Higher-dimensional multifractal analysis

By Luís Barreira, B. Saussol, J. Schmeling. 2002

156 citations recorded by the source corpus.

We establish a higher-dimensional version of multifractal analysis for several classes of hyperbolic dynamical systems. This means that we consider multifractal decompositions which are associated to multi-dimensional parameters. In particular, we obtain a conditional variational principle, which shows that the topological entropy of the level sets of pointwise dimensions, local entropies, and Lyapunov exponents can be simultaneously approximated by the entropy of ergodic measures. A similar result holds for the Hausdorff dimension. This study allows us to exhibit new nontrivial phenomena absent in the one-dimensional multifractal analysis. In particular, while the domain of definition of a one-dimensional spectrum is always an interval, we show that for higher-dimensional spectra the domain may not be convex and may even have empty interior, while still containing an uncountable number of points. Furthermore, the interior of the domain of a higher-dimensional spectrum has in general more than one connected component.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/a8be6b1490264fd8050b4337e6bedf58707ea59c

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In Mathematical Physics

1. De-Jun Feng
2. Wen Huang
3. B. Saussol
4. Luís Barreira
5. J. Schmeling
6. Jun Wu
7. Bao-Wei Wang
8. A. Fan

In Statistical and Nonlinear Physics

1. De-Jun Feng
2. Wen Huang
3. B. Saussol
4. Luís Barreira
5. Bao-Wei Wang
6. Jun Wu
7. J. Schmeling
8. A. Fan

Citations in the corpus

• Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices (2003, background): “Theorem A extends the corresponding results for the Birkho averages of scalar functions, for which one is referred to, e.g., [1, 2 , 3, 9, 10, 16, 35, 38, 40, 41, 47].”
• Almost additive multifractal analysis (2009, unknown): “Our approach builds on related arguments in former work of Barreira, Saussol and Schmeling in [5], although now in the more general context of almost additive sequences.”
• Lyapunov Spectrum of Asymptotically Sub-additive Potentials (2009, background): “In this case, the topological entropy (or the Hausdorff dimension) of the level sets of Birkhoff averages has been extensively studied in the recent two decades (see, e.g., [13, 7, 41, 9, 20, 21, 34, 27, 8, 42, 47, 23, 16, 22, 1, 2, 29] and references therein).”
• On Khintchine exponents and Lyapunov exponents of continued fractions (2008, background): “When X is compact, T and φ are continuous, the Birkhoff spectrum are well studied (see [1, 14, 15] and the references therein.”
• A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation (2013, background): “The following conditional variational principle is well known [3, 15, 16].”
• Distribution of frequencies of digits via multifractal analysis (2002, methodology): “In particular, under the assumptions of Theorem 9, it can be used to show that D is analytic (see [1, 2] for details).”