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STABLE SETS OF CERTAIN NON-UNIFORMLY HYPERBOLIC HORSESHOES HAVE THE EXPECTED DIMENSION

Part of: Smooth dynamical systems: general theory Measure-theoretic ergodic theory Low-dimensional dynamical systems

Published online by Cambridge University Press:  04 April 2019

Carlos Matheus ,
Jacob Palis  and
Jean-Christophe Yoccoz
Carlos Matheus
Affiliation:
Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), F-93439, Villetaneuse, France (matheus@impa.br)
Jacob Palis
Affiliation:
IMPA, Estrada D. Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil (jpalis@impa.br)
Jean-Christophe Yoccoz
Affiliation:
Collège de France, 3, Rue d’Ulm, Paris, CEDEX 05, France (jean-c.yoccoz@college-de-france.fr)
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Abstract

We show that the stable and unstable sets of non-uniformly hyperbolic horseshoes arising in some heteroclinic bifurcations of surface diffeomorphisms have the value conjectured in a previous work by the second and third authors of the present paper. Our results apply to first heteroclinic bifurcations associated with horseshoes with Hausdorff dimension ${<}22/21$ of conservative surface diffeomorphisms.

Keywords

heteroclinic bifurcationsnon-uniformly hyperbolic horseshoesHausdorff dimension

MSC classification

Primary: 37C29: Homoclinic and heteroclinic orbits 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces
Secondary: 28D20: Entropy and other invariants
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 1 , January 2021 , pp. 305 - 329
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Copyright
© Cambridge University Press 2019

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References

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Matheus, C., Palis, J. and Yoccoz, J.-C., The Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes, work in progress.Google Scholar
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