Renormalization of Polygon Exchange Maps arising from Corner Percolation

Details Renormalization of Polygon Exchange Maps arising from Corner Percolation Renormalization of Polygon Exchange Maps arising from Corner Percolation

2011-05-30

arxiv.org/abs/1105.6137v3

Mathematics

Dynamical Systems

48 pages, 14 figures; improved exposition and proofs

Scientific paper

We describe a 2 parameter family of polygon exchange transformations parameterized by points in a square. Whenever the two parameters are irrational, the polygon exchange has periodic orbits of arbitrarily large period. We show that for almost all parameters, the polygon exchange map has the property that almost every point is periodic. However, there is a dense set of irrational parameters for which this fails. By choosing parameters carefully, the measure of non-periodic points can be made arbitrarily close to full measure. These results are powered by a notion of renormalization which holds in a more general setting. Namely, we consider a renormalization of tilings arising from the Corner Percolation Model.

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