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

### Vadim Yu. Kaloshin and Brian R. Hunt

**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.

*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`

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