American Institute of Mathematical Sciences

Advanced
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, 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 $
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437
[2]

Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010
[3]

María Isabel Cortez, Samuel Petite. Realization of big centralizers of minimal aperiodic actions on the Cantor set. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2891-2901. doi: 10.3934/dcds.2020153
[4]

Christopher F. Novak. Discontinuity-growth of interval-exchange maps. Journal of Modern Dynamics, 2009, 3 (3) : 379-405. doi: 10.3934/jmd.2009.3.379
[5]

Sébastien Ferenczi, Pascal Hubert. Rigidity of square-tiled interval exchange transformations. Journal of Modern Dynamics, 2019, 14: 153-177. doi: 10.3934/jmd.2019006
[6]

Luca Marchese. The Khinchin Theorem for interval-exchange transformations. Journal of Modern Dynamics, 2011, 5 (1) : 123-183. doi: 10.3934/jmd.2011.5.123
[7]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[8]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[9]

Jon Chaika, Howard Masur. There exists an interval exchange with a non-ergodic generic measure. Journal of Modern Dynamics, 2015, 9: 289-304. doi: 10.3934/jmd.2015.9.289
[10]

Jon Chaika, David Damanik, Helge Krüger. Schrödinger operators defined by interval-exchange transformations. Journal of Modern Dynamics, 2009, 3 (2) : 253-270. doi: 10.3934/jmd.2009.3.253
[11]

Jacek Brzykcy, Krzysztof Frączek. Disjointness of interval exchange transformations from systems of probabilistic origin. Discrete & Continuous Dynamical Systems, 2010, 27 (1) : 53-73. doi: 10.3934/dcds.2010.27.53
[12]

Jon Chaika, Alex Eskin. Möbius disjointness for interval exchange transformations on three intervals. Journal of Modern Dynamics, 2019, 14: 55-86. doi: 10.3934/jmd.2019003
[13]

Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35
[14]

Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077
[15]

Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
[16]

Jozef Bobok, Martin Soukenka. On piecewise affine interval maps with countably many laps. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 753-762. doi: 10.3934/dcds.2011.31.753
[17]

Ye Yuan, Yan Ren, Xiaodong Liu, Jing Wang. Approach to image segmentation based on interval neutrosophic set. Numerical Algebra, Control & Optimization, 2020, 10 (1) : 1-11. doi: 10.3934/naco.2019028
[18]

A. Pedas, G. Vainikko. Smoothing transformation and piecewise polynomial projection methods for weakly singular Fredholm integral equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 395-413. doi: 10.3934/cpaa.2006.5.395
[19]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877
[20]

Antonio Pumariño, José Ángel Rodríguez, Enrique Vigil. Renormalization of two-dimensional piecewise linear maps: Abundance of 2-D strange attractors. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 941-966. doi: 10.3934/dcds.2018040

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (137)
  • HTML views (94)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences