Renormalization of polygon exchange maps arising from corner percolation |
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Authors: | W. Patrick Hooper |
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Affiliation: | 1. The City College of New York, New York, NY, 10031, USA
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Abstract: | We describe a family {Ψ α,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, Ψ α,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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