Topological entropy of continuous self-maps on a graph
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(2019) 38:154
Topological entropy of continuous self-maps on a graph Juan Luis García Guirao1
· Jaume Llibre2 · Wei Gao1,3
Received: 17 January 2019 / Revised: 21 September 2019 / Accepted: 24 September 2019 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019
Abstract Let G be a graph and f be a continuous self-map on G. Using the Lefschetz zeta function of f , we provide a sufficient condition in order that f has positive topological entropy. Moreover, for some classes of graphs we improve this condition making it easier to check. Keywords Topological graph · Discrete dynamical systems · Lefschetz numbers · Lefschetz zeta function · Periodic point · Period · Topological entropy Mathematics Subject Classification 37E25 · 37C25 · 37C30
1 Introduction and statement of the main results One of the ways to measure the complexity of a dynamical system shows that its topological entropy is positive. The topological entropy is a nonnegative real number such that the larger this number, the greater the complexity of the dynamical system. The notion of topological entropy was introduced by Adler, Konheim and McAndrew in 1965, see Adler et al. (1965). Later on this definition was modified by Kolmogorov–Sinai introducing the metric entropy (see Walters 1992), of course both definitions are related. After Bowen (1971) provided a weaker definition of topological entropy which clarifies the meaning of the topological entropy. Roughly speaking, for a system given by an iterated map (as the ones studied in this
Communicated by Carlos Hoppen.
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Juan Luis García Guirao [email protected] Jaume Llibre [email protected] Wei Gao [email protected]
1
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 Barcelona, Bellaterra, 08193 Barcelona, Catalonia, Spain
3
School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China
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paper) the topological entropy essentially is the exponential growth rate of the number of distinguishable orbits of the iterates. In this work, a graph G will be a compact connected space containing a finite set of points V such that G\V has finitely many open connected components, each one of them homeomorphic to the interval (0, 1). These components are called the edges of G, and the points of V are called the vertices of G. The edges are disjoint from the vertices, and the vertices are at the boundary of the edges. For a graph G, the degree of a vertex is the number of edges having this vertex in its boundary, if an edge has both boundaries in the same vertex then we compute this edge twice in the definition of the degree of that vertex. 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. Let G be a graph and f : G → G a continuous map. A point x ∈ G is periodic of period k if f k
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