Quantitative weak mixing for interval exchange transformations

• Published in: Geometric and functional analysis Vol. 33; no. 1; pp. 1 - 56
• Main Authors: Avila, Artur, Forni, Giovanni, Safaee, Pedram
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2023; Springer Nature B.V
• Subjects: Analysis; Decay rate; Logarithms; Lower bounds; Mathematical analysis; Mathematics; Mathematics and Statistics; Polynomials; Rotation; Transformations (mathematics); Upper bounds
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-023-00625-y

We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok–Stepin, Veech, and Avila–Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have local dimension bounded away from zero (by a constant depending only on the number of intervals). In our approach, upper bounds are obtained through estimates of twisted Birkhoff sums of Lipschitz functions, while the logarithmic lower bounds are based on the slow deviation of ergodic averages that govern the relation between rigid rotations and their induced maps.