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The entropy of $ C^1$-diffeomorphisms without a dominated splitting


Authors: Jérôme Buzzi, Sylvain Crovisier and Todd Fisher
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 37C15; Secondary 37B40, 37D05, 37D30
DOI: https://doi.org/10.1090/tran/7380
Published electronically: March 20, 2018
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Abstract: A classical construction due to Newhouse creates horseshoes from hyperbolic periodic orbits with large period and weak domination through local $ C^1$-perturbations. Our main theorem shows that when one works in the $ C^1$-topology, the entropy of such horseshoes can be made arbitrarily close to an upper bound following from Ruelle's inequality, i.e., the sum of the positive Lyapunov exponents (or the same for the inverse diffeomorphism, whichever is smaller).

This optimal entropy creation yields a number of consequences for $ C^1$-generic diffeomorphisms, especially in the absence of a dominated splitting. For instance, in the conservative settings, we find formulas for the topological entropy, deduce that the topological entropy is continuous but not locally constant at the generic diffeomorphism, and prove that these generic diffeomorphisms have no measure of maximum entropy. In the dissipative setting, we show the locally generic existence of infinitely many homoclinic classes with entropy bounded away from zero.


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Additional Information

Jérôme Buzzi
Affiliation: Laboratoire de Mathématiques d’Orsay, CNRS - UMR 8628, Université Paris-Sud 11, 91405 Orsay, France
Email: jerome.buzzi@math.u-psud.fr

Sylvain Crovisier
Affiliation: Laboratoire de Mathématiques d’Orsay, CNRS - UMR 8628, Université Paris-Sud 11, 91405 Orsay, France
Email: sylvain.crovisier@math.u-psud.fr

Todd Fisher
Affiliation: Department of Mathematics, Brigham Young University, Provo, Utah 84602
Email: tfisher@math.byu.edu

DOI: https://doi.org/10.1090/tran/7380
Keywords: Topological entropy, measure theoretic entropy, dominated splitting, homoclinic tangency, homoclinic class, Lyapunov exponent.
Received by editor(s): December 28, 2016
Published electronically: March 20, 2018
Additional Notes: The second author was partially supported by the ERC project 692925 NUHGD
The third author was supported by Simons Foundation grant #239708.
Article copyright: © Copyright 2018 American Mathematical Society

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