---
_id: '8521'
abstract:
- lang: eng
text: 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.
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 R.
full_name: Hunt, Brian R.
last_name: Hunt
citation:
ama: Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of
the number of periodic points for prevalent diffeomorphisms II. Electronic
Research Announcements of the American Mathematical Society. 2001;7(5):28-36.
doi:10.1090/s1079-6762-01-00091-9
apa: Kaloshin, V., & Hunt, B. R. (2001). A stretched exponential bound on the
rate of growth of the number of periodic points for prevalent diffeomorphisms
II. Electronic Research Announcements of the American Mathematical Society.
American Mathematical Society. https://doi.org/10.1090/s1079-6762-01-00091-9
chicago: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the
Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms
II.” Electronic Research Announcements of the American Mathematical Society.
American Mathematical Society, 2001. https://doi.org/10.1090/s1079-6762-01-00091-9.
ieee: V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of
growth of the number of periodic points for prevalent diffeomorphisms II,” Electronic
Research Announcements of the American Mathematical Society, vol. 7, no. 5.
American Mathematical Society, pp. 28–36, 2001.
ista: Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth
of the number of periodic points for prevalent diffeomorphisms II. Electronic
Research Announcements of the American Mathematical Society. 7(5), 28–36.
mla: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate
of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms II.”
Electronic Research Announcements of the American Mathematical Society,
vol. 7, no. 5, American Mathematical Society, 2001, pp. 28–36, doi:10.1090/s1079-6762-01-00091-9.
short: V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American
Mathematical Society 7 (2001) 28–36.
date_created: 2020-09-18T10:49:43Z
date_published: 2001-04-24T00:00:00Z
date_updated: 2021-01-12T08:19:51Z
day: '24'
doi: 10.1090/s1079-6762-01-00091-9
extern: '1'
intvolume: ' 7'
issue: '5'
keyword:
- General Mathematics
language:
- iso: eng
month: '04'
oa_version: None
page: 28-36
publication: Electronic Research Announcements of the American Mathematical Society
publication_identifier:
issn:
- 1079-6762
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: A stretched exponential bound on the rate of growth of the number of periodic
points for prevalent diffeomorphisms II
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2001'
...
---
_id: '8522'
abstract:
- lang: eng
text: 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.
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 R.
full_name: Hunt, Brian R.
last_name: Hunt
citation:
ama: Kaloshin V, Hunt BR. A stretched exponential bound on the rate of growth of
the number of periodic points for prevalent diffeomorphisms I. Electronic Research
Announcements of the American Mathematical Society. 2001;7(4):17-27. doi:10.1090/s1079-6762-01-00090-7
apa: Kaloshin, V., & Hunt, B. R. (2001). A stretched exponential bound on the
rate of growth of the number of periodic points for prevalent diffeomorphisms
I. Electronic Research Announcements of the American Mathematical Society.
American Mathematical Society. https://doi.org/10.1090/s1079-6762-01-00090-7
chicago: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the
Rate of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms
I.” Electronic Research Announcements of the American Mathematical Society.
American Mathematical Society, 2001. https://doi.org/10.1090/s1079-6762-01-00090-7.
ieee: V. Kaloshin and B. R. Hunt, “A stretched exponential bound on the rate of
growth of the number of periodic points for prevalent diffeomorphisms I,” Electronic
Research Announcements of the American Mathematical Society, vol. 7, no. 4.
American Mathematical Society, pp. 17–27, 2001.
ista: Kaloshin V, Hunt BR. 2001. A stretched exponential bound on the rate of growth
of the number of periodic points for prevalent diffeomorphisms I. Electronic Research
Announcements of the American Mathematical Society. 7(4), 17–27.
mla: Kaloshin, Vadim, and Brian R. Hunt. “A Stretched Exponential Bound on the Rate
of Growth of the Number of Periodic Points for Prevalent Diffeomorphisms I.” Electronic
Research Announcements of the American Mathematical Society, vol. 7, no. 4,
American Mathematical Society, 2001, pp. 17–27, doi:10.1090/s1079-6762-01-00090-7.
short: V. Kaloshin, B.R. Hunt, Electronic Research Announcements of the American
Mathematical Society 7 (2001) 17–27.
date_created: 2020-09-18T10:49:56Z
date_published: 2001-04-18T00:00:00Z
date_updated: 2021-01-12T08:19:51Z
day: '18'
doi: 10.1090/s1079-6762-01-00090-7
extern: '1'
intvolume: ' 7'
issue: '4'
keyword:
- General Mathematics
language:
- iso: eng
month: '04'
oa_version: None
page: 17-27
publication: Electronic Research Announcements of the American Mathematical Society
publication_identifier:
issn:
- 1079-6762
publication_status: published
publisher: American Mathematical Society
quality_controlled: '1'
status: public
title: A stretched exponential bound on the rate of growth of the number of periodic
points for prevalent diffeomorphisms I
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 7
year: '2001'
...