Luapunov Exponents Vary Continuously With Respect to Parameter

• Published: 21 August 2021

Lyapunov Exponents Everywhere and Rigidity

Journal of Dynamical and Control Systems volume 27,pages 819–831 (2021)Cite this article

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Abstract

In the present work, we obtain rigidity results analyzing the set of regular points, in the sense of Oseledec's Theorem. It is presented a study on the possibility of Anosov diffeomorphisms having all Lyapunov exponents defined everywhere. We prove that this condition implies local rigidity of an Anosov automorphism of the torus \(\mathbb {T}^{d}, d \geq 3,\) C ¹ −close to a linear automorphism diagonalizable over \(\mathbb {R}\) and such that its characteristic polynomial is irreducible over \(\mathbb {Q}.\)

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Acknowledgements

F. Micena appreciates the unconditional support of his family. Also, Micena is grateful to Rafael de la Llave for the opportunity to write an article with him. The authors thank the anonymous referees for their suggestions and valuable comments.

Funding

R.L. was partially supported by NSF grant DMS-1800241.

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Authors and Affiliations

1. Instituto de Matemática e Computação, UNIFEI - Universidade Federal de Itajubá, Av. BPS, 1303, Itajubá, MG, CEP 37500-903, Brazil

   Fernando Pereira Micena

2. School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332-1160, USA

   Rafael de la Llave

Corresponding author

Correspondence to Fernando Pereira Micena.

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Micena, F.P., Llave, R.d.l. Lyapunov Exponents Everywhere and Rigidity. J Dyn Control Syst 27, 819–831 (2021). https://doi.org/10.1007/s10883-021-09563-0

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• Received: 16 February 2021

• Revised: 22 June 2021

• Accepted: 13 July 2021

• Published: 21 August 2021

• Issue Date: October 2021

• DOI : https://doi.org/10.1007/s10883-021-09563-0

Keywords

• Lyapunov exponents
• Anosov diffeomorphisms
• Rigidity

Mathematics Subject Classification (2010)

• MSC 37D20
• MSC 37D25

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