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), 1727
MSC (2000):
Primary 37C20, 37C27, 37C35, 34C25, 34C27
DOI:
https://doi.org/10.1090/S1079676201000907
Published electronically:
April 18, 2001
MathSciNet review:
1826992
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Abstract  References  Similar Articles  Additional Information
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 measuretheoretic 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 measuretheoretically 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
MR Author ID:
624885
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
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