Tomasz Downarowicz (University of Maryland)

Entropy properties in topological dynamics.

Abstract. In a non-uniquely ergodic system (action of a single homeomorphism T) the seemingly most complete information about entropy is in the entropy function (assigning to each invariant measure the entropy of T with respect to this measure). However, for many aspects like asymptotic h-expansiveness, existence of a subshift cover, etc., we need a more subtle tool, namely we need to know how the entropy function is approximated by certain relative entropy functions. We will present some recent results concerning the entropy function, relative entropy functions, relative variational principles, existence of subshift covers, criteria for asymptotic h-expansiveness, and the like. We will propose a new topological invariant (which we call "entropy structure") carrying complete information about the above discussed entropy properties of the system.