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Singular potentials, rigidity and recurrence in low dimensional dynamics

Lamprinakis, Georgios LU (2024)
Abstract
In the first part of this thesis we study the behaviour of the equilibrium measure and the Birkhoff sums for a singular potential over the doubling map. A complete multifractal analysis for the the Birkhoff sums and the equilibrium measure (for the ’appropriate’ scaling) is given in Paper I. In Paper II we prove a parameter continuity property of the pressure function for a family of singular potentials. In the third paper we study some properties of the invariant sets of a general expanding Markov map of the circle and investigate a rigidity related question, proving that for a fixed such map there are not many (in the topological sense) other maps so that they share common compact invariant sets of ’small’ Hausdorff dimension. The last... (More)
In the first part of this thesis we study the behaviour of the equilibrium measure and the Birkhoff sums for a singular potential over the doubling map. A complete multifractal analysis for the the Birkhoff sums and the equilibrium measure (for the ’appropriate’ scaling) is given in Paper I. In Paper II we prove a parameter continuity property of the pressure function for a family of singular potentials. In the third paper we study some properties of the invariant sets of a general expanding Markov map of the circle and investigate a rigidity related question, proving that for a fixed such map there are not many (in the topological sense) other maps so that they share common compact invariant sets of ’small’ Hausdorff dimension. The last project deviates from the previous ones and focuses on recurrence properties of hyperbolic systems and specifically, hyperbolic automorphisms of the two-dimensional torus. For such a system, we provide a formula for the Hausdorff dimension of the so-called uniform recurrence set.
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Please use this url to cite or link to this publication:
author
supervisor
opponent
  • Prof. Kesseböhmer, Marc, University of Bremen, Germany.
organization
publishing date
type
Thesis
publication status
published
subject
pages
168 pages
publisher
Lund University / Centre for Mathematical Sciences /LTH
defense location
Lecture Hall Hörmander, Centre of Mathematical Sciences, Sölvegatan 18, Faculty of Engineering LTH, Lund University, Lund.
defense date
2024-05-28 10:00:00
ISBN
978-91-8104-043-2
978-91-8104-042-5
language
English
LU publication?
yes
id
0228c92f-0163-4d85-ae66-790a663a3e3c
date added to LUP
2024-04-30 13:38:55
date last changed
2024-05-02 08:38:30
@phdthesis{0228c92f-0163-4d85-ae66-790a663a3e3c,
  abstract     = {{In the first part of this thesis we study the behaviour of the equilibrium measure and the Birkhoff sums for a singular potential over the doubling map. A complete multifractal analysis for the the Birkhoff sums and the equilibrium measure (for the ’appropriate’ scaling) is given in Paper I. In Paper II we prove a parameter continuity property of the pressure function for a family of singular potentials. In the third paper we study some properties of the invariant sets of a general expanding Markov map of the circle and investigate a rigidity related question, proving that for a fixed such map there are not many (in the topological sense) other maps so that they share common compact invariant sets of ’small’ Hausdorff dimension. The last project deviates from the previous ones and focuses on recurrence properties of hyperbolic systems and specifically, hyperbolic automorphisms of the two-dimensional torus. For such a system, we provide a formula for the Hausdorff dimension of the so-called uniform recurrence set.<br/>}},
  author       = {{Lamprinakis, Georgios}},
  isbn         = {{978-91-8104-043-2}},
  language     = {{eng}},
  publisher    = {{Lund University / Centre for Mathematical Sciences /LTH}},
  school       = {{Lund University}},
  title        = {{Singular potentials, rigidity and recurrence in low dimensional dynamics}},
  url          = {{https://lup.lub.lu.se/search/files/181949253/Thesis_-_Georgios_Lamprinakis.pdf}},
  year         = {{2024}},
}