On essential entropy-carrying sets

In this paper we study properties of essential entropy-carrying sets of a continuous map on a compact metric space. If f : X → X is continuous on a compact metric space X, then the intersection of all essential entropy-carrying sets of f may or may not be an essential entropy-carrying set of f. When this intersection is an essential entropy-carrying set we denote it by E ( f ) , the least essential entropy-carrying set, otherwise we say that E ( f ) does not exist. We present an example where E ( f ) does not exist but also find a sufficient condition for E ( f ) to exist. If f is a piecewise monotone map, we show that E ( f ) exists and is the finite union of the entropy-carrying sets in the Nitecki Decomposition of the nonwandering set of f intersected with the closure of the periodic points of f. When E ( f ) exists we study how it relates to other entropy-carrying sets of f including subsets of itself.