On uniform recurrence for hyperbolic automorphisms of the 2-dimensional torus
(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:
https://lup.lub.lu.se/record/40da4971-16c8-4110-b3af-c9fa43817067
- author
- Lamprinakis, Georgios
LU
; Persson, Tomas
LU
and Sponheimer, Alejandro Rodriguez LU
- organization
- publishing date
- 2025-06
- 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 | > 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}}, }