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

DOI:
https://doi.org/10.1090/S1079-6762-01-00090-7

Published electronically:
April 18, 2001

MathSciNet review:
1826992

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Abstract | References | Similar Articles | Additional Information

For diffeomorphisms of smooth compact manifolds, we consider the problem of how fast the number of periodic points with period grows as a function of . 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 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 , there is a prevalent set of ( or smoother) diffeomorphisms for which the number of period points is bounded above by for some independent of . We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of . 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

Email:
kaloshin@math.princeton.edu

**Brian R. Hunt**

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

Email:
bhunt@ipst.umd.edu

DOI:
https://doi.org/10.1090/S1079-6762-01-00090-7

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