---
_id: '8504'
abstract:
- lang: eng
  text: In this paper we present a surprising example of a Cr unimodal map of an interval
    f:I→I whose number of periodic points Pn(f)=∣{x∈I:fnx=x}∣ grows faster than any
    ahead given sequence along a subsequence nk=3k. This example also shows that ‘non-flatness’
    of critical points is necessary for the Martens–de Melo–van Strien theorem [M.
    Martens, W. de Melo and S. van Strien. Julia–Fatou–Sullivan theory for real one-dimensional
    dynamics. Acta Math.168(3–4) (1992), 273–318] to hold.
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: O. S.
  full_name: KOZLOVSKI, O. S.
  last_name: KOZLOVSKI
citation:
  ama: Kaloshin V, KOZLOVSKI OS. A Cr unimodal map with an arbitrary fast growth of
    the number of periodic points. <i>Ergodic Theory and Dynamical Systems</i>. 2012;32(1):159-165.
    doi:<a href="https://doi.org/10.1017/s0143385710000817">10.1017/s0143385710000817</a>
  apa: Kaloshin, V., &#38; KOZLOVSKI, O. S. (2012). A Cr unimodal map with an arbitrary
    fast growth of the number of periodic points. <i>Ergodic Theory and Dynamical
    Systems</i>. Cambridge University Press. <a href="https://doi.org/10.1017/s0143385710000817">https://doi.org/10.1017/s0143385710000817</a>
  chicago: Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary
    Fast Growth of the Number of Periodic Points.” <i>Ergodic Theory and Dynamical
    Systems</i>. Cambridge University Press, 2012. <a href="https://doi.org/10.1017/s0143385710000817">https://doi.org/10.1017/s0143385710000817</a>.
  ieee: V. Kaloshin and O. S. KOZLOVSKI, “A Cr unimodal map with an arbitrary fast
    growth of the number of periodic points,” <i>Ergodic Theory and Dynamical Systems</i>,
    vol. 32, no. 1. Cambridge University Press, pp. 159–165, 2012.
  ista: Kaloshin V, KOZLOVSKI OS. 2012. A Cr unimodal map with an arbitrary fast growth
    of the number of periodic points. Ergodic Theory and Dynamical Systems. 32(1),
    159–165.
  mla: Kaloshin, Vadim, and O. S. KOZLOVSKI. “A Cr Unimodal Map with an Arbitrary
    Fast Growth of the Number of Periodic Points.” <i>Ergodic Theory and Dynamical
    Systems</i>, vol. 32, no. 1, Cambridge University Press, 2012, pp. 159–65, doi:<a
    href="https://doi.org/10.1017/s0143385710000817">10.1017/s0143385710000817</a>.
  short: V. Kaloshin, O.S. KOZLOVSKI, Ergodic Theory and Dynamical Systems 32 (2012)
    159–165.
date_created: 2020-09-18T10:47:33Z
date_published: 2012-02-01T00:00:00Z
date_updated: 2021-01-12T08:19:44Z
day: '01'
doi: 10.1017/s0143385710000817
extern: '1'
intvolume: '        32'
issue: '1'
keyword:
- Applied Mathematics
- General Mathematics
language:
- iso: eng
month: '02'
oa_version: None
page: 159-165
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  issn:
  - 0143-3857
  - 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: A Cr unimodal map with an arbitrary fast growth of the number of periodic points
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 32
year: '2012'
...
---
_id: '8514'
abstract:
- lang: eng
  text: We study the extent to which the Hausdorff dimension of a compact subset of
    an infinite-dimensional Banach space is affected by a typical mapping into a finite-dimensional
    space. It is possible that the dimension drops under all such mappings, but the
    amount by which it typically drops is controlled by the ‘thickness exponent’ of
    the set, which was defined by Hunt and Kaloshin (Nonlinearity12 (1999), 1263–1275).
    More precisely, let $X$ be a compact subset of a Banach space $B$ with thickness
    exponent $\tau$ and Hausdorff dimension $d$. Let $M$ be any subspace of the (locally)
    Lipschitz functions from $B$ to $\mathbb{R}^{m}$ that contains the space of bounded
    linear functions. We prove that for almost every (in the sense of prevalence)
    function $f \in M$, the Hausdorff dimension of $f(X)$ is at least $\min\{ m, d
    / (1 + \tau) \}$. We also prove an analogous result for a certain part of the
    dimension spectra of Borel probability measures supported on $X$. The factor $1
    / (1 + \tau)$ can be improved to $1 / (1 + \tau / 2)$ if $B$ is a Hilbert space.
    Since dimension cannot increase under a (locally) Lipschitz function, these theorems
    become dimension preservation results when $\tau = 0$. We conjecture that many
    of the attractors associated with the evolution equations of mathematical physics
    have thickness exponent zero. We also discuss the sharpness of our results in
    the case $\tau > 0$.
article_processing_charge: No
article_type: original
author:
- first_name: WILLIAM
  full_name: OTT, WILLIAM
  last_name: OTT
- first_name: BRIAN
  full_name: HUNT, BRIAN
  last_name: HUNT
- first_name: Vadim
  full_name: Kaloshin, Vadim
  id: FE553552-CDE8-11E9-B324-C0EBE5697425
  last_name: Kaloshin
  orcid: 0000-0002-6051-2628
citation:
  ama: OTT W, HUNT B, Kaloshin V. The effect of projections on fractal sets and measures
    in Banach spaces. <i>Ergodic Theory and Dynamical Systems</i>. 2006;26(3):869-891.
    doi:<a href="https://doi.org/10.1017/s0143385705000714">10.1017/s0143385705000714</a>
  apa: OTT, W., HUNT, B., &#38; Kaloshin, V. (2006). The effect of projections on
    fractal sets and measures in Banach spaces. <i>Ergodic Theory and Dynamical Systems</i>.
    Cambridge University Press. <a href="https://doi.org/10.1017/s0143385705000714">https://doi.org/10.1017/s0143385705000714</a>
  chicago: OTT, WILLIAM, BRIAN HUNT, and Vadim Kaloshin. “The Effect of Projections
    on Fractal Sets and Measures in Banach Spaces.” <i>Ergodic Theory and Dynamical
    Systems</i>. Cambridge University Press, 2006. <a href="https://doi.org/10.1017/s0143385705000714">https://doi.org/10.1017/s0143385705000714</a>.
  ieee: W. OTT, B. HUNT, and V. Kaloshin, “The effect of projections on fractal sets
    and measures in Banach spaces,” <i>Ergodic Theory and Dynamical Systems</i>, vol.
    26, no. 3. Cambridge University Press, pp. 869–891, 2006.
  ista: OTT W, HUNT B, Kaloshin V. 2006. The effect of projections on fractal sets
    and measures in Banach spaces. Ergodic Theory and Dynamical Systems. 26(3), 869–891.
  mla: OTT, WILLIAM, et al. “The Effect of Projections on Fractal Sets and Measures
    in Banach Spaces.” <i>Ergodic Theory and Dynamical Systems</i>, vol. 26, no. 3,
    Cambridge University Press, 2006, pp. 869–91, doi:<a href="https://doi.org/10.1017/s0143385705000714">10.1017/s0143385705000714</a>.
  short: W. OTT, B. HUNT, V. Kaloshin, Ergodic Theory and Dynamical Systems 26 (2006)
    869–891.
date_created: 2020-09-18T10:48:52Z
date_published: 2006-06-01T00:00:00Z
date_updated: 2021-01-12T08:19:48Z
day: '01'
doi: 10.1017/s0143385705000714
extern: '1'
intvolume: '        26'
issue: '3'
language:
- iso: eng
month: '06'
oa_version: None
page: 869-891
publication: Ergodic Theory and Dynamical Systems
publication_identifier:
  issn:
  - 0143-3857
  - 1469-4417
publication_status: published
publisher: Cambridge University Press
quality_controlled: '1'
status: public
title: The effect of projections on fractal sets and measures in Banach spaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 26
year: '2006'
...