Complete realization of multifractal entropy spectra

TL;DR

This paper proves a comprehensive realization theorem for multifractal entropy spectra in dynamical systems, demonstrating the ability to realize any suitable spectrum and analyzing its dependence on potentials.

Contribution

It establishes a complete realization theorem for multifractal entropy spectra and explores their potential dependence and pressure duality in dynamical systems.

Findings

Every continuous concave spectrum with maximum H can be realized by a system with topological entropy H.

Multiple non-cohomologous potentials can produce the same spectrum.

The paper proves lower semicontinuity and dense failure of upper semicontinuity of spectra in certain systems.

Abstract

We prove a complete realization theorem for multifractal entropy spectra of continuous potentials on a broad class of dynamical systems. More precisely, for every $H>0$ and every continuous concave function on a compact interval with maximum value $H$ attained at a unique point, each system in this class with topological entropy $H$ admits a continuous potential whose multifractal entropy spectrum is exactly that function. The same spectrum can moreover be realized by arbitrarily many pairwise non-cohomologous potentials. We also study the potential dependence of entropy spectra in the Hausdorff graph metric, proving general lower semicontinuity and dense failure of upper semicontinuity for non-trivial mixing subshifts of finite type. Finally, as an application of the realization theorem, we use the Legendre transform duality between multifractal entropy spectra and pressure functions to derive a pressure flexibility theorem for pressure functions defined on the whole real line, thereby giving an affirmative answer to a question of Kucherenko and Quas~\cite{KQ2022}.