Interval translation maps with weakly mixing attractors
Volume 180 / 2026
Colloquium Mathematicum 180 (2026), 99-136
MSC: Primary 37E05; Secondary 37A05, 37A25, 37A30, 58F19, 58F11, 54H20
DOI: 10.4064/cm9597-12-2025
Published online: 17 March 2026
Abstract
We study linear recurrence and weak mixing of a two-parameter family of interval translation maps $T_{\alpha ,\beta }$ for the subset of parameter space where $T_{\alpha ,\beta }$ has a Cantor attractor. For this class, there is a procedure similar to the Rauzy induction which acts as a dynamical system $G$ on parameter space, which was used previously to decide whether $T_{\alpha ,\beta }$ has an attracting Cantor set, and if so, whether $T_{\alpha ,\beta }$ is uniquely ergodic. In this paper we use properties of $G$ to decide whether $T_{\alpha ,\beta }$ is linearly recurrent or weakly mixing.
