Non-uniform hyperbolicity of maps on \(\mathbb{T}^2\)

In this paper we prove that the homotopy class of non-homothety linear endomorphisms on \(\mathbb{T}^2\) with determinant greater than 2 contains a \(C^1\) open set of non-uniformly hyperbolic endomorphisms. Furthermore, we prove that the homotopy class of non-hyperbolic elements (having either \(1\...

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Bibliographic Details
Published in:arXiv.org 2024-09
Main Authors: Ramírez, Sebastián, Vivas, Kendry J
Format: Article
Language:English
Subjects:
Eigenvalues
Online Access:Get full text
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Summary:In this paper we prove that the homotopy class of non-homothety linear endomorphisms on \(\mathbb{T}^2\) with determinant greater than 2 contains a \(C^1\) open set of non-uniformly hyperbolic endomorphisms. Furthermore, we prove that the homotopy class of non-hyperbolic elements (having either \(1\) or \(-1\) as an eigenvalue) whose degree is large enough contains non-uniformly hyperbolic endomorphisms that are also \(C^2\) stably ergodic. These results provide partial answers to certain questions posed in arXiv:2206.08295v2
ISSN:2331-8422