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Higher-dimensional multifractal analysis

Barreira, L; Saussol, B and Schmeling, Jörg LU (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)
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author
organization
publishing date
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
2007-08-08 09:24:44
date last changed
2017-10-22 04:46:36
@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},
  keyword      = {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},
  volume       = {81},
  year         = {2002},
}