Variational principles and mixed multifractal spectra

2001/06/06 by L. Barreira, Luís Barreira, B. Saussol +1 · 161 citations
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Classification of discontinuities #Curse of dimensionality #Dynamical systems theory #Entropy (arrow of time) #Fractal #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Multifractal system #Physics #Pointwise #Pure mathematics #Quantum chaos and dynamical systems #Quantum mechanics #Spectral line #Spectrum (functional analysis) #Statistical physics #Statistics #Variational principle

paper · pdf · doi:10.1090/s0002-9947-01-02844-6

published in Transactions of the American Mathematical Society 353(10), 3919-3944 (American Mathematical Society)

openalex publication_date 2001/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26

Abstract

We establish a “conditional” variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy. Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the “mixed” multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the “non-mixed” multifractal spectra.

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