# American Institute of Mathematical Sciences

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

 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 $\gamma\in[\alpha,\beta]$.
Citation: Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589
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