Renormalizing an infinite rational IET

W. Patrick Hooper; Kasra Rafi; Anja Randecker

doi:10.3934/dcds.2020220

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2020, Volume 40, Issue 9: 5105-5116. Doi: 10.3934/dcds.2020220

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Renormalizing an infinite rational IET

1.
The City College of New York New York, NY, 10031, USA
2.
CUNY Graduate Center New York, NY, 10016, USA
3.
University of Toronto Toronto, ON, M5S 2E4, Canada

Received: August 31, 2018

Revised: September 30, 2019

Published: June 2020

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Abstract

We study an interval exchange transformation of $ [0, 1] $ formed by cutting the interval at the points $ \frac{1}{n} $ and reversing the order of the intervals. We find that the transformation is periodic away from a Cantor set of Hausdorff dimension zero. On the Cantor set, the dynamics are nearly conjugate to the $ 2 $–adic odometer.

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Mathematics Subject Classification: Primary: 37E05; Secondary: 37E20, 32G15.

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Figure 1. Top: The interval $ [0,1) $ cut into intervals of the form $ [1-\frac{1}{k},1-\frac{1}{k+1}) $. Bottom: The images of these intervals under $ T_1 $

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Figure 2. The intervals $ I_{w0} $ and $ I_{w1} $ produced from $ I_w $ when $ s_{|w|} = \frac{1}{4} $

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Figure 3. The construction of the Cantor set $ {{\mathcal{C}}} $ when $ N = 1 $

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