Spectral decomposition, periods of cycles and a conjecture of M. Misiurewicz for graph maps

Abstract

We describe a spectral decomposition of the set $\omega (f) = \bigcup\limits_{x \in X} {\omega (x)}$for a continuous map f : X → X of a one-dimensional branched manifold (“graph”) into itself similar to that of Jonker-Rand [JR], Hofbauer [H] and Nitecki [N] (see also [B1–B3]; the analogous decomposition holds for the sets Θ(f), $\overline {Per(f)}$. Denoting by P(f) the set of all periods of cycles of a map f we then verify the following Misiurewicz conjecture: for a graph X there exists an interger L=L(X) such that for a continuous map f : X → X the inclusion P(f) ⊃ {1,...,L} implies that P(f)=N (we prove also that such a map f has a positive entropy). It allows us to prove the following Theorem. Let f : X → X be a continuous graph map. Then the following statements are equivalent.

• 1) The map f has positive entropy.

• 2) There exists such n that P(f) ⊃ nN={i · n|i ∈ N}.