Interval translation maps with weakly mixing attractors

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.