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), 17-27
MSC (2000):
Primary 37C20, 37C27, 37C35, 34C25, 34C27
DOI:
https://doi.org/10.1090/S1079-6762-01-00090-7
Published electronically:
April 18, 2001
MathSciNet review:
1826992
Full-text PDF Free Access
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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.
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[A]A Problems of Arnold’s seminar, FAZIS, Moscow, 2000.
[AM]AM M. Artin and B. Mazur, On periodic points, Ann. Math. 81 (1965), 82–99.
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[K3]K3 V. Yu. Kaloshin, Some prevalent properties of smooth dynamical systems, Tr. Mat. Inst. Steklova 213 (1997), 123–151.
[K4]K4 V. Yu. Kaloshin, Ph.D. thesis, Princeton University, 2001.
[KH]KH V. Kaloshin, B. Hunt, Stretched exponential bound on growth of the number of periodic points for prevalent diffeomorphisms, part 2, in preparation.
[KK]KK V. Kaloshin, O. Kozlovski, An example of a $C^2$-unimodal map with an arbitrarily fast growth of the number of periodic points, in preparation
[MMS]MMS M. Martens, W. de Melo, S. Van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, Acta Math. 168 (1992), no. 3-4, 273–318.
[PM]PM J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, 1982.
[O]O J. C. Oxtoby, Measure and Category, Springer-Verlag, 1971.
[Sac]Sac R. J. Sacker, A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech. 18 (1969), 705–762.
[VK]VK M. Vishik, S. Kuksin, Quasilinear elliptic equations and Fredholm manifolds, Moscow Univ. Math. Bull. 40 (1985), no. 6, 26–34.
[W]W H. Whitney, Differentiable manifolds, Ann. Math. 37 (1936), 645–680.
[Y]Y Y. Yomdin, A quantitative version of the Kupka-Smale Theorem, Ergod. Th. Dynam. Sys. 5 (1985), 449–472.
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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
