Linear and metric maps on trees via Markov graphs

Abstract

The main focus of combinatorial dynamics is put on the structure of periodic points (and the corresponding orbits) of topological dynamical systems. The first result in this area is the famous Sharkovsky's theorem which completely describes the coexistence of periods of periodic points for a continuous map from the closed unit interval to itself. One feature of this theorem is that it can be proved using digraphs of a special type (the so-called periodic graphs). In this paper we use Markov graphs (which are the natural generalization of periodic graphs in case of dynamical systems on trees) as a tool to study several classes of maps on trees. The emphasis is put on linear and metric maps.

Short proofs for long induced paths

“…The classical literature on Random Graphs considers many problems on finding induced subgraphs, starting from several independent papers [4,21,28] in the 1970's calculating the asymptotic independence number of G(n, p) for fixed p and large n. This was later extended by Frieze [17] to p = c/n for large constant c (as noted in [8], this extends to all p ≥ c/n). Erdős and Palka [16] initiated the study of induced trees in G(n, p), which developed a large literature [9,18,25,27] before its final resolution by de la Vega [10], showing that the size of the largest induced tree matches the asymptotics found earlier for the independence number: it is ∼ 2 log q (np) for all p ≥ c/n for large constant c. The Longest Induced Path problem in G(n, p) is also classical [26,31], and was recently resolved asymptotically by Draganić, Glock and Krivelevich [12].…”