The thermodynamic approach to multifractal analysis
arXiv:1102.5427 · doi:10.1017/etds.2014.12
Abstract
Most results in multifractal analysis are obtained using either a thermodynamic approach based on existence and uniqueness of equilibrium states or a saturation approach based on some version of the specification property. A general framework incorporating the most important multifractal spectra was introduced by Barreira and Saussol, who used the thermodynamic approach to establish the multifractal formalism in the uniformly hyperbolic setting, unifying many existing results. We extend this framework to apply to a broad class of non-uniformly hyperbolic systems, including examples with phase transitions. In the process, we compare this thermodynamic approach with the saturation approach and give a survey of many of the multifractal results in the literature.
51 pages, minor corrections, added formal statements of new results to "applications" section
References in corpus (17)
- Lyapunov spectrum of asymptotically sub-additive potentials
- Bowen's equation in the non-uniform setting
- Natural equilibrium states for multimodal maps
- Phase transitions for the multifractal analysis of self-similar measures
- Nonconcave entropies in multifractals and the thermodynamic formalism
- Topological pressure of simultaneous level sets
- Multifractal analysis of Birkhoff averages for countable Markov maps
- Dimension theory for multimodal maps
- Level sets of multiple ergodic averages
- Local multifractal analysis in metric spaces
- Limiting modular symbols and their fractal geometry
- Multifractal analysis of weak Gibbs measures for non-uniformly expanding C^1 maps
- Lyapunov spectrum for exceptional rational maps
- Multifractal formalism derived from thermodynamics
- Birkhoff spectra for one-dimensional maps with some hyperbolicity
- Multifractal analysis and localized asymptotic behavior for almost additive potentials
- Multifractal analysis of non-uniformly hyperbolic systems
Cited by in corpus (14)
- Multifractal analysis of the irregular set for almost-additive sequences via large deviations
- A variational principle for the metric mean dimension of level sets
- Quasi-multiplicativity of typical cocycles
- Recurrence times, waiting times and universal entropy production estimators
- Unique equilibrium states for Bonatti-Viana diffeomorphisms
- Equilibrium states for Mañé diffeomorphisms
- Entropy spectrum of Lyapunov exponents for typical cocycles
- Multifractal analysis of weighted ergodic averages
- Distributional chaos in multifractal analysis, recurrence and transitivity
- Transience and multifractal analysis
- Variational principles for metric mean dimension with potential of level sets
- Birkhoff spectrum for diagonally self-affine sets and digit frequencies for GLS systems with redundancy
- Multifractal analysis in non-uniformly hyperbolic interval maps
- Infinite Non-Conformal Iterated Function Systems