A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems

Abstract. We establish a "conditional" variational principle, which unifies and extends many results in the multifractal analysis of dynamical systems. Namely, instead of considering several quantities of local nature and studying separately their multifractal spectra we develop a unified approach which allows us to obtain all spectra from a new multifractal spectrum. Using the variational principle we are able to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.Another application of the variational principle is the following. The multifractal analysis of dynamical systems studies multifractal spectra such as the dimension spectrum for pointwise dimensions and the entropy spectrum for local entropies. It has been a standing open problem to effect a similar study for the "mixed" multifractal spectra, such as the dimension spectrum for local entropies and the entropy spectrum for pointwise dimensions. We show that they are analytic for several classes of hyperbolic maps. We also show that these spectra are not necessarily convex, in strong contrast with the "non-mixed" multifractal spectra.

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“…When Z = X, we recover the classical notion of topological entropy. We refer to [7,1] for references and further details.…”

Section: Theoremmentioning

confidence: 99%

“…Non-additive topological pressure. We recall the non-additive version of the topological pressure introduced by Barreira in [1]. We use the notations of Section 2.1.…”

Section: Theorem 16 If X Is a Repeller Of A Topologically Mixing Cmentioning

confidence: 99%

Variational principles and mixed multifractal spectra

Barreira¹,

Saussol

2001

Trans. Amer. Math. Soc.

135

114

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“…A Moran fractal is a Cantor-type set defined by iteration; it has a tractable structure and has been used extensively for estimating entropies and dimensions (see e.g. [1,11,17,18,21,22]). In our context, we will make use of a special dynamically defined (by the Bowen metric) Moran fractal (Definition 4.1).…”

Section: Spectrum Of Poincaré Recurrence 1919mentioning

confidence: 99%

The spectrum of Poincaré recurrence

Lau

Shu

2008

Ergod. Th. Dynam. Sys.

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Abstract. We investigate the relationship between Poincaré recurrence and topological entropy of a dynamical system (X, f ). For 0 ≤ α ≤ β ≤ ∞, let D(α, β) be the set of x with lower and upper recurrence rates α and β, respectively. Under the assumptions that the system is not minimal and that the map f is positively expansive and satisfies the specification condition, we show that for any open subset ∅ = U ⊆ X , D(α, β) ∩ U has the full topological entropy of X . This extends a result of Feng and Wu [The Hausdorff dimension of recurrence sets in symbolic spaces. Nonlinearity 14 (2001), 81-85] for symbolic spaces.

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“…We mostly have almost all type results in the self-affine case (i.e. when the local inverses are affine maps) in the sense of Falconer's paper [6] and upper estimates for the dimension of nonconformal repellers [7] , [1]. For the multifractal case there has been work on a class of examples relating to Sierpiński carpets in 2 , [10] and d , [13].…”

Section: Introductionmentioning

confidence: 99%