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

Renormalizing an infinite rational IET

W. Patrick Hooper 1,2, , Kasra Rafi 3, and  Anja Randecker 3,

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.

Keywords: Interval exchange transformation, odometer, aperiodic set, piecewise isometry, renormalization.
Mathematics Subject Classification: Primary: 37E05; Secondary: 37E20, 32G15.
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:
[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

show all references

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

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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