---
_id: '8512'
abstract:
- lang: eng
  text: "For diffeomorphisms of smooth compact finite-dimensional 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 C2 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 ρ,δ>0, there is a prevalent set of C1+ρ (or
    smoother) diffeomorphisms for which the number of periodic n points is bounded
    above by exp(Cn1+δ) for some C independent of n. We also obtain a related bound
    on the decay of hyperbolicity of the periodic points as a function of n, and obtain
    the same results for 1-dimensional endomorphisms. 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 show this to be a subtle
    and complex phenomenon, reminiscent of KAM theory. Here in Part I we state our
    results and describe the methods we use. We complete most of the proof in the
    1-dimensional C2-smooth case and outline the remaining steps, deferred to Part
    II, that are needed to establish the general case.\r\n\r\nThe novel feature of
    the approach we develop in this paper is the introduction of Newton Interpolation
    Polynomials as a tool for perturbing trajectories of iterated maps."
article_processing_charge: No
article_type: original
author:
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
- first_name: Brian
  full_name: Hunt, Brian
  last_name: Hunt
citation:
  ama: Kaloshin V, Hunt B. Stretched exponential estimates on growth of the number
    of periodic points for prevalent diffeomorphisms I. <i>Annals of Mathematics</i>.
    2007;165(1):89-170. doi:<a href="https://doi.org/10.4007/annals.2007.165.89">10.4007/annals.2007.165.89</a>
  apa: Kaloshin, V., &#38; Hunt, B. (2007). Stretched exponential estimates on growth
    of the number of periodic points for prevalent diffeomorphisms I. <i>Annals of
    Mathematics</i>. Princeton University Press. <a href="https://doi.org/10.4007/annals.2007.165.89">https://doi.org/10.4007/annals.2007.165.89</a>
  chicago: Kaloshin, Vadim, and Brian Hunt. “Stretched Exponential Estimates on Growth
    of the Number of Periodic Points for Prevalent Diffeomorphisms I.” <i>Annals of
    Mathematics</i>. Princeton University Press, 2007. <a href="https://doi.org/10.4007/annals.2007.165.89">https://doi.org/10.4007/annals.2007.165.89</a>.
  ieee: V. Kaloshin and B. Hunt, “Stretched exponential estimates on growth of the
    number of periodic points for prevalent diffeomorphisms I,” <i>Annals of Mathematics</i>,
    vol. 165, no. 1. Princeton University Press, pp. 89–170, 2007.
  ista: Kaloshin V, Hunt B. 2007. Stretched exponential estimates on growth of the
    number of periodic points for prevalent diffeomorphisms I. Annals of Mathematics.
    165(1), 89–170.
  mla: Kaloshin, Vadim, and Brian Hunt. “Stretched Exponential Estimates on Growth
    of the Number of Periodic Points for Prevalent Diffeomorphisms I.” <i>Annals of
    Mathematics</i>, vol. 165, no. 1, Princeton University Press, 2007, pp. 89–170,
    doi:<a href="https://doi.org/10.4007/annals.2007.165.89">10.4007/annals.2007.165.89</a>.
  short: V. Kaloshin, B. Hunt, Annals of Mathematics 165 (2007) 89–170.
date_created: 2020-09-18T10:48:33Z
date_published: 2007-01-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.4007/annals.2007.165.89
extern: '1'
intvolume: '       165'
issue: '1'
language:
- iso: eng
month: '01'
oa_version: None
page: 89-170
publication: Annals of Mathematics
publication_identifier:
  issn:
  - 0003-486X
publication_status: published
publisher: Princeton University Press
quality_controlled: '1'
status: public
title: Stretched exponential estimates on growth of the number of periodic points
  for prevalent diffeomorphisms I
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 165
year: '2007'
...