# American Institute of Mathematical Sciences

September  2020, 40(9): 5105-5116. doi: 10.3934/dcds.2020220

## 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  September 2018 Revised  October 2019 Published  June 2020

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.

Citation: W. Patrick Hooper, Kasra Rafi, Anja Randecker. Renormalizing an infinite rational IET. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5105-5116. doi: 10.3934/dcds.2020220
##### References:
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##### References:
 [1] S. Akiyama and E. Harriss, Pentagonal domain exchange, Discrete Contin. Dyn. Syst., 33 (2013), 4375-4400.  doi: 10.3934/dcds.2013.33.4375.  Google Scholar [2] J. P. Bowman, The complete family of Arnoux-Yoccoz surfaces, Geometriae Dedicata, 164 (2013), 113-130.  doi: 10.1007/s10711-012-9762-9.  Google Scholar [3] R. Chamanara, Affine automorphism groups of surfaces of infinite type, In the Tradition of Ahlfors and Bers, III, Contemp. Math., American Mathematical Society, Providence, RI, 355 (2004), 123-145.  doi: 10.1090/conm/355/06449.  Google Scholar [4] V. Delecroix, Package Surface_Dynamics for SageMath, the Sage Mathematics Software System, 2018, http://www.sagemath.org, https://gitlab.com/videlec/surface_dynamics. Google Scholar [5] V. Delecroix, P. Hubert and F. Valdez, Infinite Translation Surfaces in the Wild, To appear. Google Scholar [6] T. Downarowicz, Survey of odometers and Toeplitz flows, Algebraic and Topological Dynamics, Contemp. Math., American Mathematical Society, Providence, RI, 385 (2005), 7-37.  doi: 10.1090/conm/385/07188.  Google Scholar [7] A. Goetz, A self-similar example of a piecewise isometric attractor, Dynamical Systems: From Crystal to Chaos, World Scientific, River Edge, NJ, (2000), 248–258.  Google Scholar [8] A. Goetz, Piecewise isometries - an emerging area of dynamical systems, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, (2003), 135–144.  Google Scholar [9] W. P. Hooper, Renormalization of polygon exchange maps arising from corner percolation, Inventiones Mathematicae, 191 (2013), 255-320.  doi: 10.1007/s00222-012-0393-4.  Google Scholar [10] K. Lindsey and R. Treviño, Infinite type flat surface models of ergodic systems, Discrete Contin. Dyn. Syst., 36 (2016), 5509-5553.  doi: 10.3934/dcds.2016043.  Google Scholar [11] H. Masur and S. Tabachnikov, Rational billiards and flat structures, Handbook of Dynamical Systems, North-Holland, Amsterdam, 1A (2002), 1015-1089.  doi: 10.1016/S1874-575X(02)80015-7.  Google Scholar [12] P. Matilla, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511623813.  Google Scholar [13] N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002. doi: 10.1007/b13861.  Google Scholar [14] R. E. Schwartz, The Octagonal PETs, Mathematical Surveys and Monographs, 197. American Mathematical Society, Providence, RI, 2014. doi: 10.1090/surv/197.  Google Scholar [15] C. E. Silva, Invitation to Ergodic Theory, Student Mathematical Library, 42. American Mathematical Society, Providence, RI, 2008.  Google Scholar [16] R. Yi, The triple lattice PETs, Experimental Mathematics, 28, (2019), 456–474. doi: 10.1080/10586458.2017.1422159.  Google Scholar
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 $T_1$
The intervals $I_{w0}$ and $I_{w1}$ produced from $I_w$ when $s_{|w|} = \frac{1}{4}$
The construction of the Cantor set ${{\mathcal{C}}}$ when $N = 1$
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