The depths and the attracting centres for continuous maps on local dendrites with the number of branch points being finite

摘要

Let Y be a local dendrite with the number of branch points being finite and f : Y -> Y be continuous. Denote by P(f), R(f), Omega(f) and omega(x, f) the set of periodic points of f, the set of recurrent points of f, the set of non-wandering points of f and the set of omega-limit points of x under f, respectively. Write omega(f) = boolean OR(x is an element of Y)omega(x, f) and omega(n+1)(f) = boolean OR(x is an element of omega)n(f)omega(x, f) and Omega(n+1) = Omega(f vertical bar Omega(n(f))) for any positive integer n. In this paper, we show that Omega(3)(f) = <(R(f))over bar> and the depth of f is at most 3, and omega(3)(f) = omega(2)(f). Furthermore, we show that <(R(f))over bar> = R(f) boolean OR <(P(f))over bar>. (C) 2020 Elsevier B.V. All rights reserved.