Higher-dimensional multifractal analysis
(2002) In Journal des Mathématiques Pures et Appliquées 81(1). p.67-91- Abstract
- 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,... (More)
- 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. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/342030
- author
- Barreira, L ; Saussol, B and Schmeling, Jörg LU
- organization
- publishing date
- 2002
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- multifractal analysis, variational principle
- in
- Journal des Mathématiques Pures et Appliquées
- volume
- 81
- issue
- 1
- pages
- 67 - 91
- publisher
- Elsevier
- external identifiers
-
- wos:000174399800003
- scopus:0036238495
- ISSN
- 0021-7824
- DOI
- 10.1016/S0021-7824(01)01228-4
- language
- English
- LU publication?
- yes
- id
- 1d6bf666-56f8-415e-91f7-dac11c1d3f91 (old id 342030)
- date added to LUP
- 2016-04-01 16:54:43
- date last changed
- 2022-02-05 19:25:43
@article{1d6bf666-56f8-415e-91f7-dac11c1d3f91, abstract = {{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.}}, author = {{Barreira, L and Saussol, B and Schmeling, Jörg}}, issn = {{0021-7824}}, keywords = {{multifractal analysis; variational principle}}, language = {{eng}}, number = {{1}}, pages = {{67--91}}, publisher = {{Elsevier}}, series = {{Journal des Mathématiques Pures et Appliquées}}, title = {{Higher-dimensional multifractal analysis}}, url = {{http://dx.doi.org/10.1016/S0021-7824(01)01228-4}}, doi = {{10.1016/S0021-7824(01)01228-4}}, volume = {{81}}, year = {{2002}}, }