February  2010, 27(1): 369-382. doi: 10.3934/dcds.2010.27.369

A note on the coding of orbits in certain discontinuous maps

1. 

Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 4619 - 007 PORTO, Portugal

Received  March 2008 Revised  May 2009 Published  February 2010

We study certain discontinuous maps by means of a coding map defined on a special partition of the phase space which is such that the points of discontinuity of the map, $\mathcal{D}$, all belong to the union of the boundaries of elements in the partition.
   For maps acting locally as homeomorphisms in a compact space, we prove that, if the set of points whose trajectory comes arbitrarily close to the set of discontinuities is closed and not the full space then all points not in that set are rationally coded, i.e., their codings eventually settle on a repeated block of symbols.
   In particular, for piecewise isometries, which are discontinuous maps acting locally as isometries, we give a topological description of the equivalence classes of the coding map in terms of the connected components generated by the closure of the preimages of $\mathcal{D}$.
Citation: Miguel Mendes. A note on the coding of orbits in certain discontinuous maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 369-382. doi: 10.3934/dcds.2010.27.369
[1]

Marcello Trovati, Peter Ashwin, Nigel Byott. Packings induced by piecewise isometries cannot contain the arbelos. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 791-806. doi: 10.3934/dcds.2008.22.791

[2]

Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381

[3]

Hun Ki Baek, Younghae Do. Dangerous Border-Collision bifurcations of a piecewise-smooth map. Communications on Pure & Applied Analysis, 2006, 5 (3) : 493-503. doi: 10.3934/cpaa.2006.5.493

[4]

Zhiying Qin, Jichen Yang, Soumitro Banerjee, Guirong Jiang. Border-collision bifurcations in a generalized piecewise linear-power map. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 547-567. doi: 10.3934/dcdsb.2011.16.547

[5]

Arek Goetz. Dynamics of a piecewise rotation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 593-608. doi: 10.3934/dcds.1998.4.593

[6]

Changzhi Wu, Kok Lay Teo, Volker Rehbock. Optimal control of piecewise affine systems with piecewise affine state feedback. Journal of Industrial & Management Optimization, 2009, 5 (4) : 737-747. doi: 10.3934/jimo.2009.5.737

[7]

Ciprian D. Coman. Dissipative effects in piecewise linear dynamics. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 163-177. doi: 10.3934/dcdsb.2003.3.163

[8]

Claudio Buzzi, Claudio Pessoa, Joan Torregrosa. Piecewise linear perturbations of a linear center. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3915-3936. doi: 10.3934/dcds.2013.33.3915

[9]

Dag Lukkassen, Annette Meidell, Peter Wall. On the conjugate of periodic piecewise harmonic functions. Networks & Heterogeneous Media, 2008, 3 (3) : 633-646. doi: 10.3934/nhm.2008.3.633

[10]

Miguel Ângelo De Sousa Mendes. Quasi-invariant attractors of piecewise isometric systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 323-338. doi: 10.3934/dcds.2003.9.323

[11]

Viviane Baladi, Sébastien Gouëzel. Banach spaces for piecewise cone-hyperbolic maps. Journal of Modern Dynamics, 2010, 4 (1) : 91-137. doi: 10.3934/jmd.2010.4.91

[12]

Peter Ashwin, Xin-Chu Fu. Symbolic analysis for some planar piecewise linear maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1533-1548. doi: 10.3934/dcds.2003.9.1533

[13]

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

[14]

Viviane Baladi, Daniel Smania. Smooth deformations of piecewise expanding unimodal maps. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 685-703. doi: 10.3934/dcds.2009.23.685

[15]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451

[16]

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

[17]

C. Kopf. Symbol sequences and entropy for piecewise monotone transformations with discontinuities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 299-304. doi: 10.3934/dcds.2000.6.299

[18]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917

[19]

Hui Liang, Hermann Brunner. Collocation methods for differential equations with piecewise linear delays. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1839-1857. doi: 10.3934/cpaa.2012.11.1839

[20]

José Luis Bravo, Manuel Fernández, Antonio Tineo. Periodic solutions of a periodic scalar piecewise ode. Communications on Pure & Applied Analysis, 2007, 6 (1) : 213-228. doi: 10.3934/cpaa.2007.6.213

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]