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


Authors: Vadim Yu. Kaloshin and Brian R. Hunt
Journal: Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 28-36
MSC (2000): Primary 37C20, 37C27, 37C35, 34C25, 34C27
Published electronically: April 24, 2001
MathSciNet review: 1826993
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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.


References [Enhancements On Off] (What's this?)

  • [GG] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518 (49 #6269)
  • [GY] A. Grigoriev and S. Yakovenko, Topology of generic multijet preimages and blow-up via Newton interpolation, J. Differential Equations 150 (1998), no. 2, 349–362. MR 1658601 (99m:58028), http://dx.doi.org/10.1006/jdeq.1998.3493
  • [K4] V. Yu. Kaloshin, Ph.D. thesis, Princeton University, 2001.
  • [K5] V. Kaloshin, Stretched exponential bound on growth of the number of periodic points for prevalent diffeomorphisms, part 1, in preparation.
  • [KH] V. Kaloshin, B. Hunt, Stretched exponential bound on growth of the number of periodic points for prevalent diffeomorphisms, part 2, in preparation.
  • [San] Luis A. Santaló, Integral geometry and geometric probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR 0433364 (55 #6340)

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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: http://dx.doi.org/10.1090/S1079-6762-01-00091-9
PII: S 1079-6762(01)00091-9
Keywords: Periodic points, prevalence, diffeomorphisms
Received by editor(s): December 21, 2000
Published electronically: April 24, 2001
Communicated by: Svetlana Katok
Article copyright: © Copyright 2001 American Mathematical Society