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Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Persson, Tomas LU orcid (2019) In Annales Henri Lebesgue 2. p.1-37
Abstract
Generalising a construction of Falconer, we consider classes of 𝐺𝛿-subsets of ℝ𝑑 with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.

As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for 𝛼≥1,

... (More)
Generalising a construction of Falconer, we consider classes of 𝐺𝛿-subsets of ℝ𝑑 with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.

As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for 𝛼≥1,

dimH{𝑦:|𝑇𝑛𝑎(𝑥)−𝑦|<𝑛−𝛼infinitelyoften}=1𝛼,
for almost every 𝑥∈[1−𝑎,1], where 𝑇𝑎 is a quadratic map with 𝑎 in a set of parameters described by Benedicks and Carleson. (Less)
Abstract (Swedish)


Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Hausdorff dimensions, limsup sets, potentials
in
Annales Henri Lebesgue
volume
2
pages
37 pages
ISSN
2644-9463
DOI
10.5802/ahl.15
language
English
LU publication?
yes
id
74ff548a-55a6-47b3-9fa9-98f64fa38f2d
date added to LUP
2020-01-11 16:05:57
date last changed
2021-03-22 15:37:00
@article{74ff548a-55a6-47b3-9fa9-98f64fa38f2d,
  abstract     = {{Generalising a construction of Falconer, we consider classes of &#x1d43a;&#x1d6ff;-subsets of ℝ&#x1d451; with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We relate these classes to some inhomogeneous potentials and energies, thereby providing some useful tools to determine if a set belongs to one of the classes.<br>
<br>
As applications of this theory, we calculate, or at least estimate, the Hausdorff dimension of randomly generated limsup-sets, and sets that appear in the setting of shrinking targets in dynamical systems. For instance, we prove that for &#x1d6fc;≥1,<br>
<br>
dimH{&#x1d466;:|&#x1d447;&#x1d45b;&#x1d44e;(&#x1d465;)−&#x1d466;|&lt;&#x1d45b;−&#x1d6fc;infinitelyoften}=1&#x1d6fc;,<br>
for almost every &#x1d465;∈[1−&#x1d44e;,1], where &#x1d447;&#x1d44e; is a quadratic map with &#x1d44e; in a set of parameters described by Benedicks and Carleson.}},
  author       = {{Persson, Tomas}},
  issn         = {{2644-9463}},
  keywords     = {{Hausdorff dimensions; limsup sets; potentials}},
  language     = {{eng}},
  pages        = {{1--37}},
  series       = {{Annales Henri Lebesgue}},
  title        = {{Inhomogeneous potentials, Hausdorff dimension and shrinking targets}},
  url          = {{http://dx.doi.org/10.5802/ahl.15}},
  doi          = {{10.5802/ahl.15}},
  volume       = {{2}},
  year         = {{2019}},
}