MARKOV GRAPHS OF ONE–DIMENSIONAL DYNAMICAL SYSTEMS AND THEIR DISCRETE ANALOGUES AND THEIR DISCRETE ANALOGUES

Abstract

One feature of the famous Sharkovsky’s theorem is that it can be proved using digraphs of a special type (the so–called Markov graphs). The most general definition assigns a Markov graph to every continuous map from the topological graph to itself. We show that this definition is too broad, i.e. every finite digraph can be viewed as a Markov graph of some one–dimensional dynamical system on a tree. We therefore consider discrete analogues of Markov graphs for vertex maps on combinatorial trees and characterize all maps on trees whose discrete Markov graphs are of the following types: complete, complete bipartite, the disjoint union of cycles, with every arc being a loop.

Keywords

• Markov graph
• one–dimensional dynamics
• Sharkovsky’s theorem