Inventiones mathematicae

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

Renormalization of polygon exchange maps arising from corner percolation

  • W. Patrick Hooper


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.


Return Time Central Curve Stable Periodic Orbit Shift Space Horizontal Step 
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© Springer-Verlag 2012

Authors and Affiliations

  1. 1.The City College of New YorkNew YorkUSA

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