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), 1727
MSC (2000):
Primary 37C20, 37C27, 37C35, 34C25, 34C27
Published electronically:
April 18, 2001
MathSciNet review:
1826992
Fulltext 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 grows as a function of . 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 or smoother diffeomorphisms. In the present work we show that, by contrast, for a measuretheoretic notion of genericity we call ``prevalence'', the growth is not much faster than exponential. Specifically, we show that for each , there is a prevalent set of ( or smoother) diffeomorphisms for which the number of period points is bounded above by for some independent of . We also obtain a related bound on the decay of the hyperbolicity of the periodic points as a function of . The contrast between topologically generic and measuretheoretically 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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 [AM]
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 [F]
 N. Fenichel, Persistence and smoothness of invariant manifolds for flows, Indiana Univ. Math. J. 21 (1971), 193226. MR 44:4313
 [GST1]
 S. V. Gonchenko, L. P. Shil'nikov, D. V. Turaev, On models with nonrough Poincaré homoclinic curves, Physica D 62 (1993), 114. MR 94c:58098
 [GST2]
 S. Gonchenko, L. Shil'nikov, D. Turaev, Homoclinic tangencies of an arbitrary order in Newhouse regions, Preprint, in Russian.
 [G]
 M. Gromov, On entropy of holomorphic maps, Preprint.
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 [K3]
 V. Yu. Kaloshin, Some prevalent properties of smooth dynamical systems, Tr. Mat. Inst. Steklova 213 (1997), 123151. MR 99h:58100
 [K4]
 V. Yu. Kaloshin, Ph.D. thesis, Princeton University, 2001.
 [KH]
 V. Kaloshin, B. Hunt, Stretched exponential bound on growth of the number of periodic points for prevalent diffeomorphisms, part 2, in preparation.
 [KK]
 V. Kaloshin, O. Kozlovski, An example of a unimodal map with an arbitrarily fast growth of the number of periodic points, in preparation
 [MMS]
 M. Martens, W. de Melo, S. Van Strien, JuliaFatouSullivan theory for real onedimensional dynamics, Acta Math. 168 (1992), no. 34, 273318. MR 93d:58137
 [PM]
 J. Palis and W. de Melo, Geometric Theory of Dynamical Systems: An Introduction, SpringerVerlag, 1982. MR 84a:58004
 [O]
 J. C. Oxtoby, Measure and Category, SpringerVerlag, 1971. MR 52:14213
 [Sac]
 R. J. Sacker, A perturbation theorem for invariant manifolds and Hölder continuity, J. Math. Mech. 18 (1969), 705762. MR 39:578
 [VK]
 M. Vishik, S. Kuksin, Quasilinear elliptic equations and Fredholm manifolds, Moscow Univ. Math. Bull. 40 (1985), no. 6, 2634. MR 88a:35086
 [W]
 H. Whitney, Differentiable manifolds, Ann. Math. 37 (1936), 645680.
 [Y]
 Y. Yomdin, A quantitative version of the KupkaSmale Theorem, Ergod. Th. Dynam. Sys. 5 (1985), 449472. MR 87c:58091
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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/S1079676201000907
PII:
S 10796762(01)000907
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
