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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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 |
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 and Continuous Dynamical Systems,
2020, 40
(9)
: 5105-5116.
doi: 10.3934/dcds.2020220
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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. |
| [2] |
J. P. Bowman,
The complete family of Arnoux-Yoccoz surfaces, Geometriae Dedicata, 164 (2013), 113-130.
doi: 10.1007/s10711-012-9762-9. |
| [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. |
| [4] |
V. Delecroix, Package Surface_Dynamics for SageMath, the Sage Mathematics Software System, 2018, http://www.sagemath.org, https://gitlab.com/videlec/surface_dynamics. |
| [5] |
V. Delecroix, P. Hubert and F. Valdez, Infinite Translation Surfaces in the Wild, To appear. |
| [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. |
| [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. |
| [8] |
A. Goetz, Piecewise isometries - an emerging area of dynamical systems, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, (2003), 135–144. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [13] |
N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002.
doi: 10.1007/b13861. |
| [14] |
R. E. Schwartz, The Octagonal PETs, Mathematical Surveys and Monographs, 197. American Mathematical Society, Providence, RI, 2014.
doi: 10.1090/surv/197. |
| [15] |
C. E. Silva, Invitation to Ergodic Theory, Student Mathematical Library, 42. American Mathematical Society, Providence, RI, 2008. |
| [16] |
R. Yi, The triple lattice PETs, Experimental Mathematics, 28, (2019), 456–474.
doi: 10.1080/10586458.2017.1422159. |
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. |
| [2] |
J. P. Bowman,
The complete family of Arnoux-Yoccoz surfaces, Geometriae Dedicata, 164 (2013), 113-130.
doi: 10.1007/s10711-012-9762-9. |
| [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. |
| [4] |
V. Delecroix, Package Surface_Dynamics for SageMath, the Sage Mathematics Software System, 2018, http://www.sagemath.org, https://gitlab.com/videlec/surface_dynamics. |
| [5] |
V. Delecroix, P. Hubert and F. Valdez, Infinite Translation Surfaces in the Wild, To appear. |
| [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. |
| [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. |
| [8] |
A. Goetz, Piecewise isometries - an emerging area of dynamical systems, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, (2003), 135–144. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [13] |
N. P. Fogg, Substitutions in Dynamics, Arithmetics and Combinatorics, Lecture Notes in Mathematics, 1794. Springer-Verlag, Berlin, 2002.
doi: 10.1007/b13861. |
| [14] |
R. E. Schwartz, The Octagonal PETs, Mathematical Surveys and Monographs, 197. American Mathematical Society, Providence, RI, 2014.
doi: 10.1090/surv/197. |
| [15] |
C. E. Silva, Invitation to Ergodic Theory, Student Mathematical Library, 42. American Mathematical Society, Providence, RI, 2008. |
| [16] |
R. Yi, The triple lattice PETs, Experimental Mathematics, 28, (2019), 456–474.
doi: 10.1080/10586458.2017.1422159. |
Figure 2.
The intervals $ I_{w0} $ and $ I_{w1} $ produced from $ I_w $ when $ s_{|w|} = \frac{1}{4} $
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