On the periods of a continuous self-map on a graph
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(2019) 38:78
On the periods of a continuous self-map on a graph Juan Luis García Guirao1 · Jaume Llibre2 Received: 31 December 2017 / Revised: 29 October 2018 / Accepted: 22 November 2018 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract Let G be a graph and f be a continuous self-map on G. We present new and known results (from another point of view) on the periods of the periodic orbits of f using mainly the action of f on its homology, or the shape of the graph G. Keywords Topological graph · Discrete dynamical systems · Lefschetz numbers · Lefschetz zeta function · Periodic point · Period Mathematics Subject Classification 37E25 · 37C25 · 37C30
1 Introduction and statement of the main results A discrete dynamical system (G, f ) is formed by a continuous map f : G → G, where G is a topological space. A point x ∈ G is periodic of period k if f k (x) = x and f i (x) = x if 0 < i < k. If k = 1, then x is called a fixed point. Per( f ) denotes the set of periods of all the periodic points of f . The orbit of the point x ∈ G is the set {x, f (x), f 2 (x), . . . , f n (x), . . .}, whereby f n we denote the composition of f with itself n times. To knowledge the behavior of all different kinds of orbits of f is to study the dynamics of the map f . Many times the periodic points play an important role for understanding the dynamics of a discrete dynamical system. One of the best known results in this direction is the paper Period three implies chaos for continuous interval maps, see Li and Yorke (1975). Here, a graph G is a compact connected space containing a finite set V , such that G\V has finitely many open connected components, each one homeomorphic to the interval (0, 1),
Communicated by Maria Aguieiras de Freitas.
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Juan Luis García Guirao [email protected] Jaume Llibre [email protected]
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Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Hospital de Marina, 30203 Cartagena, Región de Murcia, Spain
2
Departament de Matemàtiques, Universitat Autònoma de, Bellaterra, 08193 Barcelona, Catalonia, Spain
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called edges of G, and the points of V are called the vertexes of G. The edges are disjoint from the vertexes, and the vertexes are at the boundary of the edges. In this paper, we shall work with a graph G. Our goal is to study the periods of the periodic points of the continuous maps f : G → G. Independently of the fact that to study the set of periods of these kinds of graph maps is relevant by itself for understanding their dynamics. The graph maps are relevant for studying the dynamics of some different kinds of surface maps, see, for instance (Handel and Thurston 1985; Mendes de Jesus 2017). The degree of a vertex V of a graph G is the number of edges having V in its boundary if an edge has both boundaries in V , then we count this edge twice. An endpoint of a graph G is a vertex of degree one. A branching point of a graph G is a vertex of degree at least three. The homologica
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