Fix a topological system (X,T), with its space K(X,T) of T-invariant Borel probabilities. If (Y,S) is a symbolic system (subshift) and ϕ:(Y,S)→(X,T) is a topological extension (factor map), then the function hϕext on K(X,T) which assigns to each μ the maximal entropy of a measure ν on Y mapping to μ is called the extension entropy function of ϕ. The infimum of such functions over all symbolic extensions is called the symbolic extension entropy function and is denoted by hsex. In this paper we completely characterize these functions in terms of functional analytic properties of an entropy structure on (X,T). The entropy structure ℋ is a sequence of entropy functions hk defined with respect to a refining sequence of partitions of X (or of X×Z, for some auxiliary system (Z,R) with simple dynamics) whose boundaries have measure zero for all the invariant Borel probabilities. We develop the functional analysis and computational techniques to produce many dynamical examples; for instance, we resolve in the negative the question of whether the infimum of the topological entropies of symbolic extensions of (X,T) must always be attained, and we show that the maximum value of hsex
need not be achieved at an ergodic measure. We exhibit several characterizations of the asymptotically h-expansive systems of Misiurewicz, which emerge as a fundamental natural class in the context of the entropy structure. The results of this paper are required for the Downarowicz-Newhouse results [DN] on smooth dynamical systems.