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Complete realization of multifractal entropy spectra

Dynamical Systems 2026-05-21 v1

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>0H>0 and every continuous concave function on a compact interval with maximum value HH attained at a unique point, each system in this class with topological entropy HH 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}.

Keywords

Cite

@article{arxiv.2605.20617,
  title  = {Complete realization of multifractal entropy spectra},
  author = {Xiaobo Hou and Xueting Tian},
  journal= {arXiv preprint arXiv:2605.20617},
  year   = {2026}
}

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42 pages