Topological weak mixing for certain linear involutions of hyperelliptic type

A linear involution is an injective piecewise isometry defined on a pair of disjoint intervals. They are defined by a combinatorial data given by a generalized permutation and a length vector. As it was done for interval exchange transformations, it is conjectured that, except for some combinatorial data, a typical linear involution is measure-theoretically weakly mixing. In this direction, one may first explore the question of topological weak mixing for topological models of linear involutions. In this article we prove topological weak mixing for the natural symbolic codings of typical linear involutions defined by some generalized permutations of hyperelliptic type. We consider the generalized permutation $$\sigma_{s,r} = \begin{pmatrix} 0 & A & 1 & 2 & \cdots & s & A & s+1 & s+2 & \cdots & s+r \\ s+r & \cdots & s+2 & s+1 & B & s & \cdots & 2 & 1 & B & 0 \end{pmatrix}.$$ We prove that the natural symbolic coding of a typical linear involution defined by a generalized permutation in the Rauzy class of $\sigma_{s,r}$ is topologically weakly mixing, provided that it has at least a simple letter and its associated genus is sufficiently large.