Complete realization of multifractal entropy spectra
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917514713235456 |
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| author | Hou, Xiaobo Tian, Xueting |
| author_facet | Hou, Xiaobo Tian, Xueting |
| contents | 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20617 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Complete realization of multifractal entropy spectra Hou, Xiaobo Tian, Xueting Dynamical Systems 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}. |
| title | Complete realization of multifractal entropy spectra |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.20617 |