Inside Singularity Sets of Random Gibbs Measures

We evaluate the scale at which the multifractal structure of some random Gibbs measures becomes discernible. The value of this scale is obtained through what we call the growth speed in Hölder singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.     

Dimension spectrum of axiom a diffeomorphisms. II. Gibbs measures

We compute the dimension spectrumf() of the singularity sets of a Gibbs measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. This case is the generalization of the case where the measure studied is the Bowen-Margulis measure—the one that realizes the topological entropy. We obtain similar results; for example, the functionf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we finally decompose these sets.     

Dimensions and Waiting Times for Gibbs Measures

For shifts of finite type, we relate the waiting time between two different orbits, one chosen according to an ergodic measure, the other according to a Gibbs measure, to Billingsley dimensions of generic sets. This is achieved by computing Billingsley dimensions of saturated sets in terms of a relative entropy which satisfies a pointwise ergodic result. As a by-product, a similar result is obtained for match lengths that are dual quantities of waiting times.     

Hausdorff Dimension in Stochastic Dispersion

We consider the evolution of a connected set in Euclidean space carried by a periodic incompressible stochastic flow. While for almost every realization of the random flow at time t most of the particles are at a distance of order away from the origin, there is an uncountable set of measure zero of points, which escape to infinity at the linear rate. In this paper we prove that this set of linear escape points has full Hausdorff dimension.     

A Lower Estimation of the Hausdorff Dimension for Attractors with Overlaps

We give a lower estimate of the Hausdorff dimension for attractors which can be obtained by an overlapping construction.     

Hausdorff Dimension of Non-Hyperbolic Repellers. I: Maps with Holes

This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps. Here we consider a general abstract model, that we call piecewise smooth maps with holes. We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold. Our approach also provides information on escape rates and dynamical dimension of the repeller.     

On the Hausdorff Dimension of Regular Points of Inviscid Burgers Equation with Stable Initial Data

Consider an inviscid Burgers equation whose initial data is a Lévy α-stable process Z with α>1. We show that when Z has positive jumps, the Hausdorff dimension of the set of Lagrangian regular points associated with the equation is strictly smaller than 1/α, as soon as α is close to 1. This gives a partially negative answer to a Conjecture of Janicki and Woyczynski (J. Stat. Phys. 86(1–2):277–299, 1997). Along the way, we contradict a recent Conjecture of Z. Shi () about the lower tails of integrated stable processes.     

Dimension spectrum of axiom a diffeomorphisms. I. The Bowen-Margulis measure

We compute the dimension spectrumf() of the singularity sets of the Bowen-Margulis measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. It is proved thatf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we decompose these sets.     

Hausdorff dimensions in two-dimensional maps and thermodynamic formalism

We compute numerically the Hausdorff dimensions of the Gibbs measures on the invariant sets of Axiom A systems. In particular, we stress the existence of a measure which has maximal dimension and can be relevant for the ergodic properties of the system. For hyperbolic maps of the plane with constant Jacobianj, we apply the Bowen-Ruelle formula, using the relationF(=d _H–1)=lnj, which links the Hausdorff dimensiond _H of an attractor to a free energy functionalF() defined in the thermodynamic formalism. We provide numerical evidence that this relation remains valid for some nonhyperbolic maps, such as the Hénon map.     

Hausdorff Dimensions of Zero-Entropy Sets of Dynamical Systems with Positive Entropy

Suppose that (X,T) is a compact positive entropy dynamical system which we mean that X is a compact metric space and T: X→ X is a continuous transformation of X and the topological entropy h(T)>0. A point x ∈ X is called a zero-entropy point provided , where is the forward orbit of x under T and Orb₊(x) is the closure. Let ε⁰(X, T) denote the set of all zero-entropy points. Naturally, one would like to ask the following important question: How big is ε⁰(X, T) for a dynamical system? In this paper, we answer this question. More precisely, we prove that if, furthermore, (X, T) is locally expanding, then the Hausdorff dimension of ε⁰(X, T) vanishes.     

Relative Entropy and Identification of Gibbs Measures in Dynamical Systems

In this work we explore the idea of using the relative entropy of ergodic measures for the identification of Gibbs measures in dynamical systems. The question we face is how to estimate the thermodynamic potential (together with a grammar) from a sample produced by the corresponding Gibbs state.     

Gibbs Measures for Brownian Paths Under the Effect of an External and a Small Pair Potential

We consider Brownian motion in the presence of an external and a weakly coupled pair interaction potential and show that its stationary measure is a Gibbs measure. Uniqueness of the Gibbs measure for two cases is shown. Also the typical path behaviour, the degree of mixing and some further properties are derived. We use cluster expansion in the small coupling parameter.     

Comparison of Finite Volume Canonical and Grand Canonical Gibbs Measures: The Continuous Case

We consider a continuous gas with finite range positive pair potential and we assume that the cluster expansion convergence condition holds. We prove a sharp bound on the difference between the finite volume grand canonical and canonical expectation of local observable. The bound is given in terms of the support of the observable, of its grand canonical variance and of the volume on which the system is confined.     

A note on the projecton of Gibbs measures

We give an example of a projection which maps two Gibbs measures for the same interaction into Gibbs measures for different interactions. As a corollary we find a case where by decimation a non-Gibbsian measure is transformed into a Gibbs measure.     

On the Hausdorff dimension of fractal attractors

We consider such mappingsx _n+1=F(x_n) of an interval into itself for which the attractor is a Cantor set. For the same class of mappings for which the Feigenbaum scaling laws hold, we show that the Hausdorff dimension of the attractor is universally equal toD=0.538 ...     

Equivalence Between Canonical Gibbs Measures and Stationary Measures for Stochastic Lattice-Gas Model

It is proved that, under appropriate conditions on the jump rate and potential, one- and two-dimensional stochastic lattice-gas models (exclusion process with speed change) have only canonical Gibbs measures as their stationary measures. This extends the previously known result, which treats only a special jump rate and potential.     

Hausdorff Dimension of Julia Sets of Feigenbaum Polynomials with High Criticality

We consider unimodal polynomials with Feigenbaum topological type and critical points whose orders tend to infinity. It is shown that the hyperbolic dimensions of their Julia set go to 2; furthermore, that the Hausdorff dimensions of the basins of attraction of their Feigenbaum attractors also tend to 2. The proof is based on constructing a limiting dynamics with a flat critical point.Both authors were supported by Grant No. 2002062 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.Partially supported by NSF grant DMS-0245358.     

Equivalences of the large deviation principle for Gibbs measures and critical balance in the Ising model

In this paper we obtain the equivalence of the large deviation principle for Gibbs measures with and without an external field. For the Ising model, the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube ((B/h) ash0 for variousB. We find a critical balance at a valueB ₀ ofB.     

Generalized Physical and SRB Measures for Hyperbolic Diffeomorphisms

In this paper we introduce the notion of generalized physical and SRB measures. These measures naturally generalize classical physical and SRB measures to measures which are supported on invariant sets that are not necessarily attractors. We then perform a detailed case study of these measures for hyperbolic Hènon maps. For this class of systems we are able to develop a complete theory about the existence, uniqueness, finiteness, and properties of these natural measures. Moreover, we derive a classification for the existence of a measure of full dimension. We also consider general hyperbolic surface diffeomorphisms and discuss possible extensions of, as well as the differences to, the results for Hènon maps. Finally, we study the regular dependence of the dimension of the generalized physical/SRB measure on the diffeomorphism. For the proofs we apply various techniques from smooth ergodic theory including the thermodynamic formalism. 2000 Mathematics Subject Classification. Primary: 37C45, 37D20, 37D35, Secondary: 37A35, 37E30     

Thermodynamic Formalism and Variations of the Hausdorff Dimension of Quadratic Julia Sets

Let d(c) denote the Hausdorff dimension of the Julia set of the polynomial z? z²+cz\mapsto z^2+c. The function d restricted to [0,+X) is real analytic in [0,\frac14)è(\frac14,+￥)[0,\frac{1}{4})\cup (\frac{1}{4},+\infty) ([Ru²]), is left-continuous at ¼ ([Bo,Zi]) but not continuous ([Do,Se,Zi]). We prove that c? d￠(c)c\mapsto d'(c) tends to + X from the left at ¼ as (\frac14-c)^{d(\frac14)-\frac32}(\frac{1}{4}-c)^{d(\frac{1}{4})-\frac{3}{2}}. In particular the graph of d has a vertical tangent on the left at ¼, a result which supports the numerical experiments.