2024/02/01 by Georgios Lamprinakis, Lamprinakis, Georgios, Tomas Persson +3
Mathematics · Physics and Astronomy · #28A78 #37B20 #37D20 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2402.00690
openalex publication_date 2024/02/01 · openalex created_date 2024/02/03 · openalex updated_date 2026/07/28
We are interested in studying sets of the form U(α) := \ x∈ X: ∃ M=M(x) ≥ 1 such that ∀ N≥ M, ∃ n≤ N such that d(Tnx, x) ≤ |λ|-αN \ where (X,T,d) is our metric dynamical system and |λ|>1. Although a lot of 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=\mathbbT2, T(x) = Ax \pmod1, 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.