non-uniformly hyperbolic, meaning that Lebesgue almost every point has a positive and a negative Lyapunov exponent. The integrated Lyapunov exponents vary continuously with the dynamics in the $\mathcal C^1$ topology and can be taken as far away from zero as desired. Explicit real analytic examples are obtained by deforming linear endomorphisms, including expanding ones. The technique works in nearly every homotopy class, and the examples are stably ergodic (in fact Bernoulli), provided that the linear map has no eigenvalue of modulus one.

Keywords

non-uniform hyperbolicityLyapunov exponentsstable ergodicity

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.) 37A05: Measure-preserving transformations

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 6 , June 2025 , pp. 1313 - 1356
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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