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Patterns and Minimal Dynamics for Graph Maps
Author(s) -
Alsedà Lluís,
Gautero François,
Guaschi John,
Los Jérôme,
Mañosas Francesc,
Mumbrú Pere
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611505015224
Subject(s) - mathematics , homotopy , endomorphism , pure mathematics , homotopy lifting property , homotopy group , n connected , whitehead theorem , graph , cofibration , rigidity (electromagnetism) , discrete mathematics , structural engineering , engineering
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. 2000 Mathematics Subject Classification 37E30, 37E25, 37E15.