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On uniform recurrence for hyperbolic automorphisms of the 2-dimensional torus

Lamprinakis, Georgios LU ; Persson, Tomas LU orcid and Sponheimer, Alejandro Rodriguez LU (2025) In Nonlinearity 38(6).
Abstract

We are interested in studying sets of the form U ( α ) := { x ∈ X : ∃ M = M ( x ) ⩾ 1 such that ∀ N ⩾ M , ∃ 1 ⩽ n ⩽ N such that d ( T n x , x ) ⩽ | λ | − α N } , where ( X , T , d ) is our metric dynamical system and | λ | > 1 . Although many results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider X = T 2 , T ( x ) = A x ( mod 1 ) , where A is a hyperbolic, area preserving 2 × 2 matrix with integer entries and λ is the eigenvalue of A of modulus larger than 1 and we explicitly calculate the Hausdorff dimension of this set.

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
Hausdorff dimension, hyperbolic dynamical system, uniform recurrence
in
Nonlinearity
volume
38
issue
6
article number
065002
publisher
IOP Publishing
external identifiers
  • scopus:105005740698
ISSN
0951-7715
DOI
10.1088/1361-6544/adcb7f
language
English
LU publication?
yes
id
40da4971-16c8-4110-b3af-c9fa43817067
date added to LUP
2025-07-31 10:18:58
date last changed
2025-07-31 10:20:01
@article{40da4971-16c8-4110-b3af-c9fa43817067,
  abstract     = {{<p>We are interested in studying sets of the form U ( α ) := { x ∈ X : ∃ M = M ( x ) ⩾ 1 such that ∀ N ⩾ M , ∃ 1 ⩽ n ⩽ N such that d ( T n x , x ) ⩽ | λ | − α N } , where ( X , T , d ) is our metric dynamical system and | λ | &gt; 1 . Although many results exist for the one dimensional case, not as many are known for systems in higher dimensions and especially in the hyperbolic case. We consider X = T 2 , T ( x ) = A x ( mod 1 ) , where A is a hyperbolic, area preserving 2 × 2 matrix with integer entries and λ is the eigenvalue of A of modulus larger than 1 and we explicitly calculate the Hausdorff dimension of this set.</p>}},
  author       = {{Lamprinakis, Georgios and Persson, Tomas and Sponheimer, Alejandro Rodriguez}},
  issn         = {{0951-7715}},
  keywords     = {{Hausdorff dimension; hyperbolic dynamical system; uniform recurrence}},
  language     = {{eng}},
  number       = {{6}},
  publisher    = {{IOP Publishing}},
  series       = {{Nonlinearity}},
  title        = {{On uniform recurrence for hyperbolic automorphisms of the 2-dimensional torus}},
  url          = {{http://dx.doi.org/10.1088/1361-6544/adcb7f}},
  doi          = {{10.1088/1361-6544/adcb7f}},
  volume       = {{38}},
  year         = {{2025}},
}