On the preservation of combinatorial types for maps on trees

Lluís Alsedà[1]; David Juher[2]; Pere Mumbrú[3]

  • [1] Universitat Autònoma de Barcelona, Departament de Matemàtiques, Edifici Cc,08913 Cerdanyola del Vallès, Barcelona (Espagne)
  • [2] Universitat de Girona, Departament d'Informàtica i Matemàtica Aplicada, Lluís Santaló s/n, 17071 Girona (Espagne)
  • [3] Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08071 Barcelona (Espagne)

Annales de l'institut Fourier (2005)

  • Volume: 55, Issue: 7, page 2375-2398
  • ISSN: 0373-0956

Abstract

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We study the preservation of the periodic orbits of an A -monotone tree map f : T ⟶ T in the class of all tree maps g : S ⟶ S having a cycle with the same pattern as A . We prove that there is a period-preserving injective map from the set of (almost all) periodic orbits of f into the set of periodic orbits of each map in the class. Moreover, the relative positions of the corresponding orbits in the trees T and S (which need not be homeomorphic) are essentially preserved.

How to cite

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Alsedà, Lluís, Juher, David, and Mumbrú, Pere. "On the preservation of combinatorial types for maps on trees." Annales de l'institut Fourier 55.7 (2005): 2375-2398. <http://eudml.org/doc/116257>.

@article{Alsedà2005,
abstract = {We study the preservation of the periodic orbits of an $A$-monotone tree map $f:\{T\}\longrightarrow \{T\}$ in the class of all tree maps $g:\{S\}\longrightarrow \{S\}$ having a cycle with the same pattern as $A$. We prove that there is a period-preserving injective map from the set of (almost all) periodic orbits of $f$ into the set of periodic orbits of each map in the class. Moreover, the relative positions of the corresponding orbits in the trees $T$ and $S$ (which need not be homeomorphic) are essentially preserved.},
affiliation = {Universitat Autònoma de Barcelona, Departament de Matemàtiques, Edifici Cc,08913 Cerdanyola del Vallès, Barcelona (Espagne); Universitat de Girona, Departament d'Informàtica i Matemàtica Aplicada, Lluís Santaló s/n, 17071 Girona (Espagne); Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, 08071 Barcelona (Espagne)},
author = {Alsedà, Lluís, Juher, David, Mumbrú, Pere},
journal = {Annales de l'institut Fourier},
keywords = {Tree maps; minimal dynamics; tree maps; combinatorial dynamics; periodic orbits; period-preserving map},
language = {eng},
number = {7},
pages = {2375-2398},
publisher = {Association des Annales de l'Institut Fourier},
title = {On the preservation of combinatorial types for maps on trees},
url = {http://eudml.org/doc/116257},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Alsedà, Lluís
AU - Juher, David
AU - Mumbrú, Pere
TI - On the preservation of combinatorial types for maps on trees
JO - Annales de l'institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 7
SP - 2375
EP - 2398
AB - We study the preservation of the periodic orbits of an $A$-monotone tree map $f:{T}\longrightarrow {T}$ in the class of all tree maps $g:{S}\longrightarrow {S}$ having a cycle with the same pattern as $A$. We prove that there is a period-preserving injective map from the set of (almost all) periodic orbits of $f$ into the set of periodic orbits of each map in the class. Moreover, the relative positions of the corresponding orbits in the trees $T$ and $S$ (which need not be homeomorphic) are essentially preserved.
LA - eng
KW - Tree maps; minimal dynamics; tree maps; combinatorial dynamics; periodic orbits; period-preserving map
UR - http://eudml.org/doc/116257
ER -

References

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  1. R. Adler, A. Konheim, M. McAndrew, Topological entropy, Trans. Am. Math. Soc. 114 (1965), 309-319 Zbl0127.13102MR175106
  2. Ll. Alsedà, J. Guaschi, J. Los, F. Mañosas, P. Mumbrú, Canonical representatives for patterns of tree maps, Topology 36 (1997), 1123-1153 Zbl0887.58012MR1445556
  3. Ll. Alsedà, F. Gautero, J. Guaschi, J. Los, F. Mañosas, P. Mumbrú, Patterns and minimal dynamics for graph maps, (2002) Zbl1082.37041
  4. Ll. Alsedà, J. Llibre, M. Misiurewicz, Periodic orbits of maps of Y, Trans. Amer. Math. Soc. 313 (1989), 475-538 Zbl0803.54032MR958882
  5. Ll. Alsedà, J. Llibre, M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, 5 (2002), World Scientific, second edition Zbl0963.37001MR1807264
  6. Ll. Alsedà, J. Llibre, M. Misiurewicz, C. Simó, Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point, Ergod. Th. & Dynam. Sys. 5 (1985), 501-517 Zbl0592.54037MR829854
  7. S. Baldwin, Generalizations of a theorem of Sharkovskii on orbits on continuous real -valued functions, Discrete Math. 67 (1987), 111-127 Zbl0632.06005MR913178
  8. S. Baldwin, An extension of Sharkovskii’s Theorem to the n -od, Ergod. Th. & Dynam. Sys. 11 (1991), 249-271 Zbl0741.58010MR1116640
  9. M. Misiurewicz, Z. Nitecki, Combinatorial patterns for maps of the interval, Mem. Amer. Math. Soc. 94 (1991) Zbl0745.58019MR1086562
  10. Y. Takahashi, A formula for the topological entropy of one-dimensional dynamics, Sci. Papers College Gen. Ed. Univ. Tokyo 30 (1980), 11-22 Zbl0465.58023MR581221

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