Entropy and Rotation Sets: a Toymodel Approach

Given a continuous dynamical system f on a compact metric space X and a continuous potential Φ : X → R, the generalized rotation set is the subset of R consisting of all integrals of Φ with respect to all invariant probability measures. The localized entropy at a point in the rotation set is defined as the supremum of the measuretheoretic entropies over all invariant measures whose integrals produce that point. In this paper we provide an introduction to the theory of rotation sets and localized entropies. Moreover, we consider a shift map and construct a Lipschitz continuous potential, for which we are able to explicitly compute the geometric shape of the rotation set and its boundary measures. We show that at an exposed point on the boundary there are exactly two ergodic localized measures of maximal entropy. 1. Motivation An important goal in dynamical systems is to understand the various typical dynamical behaviors of a given system. By Birkhoff’s Ergodic Theorem we can associate typical dynamical behavior with an invariant ergodic probability measure. However, for many systems the set of invariant measures is rather large. This raises the question as to which invariant measure is the natural choice to consider. One natural candidate to consider is a measure that maximizes a certain topological complexity, i.e. entropy, among all invariant probability measures. Such a measure (if it exists) is called a measure of maximal entropy. On the other hand, many systems exhibit an abundance of invariant measures in which case the restriction to one invariant measure may result in the loss of other relevant dynamical information associated with other measures. A natural approach to overcome this problem is to partition the space of invariant measures into smaller classes and then to consider entropy maximizing measures within these classes. In this paper we partition the space of invariant measures by identifying measures according to their integral averages on a given predescribed set of continuous potentials. While