Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling

We study piecewise rational rotations of convex polygons with a recursive tiling property. For these dynamical systems, the set Σ of discontinuity-avoiding aperiodic orbits decomposes into invariant subsets endowed with a hierarchical symbolic dynamics (Vershik map on a Bratteli diagram). Under conditions which guarantee a form of asymptotic temporal scaling, we prove minimality and unique ergodicity for each invariant component. We study the multifractal properties of the model with respect to recurrence times, deriving a method of successive approximations for the generalized dimensions α(q). We consider explicit examples in which the trace of the rotation matrix is a quadratic or cubic irrational, and evaluate numerically, with high precision, the function α(q) and its Legendre transform.