Uniqueness of maximal entropy measure on essential spanning forests

Uniqueness of maximal entropy measure on essential spanning forests

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for which ∪Gn = G. Pemantle’s arguments (1991) imply that the uniform measures on spanning trees of Gn converge weakly to an Aut(G)-invariant measure μG on essential spanning forests of G. We show that...

Measures of maximal entropy

We extend the results of Walters on the uniqueness of invariant measures with maximal entropy on compact groups to an arbitrary locally compact group. We show that the maximal entropy is attained at the left Haar measure and the measure of maximal entropy is unique.

Entropy Estimate for Maps on Forests

A 1993 result of J. Llibre, and M. Misiurewicz, (Theorem A [5]), states that if a continuous map f of a graph into itself has an s-horseshoe, then the topological entropy of f is greater than or equal to logs, that is h( f ) ? logs. Also a 1980 result of L.S. Block, J. Guckenheimer, M. Misiurewicz and L.S. Young (Lemma 1.5 [3]) states that if G is an A-graph of f then h(G) ? h( f ). In this pap...

A Measure of Maximal Entropy for Nonuniformly Hyperbolic Hilbert Geometries

Some Transformations Having a Unique Measure with Maximal Entropy

1. Introduction Let X be a compact metric space and T: X-> X a homeomorphism of X onto X. Let M(T) denote the collection of all T-invariant Borel probability measures on X. By Krylov and Bogolioubov's work we know M(T) is non-empty (see [10]). M{T) is a convex set and closed in the weak topology. For /x e M(T), h(T, p) will denote the measure-theoretic entropy of T with respect to p. Ifh top (T...