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. We develop their higher-genus counterpart on compact surfaces, where generalized interval exchange transformations (GIETs) with flat pieces arise naturally as first return maps. We construct families of $C^\infty$ Cherry flows realizing, up to semi-conjugacy, every interval exchange transformation satisfying the Keane property and whose combinatorics is compatible with the topology of the surface. In particular, for every genus g, every prescribed collection of saddle indices, and every $k=1,\dots,g$, we obtain a Cherry flow whose unique quasi-minimal set supports exactly $k$ ergodic invariant measures. The one-dimensional mechanism underlying this construction is a Full Family Theorem for GIETs with flat pieces, which realizes every admissible Rauzy renormalization path within a finite-dimensional parameter family with the optimal number of parameters.