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

V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American Mathematical Society 7 (2001) 17–27.

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Journal Article | Published | English
Author
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.
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/s1079-6762-01-00090-7
Kaloshin, 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, 7(4), 17–27. https://doi.org/10.1090/s1079-6762-01-00090-7
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 7, no. 4 (2001): 17–27. 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, pp. 17–27, 2001.
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.
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.

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