DEVIATION OF ERGODIC AVERAGES FOR SUBSTITUTION DYNAMICAL SYSTEMS WITH EIGENVALUES OF MODULUS ONE

Xavier Bressaud (1) , Alexander I. Bufetov (2) , Pascal Hubert (3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 I2M - Institut de Mathématiques de Marseille
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod-ulus 1. The functions γ we consider are the corresponding eigen-functions. In Theorem 1.1 we prove that the limit inferior of the ergodic sums (n, γ(x 0) +. .. + γ(x n−1)) n∈N is bounded for every point x in the phase space. In Theorem 1.2, we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length. Under additional assumptions, we prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3, Theorem 1.9).

Domaines: Systèmes dynamiques [math.DS]

Xavier Bressaud, Alexander I. Bufetov, Pascal Hubert. DEVIATION OF ERGODIC AVERAGES FOR SUBSTITUTION DYNAMICAL SYSTEMS WITH EIGENVALUES OF MODULUS ONE. Proceedings of the London Mathematical Society, 2014. ⟨hal-01256152⟩