Dynamics Seminar

Event Information Entropy and dynamics of connected sets in dimension one
15:10 on Monday November 04, 2013
16:00 on Monday November 04, 2013
BA6183, Bahen Center, 40 St. George St.
Mykola Matviichuk

University of Toronto

By a tree we mean a connected space that is a finite union of closed intervals and contains no circle. The simplest example to keep in mind is a closed interval itself. Given a continuous map f from a tree into itself, we investigate how closed connected subsets of the tree behave under the action of f. We prove that every such a set, when iterated under f, is either asymtotically periodic or asymtotically degenerate. As an important corollary, we prove that whenever f has zero entropy, so does the functional envelope of f. By the functional envelope of f we mean the dynamical system on the set of all maps from the tree into itself, where each map g is sent to the composition f(g).