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

Pentagonal domain exchange

Shigeki Akiyama 1, and  Edmund Harriss 2,

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.
Keywords: numeration systems., pisot number, piecewise isometry, symbolic dynamics, Pentagon.
Mathematics Subject Classification: 37E99, 11K16, 11A6.
Citation: Shigeki Akiyama, Edmund Harriss. Pentagonal domain exchange. Discrete and Continuous Dynamical Systems, 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), 197-251.

[2]

Peter Ashwin and Xin-Chu Fu, On the geometry of orientation-preserving planar piecewise isometries, J. Nonlinear Sci., 12 (2002), 207-240. doi: 10.1007/s00332-002-0477-1.

[3]

Roy Adler, Bruce Kitchens and Charles Tresser, Report on the dynamics of certain piecewise isometries of the torus, in "Dynamical systems (Luminy-Marseille, 1998)," World Sci. Publ., River Edge, NJ, (2000), 231-247.

[4]

Pierre Arnoux and Gérard Rauzy, Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. France, 119 (1991), 199-215.

[5]

Nicolas Bedaride and Julien Cassaigne, Outer billiard outside regular polygons, J. Lond. Math. Soc. (2), 84 (2011), 303-324. doi: 10.1112/jlms/jdr010.

[6]

Xavier Bressaud and Guillaume Poggiaspalla, A tentative classification of bijective polygonal piecewise isometries, Experiment. Math., 16 (2007), 77-99. doi: 10.1080/10586458.2007.10128987.

[7]

Enrico Bombieri and Jean E Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections, in "The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985)," Amer. Math. Soc., Providence, RI, (1987), 241-264.

[8]

Arek Goetz, A self-similar example of a piecewise isometric attractor, in "Dynamical systems (Luminy-Marseille, 1998)," World Sci. Publ., River Edge, NJ, (2000), 248-258.

[9]

Arek Goetz, Piecewise isometries-an emerging area of dynamical systems, in "Fractals in Graz (2001)," Trends Math. Birkhäuser, Basel, (2003), 135-144.

[10]

Arek Goetz, Return maps in cyclotomic piecewise similarities, Dyn. Syst., 20 (2005), 255-265. doi: 10.1080/14689360500092918.

[11]

John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.

[12]

Shunji Ito and Hui Rao, Purely periodic $\beta$-expansions with Pisot unit base, Proc. Amer. Math. Soc., 133 (2005), 953-964. doi: 10.1090/S0002-9939-04-07794-9.

[13]

Teturo Kamae, Numeration systems, fractals and stochastic processes, Israel J. Math., 149 (2005), 87-135. doi: 10.1007/BF02772537.

[14]

Konstantin L. Kouptsov, John Lowenstein and Franco Vivaldi, Quadratic rational rotations of the torus and dual lattice maps, Nonlinearity, 15 (2002), 1795-1842. doi: 10.1088/0951-7715/15/6/306.

[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), 49-66. doi: 10.1063/1.166240.

[16]

John H. Lowenstein, Konstantin L. Kouptsov and Franco Vivaldi, Recursive tiling and geometry of piecewise rotations by $\pi/7$, Nonlinearity, 17 (2004), 371-395. doi: 10.1088/0951-7715/17/2/001.

[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), 73-106. doi: 10.1080/14689360601028126.

[18]

Jeong-Yup Lee and Boris Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338. doi: 10.1007/s00454-008-9054-1.

[19]

John H. Lowenstein and Franco Vivaldi, Scaling dynamics of a cubic interval-exchange transformation, Dyn. Syst., 23 (2008), 283-298. doi: 10.1080/14689360802253291.

[20]

Miguel Ângelo de Sousa Mendes, Stability of periodic points in piecewise isometries of Euclidean spaces, Ergodic Theory Dynam. Systems, 27 (2007), 183-197. doi: 10.1017/S0143385706000460.

[21]

Guillaume Poggiaspalla, John H. Lowenstein and Franco Vivaldi, Geometric representation of interval exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.

[22]

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

[23]

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

[24]

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

[25]

Gérard Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[26]

Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738.

[27]

Marcello Trovati and Peter Ashwin, Tangency properties of a pentagonal tiling generated by a piecewise isometry, Chaos, 17 (2007), pp. 11, 043129. doi: 10.1063/1.2825291.

[28]

Serge L. Tabachnikov, On the dual billiard problem, Adv. Math., 115 (1995), 221-249. doi: 10.1006/aima.1995.1055.

[29]

Wolfgang Thomas, Automata on infinite objects, in "Handbook of theoretical computer science, Vol. B," Elsevier, Amsterdam, (1990), 133-191.

[30]

William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

[31]

Jean-Christophe Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction, in "Frontiers in Number Theory, Physics, and Geometry. I," Springer, Berlin, (2006), 401-435. doi: 10.1007/978-3-540-31347-2_12.

[32]

Anton Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.

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), 197-251.

[2]

Peter Ashwin and Xin-Chu Fu, On the geometry of orientation-preserving planar piecewise isometries, J. Nonlinear Sci., 12 (2002), 207-240. doi: 10.1007/s00332-002-0477-1.

[3]

Roy Adler, Bruce Kitchens and Charles Tresser, Report on the dynamics of certain piecewise isometries of the torus, in "Dynamical systems (Luminy-Marseille, 1998)," World Sci. Publ., River Edge, NJ, (2000), 231-247.

[4]

Pierre Arnoux and Gérard Rauzy, Représentation géométrique de suites de complexité $2n+1$, Bull. Soc. Math. France, 119 (1991), 199-215.

[5]

Nicolas Bedaride and Julien Cassaigne, Outer billiard outside regular polygons, J. Lond. Math. Soc. (2), 84 (2011), 303-324. doi: 10.1112/jlms/jdr010.

[6]

Xavier Bressaud and Guillaume Poggiaspalla, A tentative classification of bijective polygonal piecewise isometries, Experiment. Math., 16 (2007), 77-99. doi: 10.1080/10586458.2007.10128987.

[7]

Enrico Bombieri and Jean E Taylor, Quasicrystals, tilings, and algebraic number theory: Some preliminary connections, in "The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985)," Amer. Math. Soc., Providence, RI, (1987), 241-264.

[8]

Arek Goetz, A self-similar example of a piecewise isometric attractor, in "Dynamical systems (Luminy-Marseille, 1998)," World Sci. Publ., River Edge, NJ, (2000), 248-258.

[9]

Arek Goetz, Piecewise isometries-an emerging area of dynamical systems, in "Fractals in Graz (2001)," Trends Math. Birkhäuser, Basel, (2003), 135-144.

[10]

Arek Goetz, Return maps in cyclotomic piecewise similarities, Dyn. Syst., 20 (2005), 255-265. doi: 10.1080/14689360500092918.

[11]

John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747. doi: 10.1512/iumj.1981.30.30055.

[12]

Shunji Ito and Hui Rao, Purely periodic $\beta$-expansions with Pisot unit base, Proc. Amer. Math. Soc., 133 (2005), 953-964. doi: 10.1090/S0002-9939-04-07794-9.

[13]

Teturo Kamae, Numeration systems, fractals and stochastic processes, Israel J. Math., 149 (2005), 87-135. doi: 10.1007/BF02772537.

[14]

Konstantin L. Kouptsov, John Lowenstein and Franco Vivaldi, Quadratic rational rotations of the torus and dual lattice maps, Nonlinearity, 15 (2002), 1795-1842. doi: 10.1088/0951-7715/15/6/306.

[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), 49-66. doi: 10.1063/1.166240.

[16]

John H. Lowenstein, Konstantin L. Kouptsov and Franco Vivaldi, Recursive tiling and geometry of piecewise rotations by $\pi/7$, Nonlinearity, 17 (2004), 371-395. doi: 10.1088/0951-7715/17/2/001.

[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), 73-106. doi: 10.1080/14689360601028126.

[18]

Jeong-Yup Lee and Boris Solomyak, Pure point diffractive substitution Delone sets have the Meyer property, Discrete Comput. Geom., 39 (2008), 319-338. doi: 10.1007/s00454-008-9054-1.

[19]

John H. Lowenstein and Franco Vivaldi, Scaling dynamics of a cubic interval-exchange transformation, Dyn. Syst., 23 (2008), 283-298. doi: 10.1080/14689360802253291.

[20]

Miguel Ângelo de Sousa Mendes, Stability of periodic points in piecewise isometries of Euclidean spaces, Ergodic Theory Dynam. Systems, 27 (2007), 183-197. doi: 10.1017/S0143385706000460.

[21]

Guillaume Poggiaspalla, John H. Lowenstein and Franco Vivaldi, Geometric representation of interval exchange maps over algebraic number fields, Nonlinearity, 21 (2008), 149-177. doi: 10.1088/0951-7715/21/1/009.

[22]

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

[23]

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

[24]

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

[25]

Gérard Rauzy, Échanges d'intervalles et transformations induites, Acta Arith., 34 (1979), 315-328.

[26]

Boris Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems, 17 (1997), 695-738.

[27]

Marcello Trovati and Peter Ashwin, Tangency properties of a pentagonal tiling generated by a piecewise isometry, Chaos, 17 (2007), pp. 11, 043129. doi: 10.1063/1.2825291.

[28]

Serge L. Tabachnikov, On the dual billiard problem, Adv. Math., 115 (1995), 221-249. doi: 10.1006/aima.1995.1055.

[29]

Wolfgang Thomas, Automata on infinite objects, in "Handbook of theoretical computer science, Vol. B," Elsevier, Amsterdam, (1990), 133-191.

[30]

William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2), 115 (1982), 201-242. doi: 10.2307/1971391.

[31]

Jean-Christophe Yoccoz, Continued fraction algorithms for interval exchange maps: an introduction, in "Frontiers in Number Theory, Physics, and Geometry. I," Springer, Berlin, (2006), 401-435. doi: 10.1007/978-3-540-31347-2_12.

[32]

Anton Zorich, Finite Gauss measure on the space of interval exchange transformations. Lyapunov exponents, Ann. Inst. Fourier (Grenoble), 46 (1996), 325-370.

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