The construction of chaotic maps in the sense of Devaney on dendrites which commute to continuous maps on the unit interval
Let \(f: I \to I\) be a continuous map from the unit interval \(I\) into itself. For a periodic orbit of \(p\) with a primitive period \(n\) we denote this orbit by \(P = \{p_0, \dots, p_{n-1}\}\) assuming that \(0 \leq p_0 < p_1 < \dots < p_{n-1} \leq 1\), i.e. \(p_i\) need not be equal to \(f(p_{i-1})\). Then \(S(I,P) = (p_0, p_1) \cup (p_1, p_2) \cup \dots \cup (p_{n-2},p_{n-1})\). The map \(f\) is said to be pointwise \(P-\)expansive if for any interval \((p_k,p_{k+1})\) there exists a positive integer \(l\) such that \((f^l(p_k),f^l(p_{k+1})) \cap P \neq \emptyset\). The first main theorem of the paper says that a continuous map \(f: I \to I\) has positive topological entropy if and only if there exists a periodic orbit \(P\) of \(f\) such that \(f\) is pointwise \(P\)-expansive. The second part of the paper is devoted to the semiconjugacy between a given continuous map and a suitably constructed map on a dendrite. A dendrite is a locally connected, uniquely arcwise continuum. For a continuous map \(f: I \to I\) and a periodic orbit \(P\) of \(f\) the authors construct the dendrite \(Z(f,P)\) and a continuous surjective map \(\pi: I \to Z(f,P)\). Also, a continuous map \(g: Z(f,P) \to Z(f,P)\) is constructed such that \(g\) is semiconjugate to \(f\), i.e. \(\pi \circ f = g \circ \pi\). The second main theorem of the paper says that if the period \(n\) of \(P\) is an odd number which is a prime or the supremum in the Sharkovsky ordering, then \(g\) is topologically mixing and chaotic in the sense of Devaney; moreover, \(g\) has positive topological entropy. Some other lemmas which lead to main theorems and some corollaries are presented. Several examples are also given.
scientific article; zbMATH DE number 2110772