A Family of Minimal and Renormalizable Rectangle Exchange Maps
Dynamical Systems
2021-03-19 v2
Abstract
A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.
Cite
@article{arxiv.1803.06369,
title = {A Family of Minimal and Renormalizable Rectangle Exchange Maps},
author = {Ian Alevy and Richard Kenyon and Ren Yi},
journal= {arXiv preprint arXiv:1803.06369},
year = {2021}
}
Comments
33 pages, 12 figures. Added a connection to the vertical flow which we use to prove that our DEMs have equidistributed orbits