Abundance of minimal measures via entropy and multifractal analysis

This paper investigates the distribution and abundance of minimal measures (measures supported on minimal sets) in various dynamical systems, extending the well-known density results for general ergodic measures. We introduce the conditional minimal-intermediate-entropy property, which asserts that for any given entropy $h$ and potential integral $a$, the set of ergodic minimal measures satisfying $h_{\mu }(f)=h$ and $\int \phi d\mu =a$ is dense in the set of invariant measures satisfying these conditions. We establish that the conditional minimal-intermediate-entropy property 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. Our proofs rely on a constructive multi-horseshoe technique adapted to handle challenges of non-compactness and non-uniformity.