Complete realization of multifractal entropy spectra and pressure functions

Abstract

We give a complete characterization of the multifractal entropy spectra arising from continuous vector-valued potentials on transitive two-sided shifts of finite type. We prove that, in every finite dimension, any nonnegative upper semicontinuous concave function on a compact convex set that attains the topological entropy as its maximum at a unique point is realized as the entropy spectrum of a potential whose rotation set is precisely that set. Every such spectrum moreover admits arbitrarily many pairwise non-cohomologous realizations. Via Legendre--Fenchel duality, this characterization yields complete pressure flexibility over the entire parameter space. In particular, it resolves the whole-space problem posed by Kucherenko and Quas \cite{KQ2022}. A separate construction based on entropy paths extends scalar spectrum and pressure realization to a substantially broader class of dynamical systems. Finally, with respect to the closed-graph Hausdorff metric, we prove that the spectrum map is lower semicontinuous in every finite dimension, whereas upper semicontinuity fails on a dense set for scalar potentials on transitive shifts of finite type.