Higher genus Cherry flows and full families of GIETs
Abstract
Cherry flows are classical examples of C∞ flows on the two-dimensional torus exhibiting non-trivial recurrent dynamics. For every genus g≥1 we construct a C∞ 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,…,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.
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BibTeX
@misc{marchese2026higher,
title = {{Higher genus Cherry flows and full families of GIETs}},
author = {Luca Marchese and Liviana Palmisano},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30965},
eprint = {2608.30965},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30965},
}
RIS
TY - UNPB
TI - Higher genus Cherry flows and full families of GIETs
AU - Luca Marchese
AU - Liviana Palmisano
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2608.30965
UR - https://arxiv.org/abs/2608.30965
AB - Cherry flows are classical examples of C^∞ flows on the two-dimensional torus exhibiting non-trivial recurrent dynamics. For every genus g≥1 we construct a C^∞ 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,…,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.
ER -