Conditional entropy realization and approximation by uniquely ergodic measures

Abstract

This paper studies conditional entropy realization and weak* approximation by uniquely ergodic measures with compact support. We prove that, after fixing an admissible potential average and an entropy strictly below the entropy supremum over the corresponding average fiber, every invariant measure satisfying these two exact constraints can be approximated weakly* by uniquely ergodic measures with compact support satisfying the same constraints. Each approximating measure is the unique invariant measure on its minimal support, whose topological entropy equals the prescribed metric entropy. This result holds for three broad classes of systems: topologically expanding maps (including topologically Anosov systems), transitive countable Markov shifts, and symbolic systems with non-uniform structure. The proof uses a nested multi-horseshoe construction, with separate arguments addressing non-invertibility, non-compactness and non-uniformity.

58 pages