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Patterns and minimal dynamics for graph maps

Published online by Cambridge University Press:  23 August 2005

Lluís Alsedà ,
François Gautero ,
John Guaschi ,
Jérôme Los ,
Francesc Mañosas  and
Pere Mumbrú
Lluís Alsedà
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain. E-mail: alseda@mat.uab.es, manyosas@mat.uab.es
François Gautero
Affiliation:
Université Blaise Pascal, Clermont-Ferrand II, Campus des Cézeaux, Laboratoire de Mathématiques, 63177 Aubière, France. E-mail: Francois.Gautero@math.univ-bpclermont.fr
John Guaschi
Affiliation:
Laboratoire de Mathématiques Emile Picard, U.M.R. CNRS 5580, Université Toulouse III, 118 route de Narbonne, 31062 Toulouse Cedex 4, France. E-mail: guaschi@picard.ups-tlse.fr
Jérôme Los
Affiliation:
Université Aix-Marseille I, Centre de Mathématiques et Informatique, 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France. E-mail: los@gyptis.univ-mrs.fr
Francesc Mañosas
Affiliation:
Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08913 Cerdanyola del Vallès, Barcelona, Spain. E-mail: alseda@mat.uab.es, manyosas@mat.uab.es
Pere Mumbrú
Affiliation:
Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain. E-mail: mumbru@mat.ub.es
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Abstract

We study the rigidity problem for periodic orbits of (continuous) graph maps belonging to the same homotopy equivalence class. Since the underlying spaces are not necessarily homeomorphic, we define a new notion of pattern which enables us to compare periodic orbits of self-maps of homotopy-equivalent spaces. This definition unifies the known notions of pattern for other spaces. The two main results of the paper are as follows: given a free group endomorphism, we study the persistence under homotopy of the periodic orbits of its topological representatives, and in the irreducible case, we prove the minimality (within the homotopy class) of the set of periodic orbits of its efficient representatives.

Keywords

graph mapsminimal dynamicsefficient representativescombinatorial patterns37E3037E2537E15
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 91 , Issue 2 , September 2005 , pp. 414 - 442
Copyright
2005 London Mathematical Society

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