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July  2008, 20(3): 589-604. doi: 10.3934/dcds.2008.20.589

Super-exponential growth of the number of periodic orbits inside homoclinic classes

Christian Bonatti 1, , Lorenzo J. Díaz 2, and  Todd Fisher 3,

1. 

Université de Bourgogne, Laboratoire de Topologie, UMR 5584 du CNRS, BP 47 870, 21078 Dijon Cedex, France
2. 

Dep. Matemática PUC-Rio, Marquês de São Vicente 225 22453-900, Rio de Janeiro, Brazil
3. 

Department of Mathematics, Brigham Young University, Provo, UT 84602, United States

Received  January 2007 Revised  August 2007 Published  December 2007

We show that there is a residual subset $\S (M)$ of Diff$^1$ (M) such that, for every $f\in \S(M)$, any homoclinic class of $f$ containing periodic saddles $p$ and $q$ of indices $\alpha$ and $\beta$ respectively, where $\alpha< \beta$, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown that the super-exponential growth occurs for hyperbolic periodic points of index $\gamma$ inside the homoclinic class for every $\gamma\in[\alpha,\beta]$.
Keywords: index of a saddle, Artin-Mazur diffeomorphism, heterodimensional cycle, chain recurrence class, homoclinic class, symbolic extensions..
Mathematics Subject Classification: Primary: 37C05, 37C20, 37C25, 37C29, 37D3.
Citation: Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589
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