The thermodynamic approach to multifractal analysis

Climenhaga, Vaughn

doi:10.1017/etds.2014.12

Cited by 26 publications

(23 citation statements)

References 131 publications

(247 reference statements)

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“…3) 13 which proves (i). Now, since I f,ϕε,gε is a non-negative convex function for all ε ∈ (0, 1) and is pointwise convergent to I f,Φ,Ψ this is also a non-negative convex function.…”

Section: Proof Of Theorem Asupporting

confidence: 61%

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Bomfim

Varandas

2015

Nonlinearity

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In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman's sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and Shannon-McMillan-Breiman theorem for Gibbs measures.

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Section: Proof Of Theorem Asupporting

confidence: 61%

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Bomfim

Varandas

2015

Nonlinearity

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“…The advantage of the thermodynamic approach over the orbit-gluing approach is that it establishes the conditional variational principle that is the final equality in (1.2); the orbit-gluing approach gives no information on invariant measures supported on the level sets [Cli12].…”

Section: Introductionmentioning

confidence: 99%

Topological pressure of simultaneous level sets

Climenhaga

2012

Nonlinearity

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Multifractal analysis studies level sets of asymptotically defined quantities in a topological dynamical system. We consider the topological pressure function on such level sets, relating it both to the pressure on the entire phase space and to a conditional variational principle. We use this to recover information on the topological entropy and Hausdorff dimension of the level sets.Our approach is thermodynamic in nature, requiring only existence and uniqueness of equilibrium states for a dense subspace of potential functions. Using an idea of Hofbauer, we obtain results for all continuous potentials by approximating them with functions from this subspace.This technique allows us to extend a number of previous multifractal results from the C 1+ε case to the C 1 case. We consider ergodic ratios Snϕ/Snψ where the function ψ need not be uniformly positive, which lets us study dimension spectra for non-uniformly expanding maps. Our results also cover coarse spectra and level sets corresponding to more general limiting behaviour.Date: October 9, 2018.

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“…Consider the potential function ϕ geo (x) = − log det(Df | E cu (x) ), where E cu (x) ⊂ T x T 4 is the two-dimensional centre-unstable subspace at x. We refer to this as the geometric potential ; see for example [5,26,23,16] for the terminology and for applications of the family {tϕ geo : t ∈ R} including multifractal analysis of Lyapunov exponents.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Unique equilibrium states for Bonatti–Viana diffeomorphisms

2018

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We show that the robustly transitive diffeomorphisms constructed by Bonatti and Viana have unique equilibrium states for natural classes of potentials. In particular, we characterize the SRB measure as the unique equilibrium state for a suitable geometric potential. The techniques developed are applicable to a wide class of DA diffeomorphisms, and persist under C 1 perturbations of the map. These results are an application of general machinery developed by the first and last named authors.

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The thermodynamic approach to multifractal analysis

Cited by 26 publications

References 131 publications

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

Topological pressure of simultaneous level sets

Unique equilibrium states for Bonatti–Viana diffeomorphisms

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