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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.

Copyright 2001 American Mathematical Society

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Article Info

• ERA Amer. Math. Soc. 07 (2001), pp. 17-27
• Publisher Identifier: S 1079-6762(01)00090-7
• 2000 Mathematics Subject Classification. Primary 37C20, 37C27, 37C35, 34C25, 34C27
• Key words and phrases. Periodic points, prevalence, diffeomorphisms
• Received by the editors December 21, 2000
• Posted on April 18, 2001
• Communicated by Svetlana Katok
• Comments (When Available)

Vadim Yu. Kaloshin

Fine Hall, Princeton University, Princeton, NJ 08544

E-mail address: kaloshin@math.princeton.edu

Brian R. Hunt

Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, MD 20742

E-mail address: bhunt@ipst.umd.edu