Diffuse Behaviour of Ergodic Sums Over Rotations

2017/05/30 by Jean-Pierre Conze, Conze, Jean-Pierre, Stefano Isola +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1705.10550

openalex publication_date 2017/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a rotation by an irrational α on the circle and a BV function φ, we study the variance of the ergodic sums S_L φ(x) := ∑_j=0^{L -1} φ(x + jα). When α is not of constant type, we construct sequences (L_N) such that, at some scale, the ergodic sums S_LN φ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane.

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