Inventiones mathematicae

, Volume 191, Issue 2, pp 255–320 | Cite as

Renormalization of polygon exchange maps arising from corner percolation

Article

Abstract

We describe a family {Ψα,β} of polygon exchange transformations parameterized by points (α,β) in the square \([0, {\frac{1}{2}}]\times[0, {\frac{1}{2}}]\). Whenever α and β are irrational, Ψα,β 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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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.The City College of New YorkNew YorkUSA

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