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

Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements of the American Mathematical Society. 7(4), 17–27.

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*Journal Article*|

*Published*|

*English*

Author

Kaloshin, Vadim

^{IST Austria}^{}; Hunt, Brian R.Abstract

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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Publishing Year

Date Published

2001-04-18

Journal Title

Electronic Research Announcements of the American Mathematical Society

Volume

7

Issue

4

Page

17-27

ISSN

IST-REx-ID

### Cite this

Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I.

*Electronic Research Announcements of the American Mathematical Society*. 2001;7(4):17-27. doi:10.1090/s1079-6762-01-00090-7Kaloshin, V., & Hunt, B. R. (2001). A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I.

*Electronic Research Announcements of the American Mathematical Society*. American Mathematical Society. https://doi.org/10.1090/s1079-6762-01-00090-7Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.”

*Electronic Research Announcements of the American Mathematical Society*. American Mathematical Society, 2001. https://doi.org/10.1090/s1079-6762-01-00090-7.V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I,”

*Electronic Research Announcements of the American Mathematical Society*, vol. 7, no. 4. American Mathematical Society, pp. 17–27, 2001.Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.”

*Electronic Research Announcements of the American Mathematical Society*, vol. 7, no. 4, American Mathematical Society, 2001, pp. 17–27, doi:10.1090/s1079-6762-01-00090-7.