Higher genus Cherry flows and full families of GIETs

Abstract

Cherry flows are classical examples of $C^\infty$ flows on the two-dimensional torus exhibiting non-trivial recurrent dynamics. For every higher genus $g\geq1$ we construct a $C^\infty$ parameter family of flows whose first return map is a generalized interval exchange transformation (GIET) with flat pieces. In such family, for every interval exchange transformation $T$ satisfying the Keane property, there are parameters corresponding to a Cherry flow whose return map is semi-conjugate to $T$. In particular, for each $k=1,\ldots,g$, our family contains a Cherry flow whose unique quasi-minimal set supports exactly $k$ ergodic invariant measures. The construction relies on a Full Family Theorem for GIETs with flat pieces, which establishes the realization of every admissible Rauzy renormalization path within a specific finite-dimensional family with the optimal number of parameters.