-
Previous Article
On the large deviation rates of non-entropy-approachable measures
- DCDS Home
- This Issue
-
Next Article
An introduction to migration-selection PDE models
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 |
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. |
[1] |
Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739 |
[2] |
David Ralston. Heaviness in symbolic dynamics: Substitution and Sturmian systems. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 287-300. doi: 10.3934/dcdss.2009.2.287 |
[3] |
Song-Mei Huan, Xiao-Song Yang. On the number of limit cycles in general planar piecewise linear systems. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2147-2164. doi: 10.3934/dcds.2012.32.2147 |
[4] |
Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 |
[5] |
Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533 |
[6] |
Jim Wiseman. Symbolic dynamics from signed matrices. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621 |
[7] |
George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete and Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43 |
[8] |
Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301 |
[9] |
P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1 |
[10] |
Arek Goetz. Dynamics of a piecewise rotation. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593 |
[11] |
Jose S. Cánovas, Tönu Puu, Manuel Ruiz Marín. Detecting chaos in a duopoly model via symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 269-278. doi: 10.3934/dcdsb.2010.13.269 |
[12] |
Nicola Soave, Susanna Terracini. Symbolic dynamics for the $N$-centre problem at negative energies. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3245-3301. doi: 10.3934/dcds.2012.32.3245 |
[13] |
Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581 |
[14] |
Frédéric Naud. Birkhoff cones, symbolic dynamics and spectrum of transfer operators. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 581-598. doi: 10.3934/dcds.2004.11.581 |
[15] |
Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487 |
[16] |
Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082 |
[17] |
Mike Boyle. The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics. Journal of Modern Dynamics, 2019, 15: 427-435. doi: 10.3934/jmd.2019026 |
[18] |
Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete and Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163 |
[19] |
Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020 |
[20] |
Xiaolei Zhang, Yanqin Xiong, Yi Zhang. The number of limit cycles by perturbing a piecewise linear system with three zones. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1833-1855. doi: 10.3934/cpaa.2022049 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]