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A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I

Authors: Vadim Yu. Kaloshin and Brian R. Hunt
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 17-27
MSC (2000): Primary 37C20, 37C27, 37C35, 34C25, 34C27
Published electronically: April 18, 2001
MathSciNet review: 1826992
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For diffeomorphisms of smooth compact manifolds, we consider the problem of how fast the number of periodic points with period $n$grows as a function of $n$. In many familiar cases (e.g., Anosov systems) the growth is exponential, but arbitrarily fast growth is possible; in fact, the first author has shown that arbitrarily fast growth is topologically (Baire) generic for $C^2$ or smoother diffeomorphisms. In the present work we show that, by contrast, for a measure-theoretic notion of genericity we call ``prevalence'', the growth is not much faster than exponential. Specifically, we show that for each $\delta > 0$, there is a prevalent set of ( $C^{1+\rho}$ or smoother) diffeomorphisms for which the number of period $n$ points is bounded above by $\operatorname{exp}(C n^{1+\delta})$ for some $C$ independent of $n$. We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of $n$. The contrast between topologically generic and measure-theoretically generic behavior for the growth of the number of periodic points and the decay of their hyperbolicity shows this to be a subtle and complex phenomenon, reminiscent of KAM theory.

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Additional Information

Vadim Yu. Kaloshin
Affiliation: Fine Hall, Princeton University, Princeton, NJ 08544

Brian R. Hunt
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

Keywords: Periodic points, prevalence, diffeomorphisms
Received by editor(s): December 21, 2000
Published electronically: April 18, 2001
Communicated by: Svetlana Katok
Article copyright: © Copyright 2001 American Mathematical Society