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Renormalizing an infinite rational IET

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  • 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.

    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 $

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

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

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