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

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

**Abstract.**
We continue the previous article's discussion of bounds, for prevalent
diffeomorphisms of smooth compact manifolds, on the growth of the
number of periodic points and the decay of their hyperbolicity as a
function of their period $n$. In that article we reduced the main
results to a problem, for certain families of diffeomorphisms, of
bounding the measure of parameter values for which the diffeomorphism
has (for a given period $n$) an almost periodic point that is almost
nonhyperbolic. We also formulated our results for $1$-dimensional
endomorphisms on a compact interval. In this article we describe some
of the main techniques involved and outline the rest of the proof. To
simplify notation, we concentrate primarily on the $1$-dimensional case.

*Copyright 2001 American Mathematical Society*

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#### Article Info

- ERA Amer. Math. Soc.
**07** (2001), pp. 28-36
- Publisher Identifier: S 1079-6762(01)00091-9
- 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 24, 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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