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Chain recurrent points and topological entropy of a tree map. (English) Zbl 1015.37014

Let \(T\) be a tree (i.e. a one-dimensional compactly connected branch manifold without cycles) and \(f\in C^0(T)\) be a continuous tree map. In the paper, the notion of division for invariant closed subsets of a tree map is introduced. The main results are as follows.
Theorem 1. If \(f\in C^0(T)\), then \(f\) has zero topological entropy if and only if for any \(x\in CR(f)-P(f)\) (\(CR(f)\) and \(P(f)\) are sets of chain recurrent and periodic orbits of \(f\), respectively) and each natural \(s\) the orbit of \(x\) under \(f^s\) has a division.
Theorem 2. If \(f\in C^0(T)\) has zero topological entropy, then for any \(x\in CR(f)-P(f)\) the \(\omega\)-limit set of \(x\) is an infinite minimal set.

MSC:

37B40 Topological entropy
37E25 Dynamical systems involving maps of trees and graphs
37B20 Notions of recurrence and recurrent behavior in topological dynamical systems
54H20 Topological dynamics (MSC2010)
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