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On eventually always hitting points

Ganotaki, Charis LU and Persson, Tomas LU orcid (2021) In Monatshefte fur Mathematik 196(4). p.763-784
Abstract

We consider dynamical systems (X, T, μ) which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls (Bn)n=1∞, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a k< m with Tk(x) ∈ Bm. We also give an asymptotic estimate as m→ ∞ on the number of k< m with Tk(x) ∈ Bm. As an application, we prove for almost every point x an asymptotic estimate on the number of k≤ m such that ak≥ mt, where t∈ (0 , 1) and ak are the continued fraction coefficients of x.

Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Continued fractions, Eventually always hitting points, Shrinking targets
in
Monatshefte fur Mathematik
volume
196
issue
4
pages
763 - 784
publisher
Springer
external identifiers
  • scopus:85113817998
ISSN
0026-9255
DOI
10.1007/s00605-021-01616-7
language
English
LU publication?
yes
id
ba79b8c5-fa33-46c0-ad5c-f56d2f4af7a3
date added to LUP
2021-09-27 13:06:54
date last changed
2025-10-14 11:33:48
@article{ba79b8c5-fa33-46c0-ad5c-f56d2f4af7a3,
  abstract     = {{<p>We consider dynamical systems (X, T, μ) which have exponential decay of correlations for either Hölder continuous functions or functions of bounded variation. Given a sequence of balls (Bn)n=1∞, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points x such that for all large enough m, there is a k&lt; m with T<sup>k</sup>(x) ∈ B<sub>m</sub>. We also give an asymptotic estimate as m→ ∞ on the number of k&lt; m with T<sup>k</sup>(x) ∈ B<sub>m</sub>. As an application, we prove for almost every point x an asymptotic estimate on the number of k≤ m such that a<sub>k</sub>≥ m<sup>t</sup>, where t∈ (0 , 1) and a<sub>k</sub> are the continued fraction coefficients of x.</p>}},
  author       = {{Ganotaki, Charis and Persson, Tomas}},
  issn         = {{0026-9255}},
  keywords     = {{Continued fractions; Eventually always hitting points; Shrinking targets}},
  language     = {{eng}},
  month        = {{08}},
  number       = {{4}},
  pages        = {{763--784}},
  publisher    = {{Springer}},
  series       = {{Monatshefte fur Mathematik}},
  title        = {{On eventually always hitting points}},
  url          = {{http://dx.doi.org/10.1007/s00605-021-01616-7}},
  doi          = {{10.1007/s00605-021-01616-7}},
  volume       = {{196}},
  year         = {{2021}},
}