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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