The complete L^q-spectrum and large deviations for return times for equilibrium states with summable potentials

2019/02/09 by Miguel Abadi, Abadi, M., Vitor Gustavo de Amorim +5 · 1 citation
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1902.03441

openalex publication_date 2019/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (X_k)_{k≥ 0} be a stationary and ergodic process with joint distribution μ where the random variables X_k take values in a finite set A. Let R_n be the first time this process repeats its first n symbols of output. It is well-known that (1)/(n)log R_n converges almost surely to the entropy of the process. Refined properties of R_n (large deviations, multifractality, etc) are encoded in the return-time L^q-spectrum defined as R(q)=lim_n(1)/(n)log∫ R_n^q dμ (q∈ℝ) provided the limit exists. We consider the case where (X_k)_{k≥ 0} is distributed according to the equilibrium state of a potential φ:A^ℕ→ℝ with summable variation, and we prove that \[ R(q) = \begincases P((1-q)φ) & for q≥ q_φ^* sup_η∫ φ dη& for q

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