On Hausdorff Dimension of Unimodal Attractors

There exists a universal constant σ<1 such that every attractor of every C⁴ unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.     

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.     

Towards a Kneading Theory for Lozi Mappings. II: Monotonicity of the Topological Entropy and Hausdorff Dimension of Attractors

We construct a kneading theory à la Milnor–Thurston for Lozi mappings (piecewise affine homeomorphisms of the plane). In the first article a two-dimensional analogue of the kneading sequence called the pruning pair is defined, and a topological model of a Lozi mapping is constructed in terms of the pruning pair only. As an application of this result, in the current paper we show the partial monotonicity of the topological entropy and of bifurcations for the Lozi family near horseshoes. Upper and lower bounds for the Hausdorff dimension of the Lozi attractor are also given in terms of parameters. Dédié au Professeur A. Douady pour son 60ème anniversaire Received: 1 September 1996 / Accepted: 16 April 1997     

Kinetic Models with Randomly Perturbed Binary Collisions

We introduce a class of Kac-like kinetic equations on the real line, with general random collisional rules which, in some special cases, identify models for granular gases with a background heat bath (Carrillo et al. in Discrete Contin. Dyn. Syst. 24(1):59–81, 2009), and models for wealth redistribution in an agent-based market (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009). Conditions on these collisional rules which guarantee both the existence and uniqueness of equilibrium profiles and their main properties are found. The characterization of these stationary states is of independent interest, since we show that they are stationary solutions of different evolution problems, both in the kinetic theory of rarefied gases (Cercignani et al. in J. Stat. Phys. 105:337–352, 2001; Villani in J. Stat. Phys. 124:781–822, 2006) and in the econophysical context (Bisi et al. in Commun. Math. Sci. 7:901–916, 2009).     

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.     

Zero Hausdorff Dimension Spectrum for the Almost Mathieu Operator

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense \({G_\delta}\) set of frequencies for which the spectrum is of zero Hausdorff dimension.     

Diffusion of Power in Randomly Perturbed Hamiltonian Partial Differential Equations

We study the evolution of the energy (mode-power) distribution for a class of randomly perturbed Hamiltonian partial differential equations and derive master equations for the dynamics of the expected power in the discrete modes. In the case where the unperturbed dynamics has only discrete frequencies (finitely or infinitely many) the mode-power distribution is governed by an equation of discrete diffusion type for times of order (^–2). Here denotes the size of the random perturbation. If the unperturbed system has discrete and continuous spectrum the mode-power distribution is governed by an equation of discrete diffusion-damping type for times of order (^–2). The methods involve an extension of the authors work on deterministic periodic and almost periodic perturbations, and yield new results which complement results of others, derived by probabilistic methods.Acknowledgement We would like to thank G. C. Papanicolaou, J.L. Lebowitz and S.E. Golowich for helpful discussions concerning this work. E.K. was supported in part by the ASCI Flash Center at the University of Chicago. M.I.W. was supported in part by a grant from the National Science Foundation.     

Hausdorff Dimension of Sets of Generic Points for Gibbs Measures

For a translation invariant Gibbs measure on the configuration space X of a lattice finite spin system, we consider the set X of generic points. Using a Breiman type convergence theorem on the set X of generic points of an arbitrary translation invariant probability measure on X, we evaluate the Hausdorff dimension of the set X with respect to any metric out of a wide class of scale metrics on X (including Billingsley metrics generated by Gibbs measures).     

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.     

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.     

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.     

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.     

四涡卷超混沌吸引子的DSP实现

提出了基于数字信号处理(DSP)实现四涡卷超混沌吸引子的方法.利用四阶龙格库塔法对连续四维超混沌系统进行离散化处理,用C语言编写源程序,并在CCS编译器上编译、调试和仿真成功后,下载固化到DSP芯片内,实现四涡卷超混沌吸引子,硬件实验与软件仿真结果完全一致.     

Linear Response Function for Coupled Hyperbolic Attractors

The SRB measures of a hyperbolic system are widely accepted as the measures that are physically relevant. It has been shown by Ruelle that they depend smoothly on the system. Furthermore, Ruelle showed by a separate argument that the first derivative, i.e., the linear response function, admits a geometric interpretation. In this paper, we consider thermodynamic limits of SRB measures in lattices of coupled hyperbolic attractors. In a previous paper, using Markov partitions and thermodynamic formalism, we had established the smooth dependence of thermodynamic limits of SRB measures. Here, we establish that the linear response function admits a geometric interpretation. The formula is analogous to the one found by Ruelle for finite dimensional systems if one term is reinterpreted appropriately. We show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume. We also obtain similar results for the derivatives of the entropy. Supported in part by NSF grants.     

Quantum Reduction for Affine Superalgebras

We extend the homological method of quantization of generalized Drinfeld–Sokolov reductions to affine superalgebras. This leads, in particular, to a unified representation theory of superconformal algebras.     

Lifshits Tails for Randomly Twisted Quantum Waveguides

We consider the Dirichlet Laplacian \(H_\gamma \) on a 3D twisted waveguide with random Anderson-type twisting \(\gamma \). We introduce the integrated density of states \(N_\gamma \) for the operator \(H_\gamma \), and investigate the Lifshits tails of \(N_\gamma \), i.e. the asymptotic behavior of \(N_\gamma (E)\) as \(E \downarrow \inf \mathrm{supp}\, dN_\gamma \). In particular, we study the dependence of the Lifshits exponent on the decay rate of the single-site twisting at infinity.     

Stochastic Attractors for Shell Phenomenological Models of Turbulence

Recently, it has been proposed that the Navier–Stokes equations and a relevant linear advection model have the same long-time statistical properties, in particular, they have the same scaling exponents of their structure functions. This assertion has been investigate rigorously in the context of certain nonlinear deterministic phenomenological shell model, the Sabra shell model, of turbulence and its corresponding linear advection counterpart model. This relationship has been established through a “homotopy-like” coefficient λ which bridges continuously between the two systems. That is, for λ=1 one obtains the full nonlinear model, and the corresponding linear advection model is achieved for λ=0. In this paper, we investigate the validity of this assertion for certain stochastic phenomenological shell models of turbulence driven by an additive noise. We prove the continuous dependence of the solutions with respect to the parameter λ. Moreover, we show the existence of a finite-dimensional random attractor for each value of λ and establish the upper semicontinuity property of this random attractors, with respect to the parameter λ. This property is proved by a pathwise argument. Our study aims toward the development of basic results and techniques that may contribute to the understanding of the relation between the long-time statistical properties of the nonlinear and linear models.     

An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension

The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462–465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797–808, 1998) proposed to look for an example of a Hamiltonian near with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.     

Regularity of SRB Entropy for Geometric Lorenz Attractors

Journal of Statistical Physics - We consider the classical geometric Lorenz attractors, showing that the SRB entropy admits $$\gamma $$ -Hölder continuity for any $$0&lt;\gamma &lt;1$$ .