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68
Sets of "NonTypical" Points Have Full Topological Entropy and Full Hausdorff Dimension
 ISRAEL J. MATH
, 2000
"... For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This ..."
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Cited by 77 (18 self)
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For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a nontrivial mathematical application of this theory.
Variational Principles and Mixed Multifractal Spectra
, 2001
"... 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 w ..."
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Cited by 40 (6 self)
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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 semicontinuous 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 "nonmixed" multifractal spectra.
NONADDITIVE THERMODYNAMIC FORMALISM: EQUILIBRIUM AND GIBBS MEASURES
"... Abstract. The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function ϕ is replaced by the topological pressure of a sequence of functions Φ = (ϕn)n. The theory also includes a variational principle for ..."
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Cited by 20 (1 self)
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Abstract. The nonadditive thermodynamic formalism is a generalization of the classical thermodynamic formalism, in which the topological pressure of a single function ϕ is replaced by the topological pressure of a sequence of functions Φ = (ϕn)n. The theory also includes a variational principle for the topological pressure, although with restrictive assumptions on Φ. Our main objective is to provide a new class of sequences, the socalled almost additive sequences, for which it is possible not only to establish a variational principle, but also to discuss the existence and uniqueness of equilibrium and Gibbs measures. In addition, we give several characterizations of the invariant Gibbs measures, also in terms of an averaging procedure over the periodic points. 1.
THE THERMODYNAMIC FORMALISM FOR SUBADDITIVE POTENTIALS
"... The topological pressure is defined for subadditive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the top ..."
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Cited by 20 (7 self)
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The topological pressure is defined for subadditive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.
Lyapunov spectrum of asymptotically subadditive potentials
, 2009
"... For general asymptotically subadditive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measuretheoretic entropies and topological pressu ..."
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Cited by 20 (6 self)
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For general asymptotically subadditive potentials (resp. asymptotically additive potentials) on general topological dynamical systems, we establish some variational relations between the topological entropy of the level sets of Lyapunov exponents, measuretheoretic entropies and topological pressures in this general situation. Most of our results are obtained without the assumption of the existence of unique equilibrium measures or the differentiability of pressure functions. Some examples are constructed to illustrate the irregularity and the complexity of multifractal behaviors in the subadditive case and in the case that the entropy map that is not uppersemi continuous.
Dimension and product structure of hyperbolic measures
 ANNALS OF MATHEMATICS, 149 (1999), 755–783
, 1999
"... We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. Th ..."
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Cited by 17 (0 self)
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We prove that every hyperbolic measure invariant under a C 1+α diffeomorphism of a smooth Riemannian manifold possesses asymptotically “almost” local product structure, i.e., its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials. This has not been known even for measures supported on locally maximal hyperbolic sets. Using this property of hyperbolic measures we prove the longstanding EckmannRuelle conjecture in dimension theory of smooth dynamical systems: the pointwise dimension of every hyperbolic measure invariant under a C 1+α diffeomorphism exists almost everywhere. This implies the crucial fact that virtually all the characteristics of dimension type of the measure (including the Hausdorff dimension, box dimension, and information dimension) coincide. This provides the rigorous mathematical justification of the concept of fractal dimension for hyperbolic measures.
SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS ARISING IN THE STUDY OF QUASICRYSTALS
, 2012
"... We survey results that have been obtained for selfadjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the onedimensional case, and in particula ..."
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Cited by 14 (7 self)
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We survey results that have been obtained for selfadjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the onedimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schrödinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of
MULTIFRACTAL ANALYSIS FOR LYAPUNOV EXPONENTS ON NONCONFORMAL REPELLERS
 COMM. MATH. PHYS. 267 (2006), 393–418.
, 2006
"... For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus ..."
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Cited by 14 (6 self)
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For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem neither the classical thermodynamic formalism. We use instead a nonadditive topological pressure to characterize the topological entropy of each level set. This prevents us from estimating the complexity of the level sets using the classical Gibbs measures, which are often one of the main ingredients of multifractal analysis. Instead, we avoid even equilibrium measures, and thus in particular gmeasures, by constructing explicitly ergodic measures, although not necessarily invariant, which play the corresponding role in our work.
Generalised dimensions of measures on almost selfaffine sets. Nonlinearity 23
, 2010
"... We establish a generic formula for the generalised qdimensions of measures supported by almost selfaffine sets, for all q> 1. These qdimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves ..."
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Cited by 12 (5 self)
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We establish a generic formula for the generalised qdimensions of measures supported by almost selfaffine sets, for all q> 1. These qdimensions may exhibit phase transitions as q varies. We first consider general measures and then specialise to Bernoulli and Gibbs measures. Our method involves estimating expectations of moment expressions in terms of ‘multienergy ’ integrals which we then bound using induction on families of trees. AMS classification scheme numbers: 28A80, 37C45 1