October  2013, 33(10): 4375-4400. doi: 10.3934/dcds.2013.33.4375

Pentagonal domain exchange

1. 

Institute of Mathematics, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki, 350-8571, Japan

2. 

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, United States

Received  April 2012 Revised  March 2013 Published  April 2013

Self-inducing structure of pentagonal piecewise isometry is applied to show detailed description of periodic and aperiodic orbits, and further dynamical properties. A Pisot number appears as a scaling constant and plays a crucial role in the proof. Further generalization is discussed in the last section.
Citation: Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4375-4400. doi: 10.3934/dcds.2013.33.4375
References:
[1]

Shigeki Akiyama, Horst Brunotte, Attila Pethő and Wolfgang Steiner, Periodicity of certain piecewise affine planar maps,, Tsukuba J. Math., 32 (2008), 197.   Google Scholar

[2]

Peter Ashwin and Xin-Chu Fu, On the geometry of orientation-preserving planar piecewise isometries,, J. Nonlinear Sci., 12 (2002), 207.  doi: 10.1007/s00332-002-0477-1.  Google Scholar

[3]

Roy Adler, Bruce Kitchens and Charles Tresser, Report on the dynamics of certain piecewise isometries of the torus,, in, (2000), 231.   Google Scholar

[4]

Pierre Arnoux and Gérard Rauzy, Représentation géométrique de suites de complexité $2n+1$,, Bull. Soc. Math. France, 119 (1991), 199.   Google Scholar

[5]

Nicolas Bedaride and Julien Cassaigne, Outer billiard outside regular polygons,, J. Lond. Math. Soc. (2), 84 (2011), 303.  doi: 10.1112/jlms/jdr010.  Google Scholar

[6]

Xavier Bressaud and Guillaume Poggiaspalla, A tentative classification of bijective polygonal piecewise isometries,, Experiment. Math., 16 (2007), 77.  doi: 10.1080/10586458.2007.10128987.  Google Scholar

[7]

Enrico Bombieri and Jean E Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections,, in, (1987), 241.   Google Scholar

[8]

Arek Goetz, A self-similar example of a piecewise isometric attractor,, in, (2000), 248.   Google Scholar

[9]

Arek Goetz, Piecewise isometries-an emerging area of dynamical systems,, in, (2001), 135.   Google Scholar

[10]

Arek Goetz, Return maps in cyclotomic piecewise similarities,, Dyn. Syst., 20 (2005), 255.  doi: 10.1080/14689360500092918.  Google Scholar

[11]

John E. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[12]

Shunji Ito and Hui Rao, Purely periodic $\beta$-expansions with Pisot unit base,, Proc. Amer. Math. Soc., 133 (2005), 953.  doi: 10.1090/S0002-9939-04-07794-9.  Google Scholar

[13]

Teturo Kamae, Numeration systems, fractals and stochastic processes,, Israel J. Math., 149 (2005), 87.  doi: 10.1007/BF02772537.  Google Scholar

[14]

Konstantin L. Kouptsov, John Lowenstein and Franco Vivaldi, Quadratic rational rotations of the torus and dual lattice maps,, Nonlinearity, 15 (2002), 1795.  doi: 10.1088/0951-7715/15/6/306.  Google Scholar

[15]

John H. Lowenstein, Spyros Hatjispyros and Franco Vivaldi, Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off,, Chaos, 7 (1997), 49.  doi: 10.1063/1.166240.  Google Scholar

[16]

John H. Lowenstein, Konstantin L. Kouptsov and Franco Vivaldi, Recursive tiling and geometry of piecewise rotations by $\pi/7$,, Nonlinearity, 17 (2004), 371.  doi: 10.1088/0951-7715/17/2/001.  Google Scholar

[17]

John H. Lowenstein, Guillaume Poggiaspalla and Franco Vivaldi, Interval exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model,, Dyn. Syst., 22 (2007), 73.  doi: 10.1080/14689360601028126.  Google Scholar

[18]

Jeong-Yup Lee and Boris Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319.  doi: 10.1007/s00454-008-9054-1.  Google Scholar

[19]

John H. Lowenstein and Franco Vivaldi, Scaling dynamics of a cubic interval-exchange transformation,, Dyn. Syst., 23 (2008), 283.  doi: 10.1080/14689360802253291.  Google Scholar

[20]

Miguel Ângelo de Sousa Mendes, Stability of periodic points in piecewise isometries of Euclidean spaces,, Ergodic Theory Dynam. Systems, 27 (2007), 183.  doi: 10.1017/S0143385706000460.  Google Scholar

[21]

Guillaume Poggiaspalla, John H. Lowenstein and Franco Vivaldi, Geometric representation of interval exchange maps over algebraic number fields,, Nonlinearity, 21 (2008), 149.  doi: 10.1088/0951-7715/21/1/009.  Google Scholar

[22]

Dominique Perrin and Jean-Éric Pin, "Infinite Words: Automata, Semigroups, Logic and Games,", 141 of Pure and Applied Mathematics. Elsevier, 141 (2004).   Google Scholar

[23]

N Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", 1794 of Lecture Notes in Mathematics. Springer-Verlag, 1794 (2002).   Google Scholar

[24]

Martine Queffélec, "Substitution Dynamical Systems-Spectral Analysis,", 1294 of Lecture Notes in Mathematics. Springer-Verlag, 1294 (1987).   Google Scholar

[25]

Gérard Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.   Google Scholar

[26]

Boris Solomyak, Dynamics of self-similar tilings,, Ergodic Theory Dynam. Systems, 17 (1997), 695.   Google Scholar

[27]

Marcello Trovati and Peter Ashwin, Tangency properties of a pentagonal tiling generated by a piecewise isometry,, Chaos, 17 (2007).  doi: 10.1063/1.2825291.  Google Scholar

[28]

Serge L. Tabachnikov, On the dual billiard problem,, Adv. Math., 115 (1995), 221.  doi: 10.1006/aima.1995.1055.  Google Scholar

[29]

Wolfgang Thomas, Automata on infinite objects,, in, (1990), 133.   Google Scholar

[30]

William A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201.  doi: 10.2307/1971391.  Google Scholar

[31]

Jean-Christophe Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction,, in, (2006), 401.  doi: 10.1007/978-3-540-31347-2_12.  Google Scholar

[32]

Anton Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325.   Google Scholar

show all references

References:
[1]

Shigeki Akiyama, Horst Brunotte, Attila Pethő and Wolfgang Steiner, Periodicity of certain piecewise affine planar maps,, Tsukuba J. Math., 32 (2008), 197.   Google Scholar

[2]

Peter Ashwin and Xin-Chu Fu, On the geometry of orientation-preserving planar piecewise isometries,, J. Nonlinear Sci., 12 (2002), 207.  doi: 10.1007/s00332-002-0477-1.  Google Scholar

[3]

Roy Adler, Bruce Kitchens and Charles Tresser, Report on the dynamics of certain piecewise isometries of the torus,, in, (2000), 231.   Google Scholar

[4]

Pierre Arnoux and Gérard Rauzy, Représentation géométrique de suites de complexité $2n+1$,, Bull. Soc. Math. France, 119 (1991), 199.   Google Scholar

[5]

Nicolas Bedaride and Julien Cassaigne, Outer billiard outside regular polygons,, J. Lond. Math. Soc. (2), 84 (2011), 303.  doi: 10.1112/jlms/jdr010.  Google Scholar

[6]

Xavier Bressaud and Guillaume Poggiaspalla, A tentative classification of bijective polygonal piecewise isometries,, Experiment. Math., 16 (2007), 77.  doi: 10.1080/10586458.2007.10128987.  Google Scholar

[7]

Enrico Bombieri and Jean E Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections,, in, (1987), 241.   Google Scholar

[8]

Arek Goetz, A self-similar example of a piecewise isometric attractor,, in, (2000), 248.   Google Scholar

[9]

Arek Goetz, Piecewise isometries-an emerging area of dynamical systems,, in, (2001), 135.   Google Scholar

[10]

Arek Goetz, Return maps in cyclotomic piecewise similarities,, Dyn. Syst., 20 (2005), 255.  doi: 10.1080/14689360500092918.  Google Scholar

[11]

John E. Hutchinson, Fractals and self-similarity,, Indiana Univ. Math. J., 30 (1981), 713.  doi: 10.1512/iumj.1981.30.30055.  Google Scholar

[12]

Shunji Ito and Hui Rao, Purely periodic $\beta$-expansions with Pisot unit base,, Proc. Amer. Math. Soc., 133 (2005), 953.  doi: 10.1090/S0002-9939-04-07794-9.  Google Scholar

[13]

Teturo Kamae, Numeration systems, fractals and stochastic processes,, Israel J. Math., 149 (2005), 87.  doi: 10.1007/BF02772537.  Google Scholar

[14]

Konstantin L. Kouptsov, John Lowenstein and Franco Vivaldi, Quadratic rational rotations of the torus and dual lattice maps,, Nonlinearity, 15 (2002), 1795.  doi: 10.1088/0951-7715/15/6/306.  Google Scholar

[15]

John H. Lowenstein, Spyros Hatjispyros and Franco Vivaldi, Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off,, Chaos, 7 (1997), 49.  doi: 10.1063/1.166240.  Google Scholar

[16]

John H. Lowenstein, Konstantin L. Kouptsov and Franco Vivaldi, Recursive tiling and geometry of piecewise rotations by $\pi/7$,, Nonlinearity, 17 (2004), 371.  doi: 10.1088/0951-7715/17/2/001.  Google Scholar

[17]

John H. Lowenstein, Guillaume Poggiaspalla and Franco Vivaldi, Interval exchange transformations over algebraic number fields: The cubic Arnoux-Yoccoz model,, Dyn. Syst., 22 (2007), 73.  doi: 10.1080/14689360601028126.  Google Scholar

[18]

Jeong-Yup Lee and Boris Solomyak, Pure point diffractive substitution Delone sets have the Meyer property,, Discrete Comput. Geom., 39 (2008), 319.  doi: 10.1007/s00454-008-9054-1.  Google Scholar

[19]

John H. Lowenstein and Franco Vivaldi, Scaling dynamics of a cubic interval-exchange transformation,, Dyn. Syst., 23 (2008), 283.  doi: 10.1080/14689360802253291.  Google Scholar

[20]

Miguel Ângelo de Sousa Mendes, Stability of periodic points in piecewise isometries of Euclidean spaces,, Ergodic Theory Dynam. Systems, 27 (2007), 183.  doi: 10.1017/S0143385706000460.  Google Scholar

[21]

Guillaume Poggiaspalla, John H. Lowenstein and Franco Vivaldi, Geometric representation of interval exchange maps over algebraic number fields,, Nonlinearity, 21 (2008), 149.  doi: 10.1088/0951-7715/21/1/009.  Google Scholar

[22]

Dominique Perrin and Jean-Éric Pin, "Infinite Words: Automata, Semigroups, Logic and Games,", 141 of Pure and Applied Mathematics. Elsevier, 141 (2004).   Google Scholar

[23]

N Pytheas Fogg, "Substitutions in Dynamics, Arithmetics and Combinatorics,", 1794 of Lecture Notes in Mathematics. Springer-Verlag, 1794 (2002).   Google Scholar

[24]

Martine Queffélec, "Substitution Dynamical Systems-Spectral Analysis,", 1294 of Lecture Notes in Mathematics. Springer-Verlag, 1294 (1987).   Google Scholar

[25]

Gérard Rauzy, Échanges d'intervalles et transformations induites,, Acta Arith., 34 (1979), 315.   Google Scholar

[26]

Boris Solomyak, Dynamics of self-similar tilings,, Ergodic Theory Dynam. Systems, 17 (1997), 695.   Google Scholar

[27]

Marcello Trovati and Peter Ashwin, Tangency properties of a pentagonal tiling generated by a piecewise isometry,, Chaos, 17 (2007).  doi: 10.1063/1.2825291.  Google Scholar

[28]

Serge L. Tabachnikov, On the dual billiard problem,, Adv. Math., 115 (1995), 221.  doi: 10.1006/aima.1995.1055.  Google Scholar

[29]

Wolfgang Thomas, Automata on infinite objects,, in, (1990), 133.   Google Scholar

[30]

William A. Veech, Gauss measures for transformations on the space of interval exchange maps,, Ann. of Math. (2), 115 (1982), 201.  doi: 10.2307/1971391.  Google Scholar

[31]

Jean-Christophe Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction,, in, (2006), 401.  doi: 10.1007/978-3-540-31347-2_12.  Google Scholar

[32]

Anton Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents,, Ann. Inst. Fourier (Grenoble), 46 (1996), 325.   Google Scholar

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