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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Extended flexibility of Lyapunov exponents for Anosov diffeomorphisms
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by Pablo D. Carrasco and Radu Saghin PDF
Trans. Amer. Math. Soc. 375 (2022), 3411-3449 Request permission


Bochi-Katok-Rodriguez Hertz proposed recently a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples of Anosov diffeomorphisms with the strong unstable exponent larger than the strong unstable exponent of the linear part. We also give examples of derived from Anosov diffeomorphisms with the metric entropy larger than the entropy of the linear part.

These results rely on a new type of deformation which goes beyond the previous Shub-Wilkinson and Baraviera-Bonatti techniques for conservative systems having some invariant directions. In order to estimate the Lyapunov exponents even after breaking the invariant bundles, we obtain an abstract result which gives bounds on exponents of some specific cocycles and which can be applied in various other settings. We also include various interesting comments in the appendices: our examples are $\mathcal {C}^2$ robust, the Lyapunov exponents are continuous (with respect to the map) even after breaking the invariant bundles, a similar construction can be obtained for the case of multiple eigenvalues.

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Additional Information
  • Pablo D. Carrasco
  • Affiliation: ICEx-UFMG, Avda. Presidente Antonio Carlos 6627, Belo Horizonte-MG BR31270-90, Brazil
  • MR Author ID: 1063404
  • ORCID: 0000-0002-1442-9436
  • Email:
  • Radu Saghin
  • Affiliation: Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
  • MR Author ID: 784425
  • Email:
  • Received by editor(s): April 6, 2021
  • Received by editor(s) in revised form: September 22, 2021, and October 20, 2021
  • Published electronically: January 12, 2022
  • Additional Notes: The second author was supported by Fondecyt Regular 1171477, 1210168
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 3411-3449
  • MSC (2020): Primary 37D25, 37D20, 37D30; Secondary 37A35
  • DOI:
  • MathSciNet review: 4402666