Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Thermodynamic Formalism and Multifractal Analysis of Conformal Infinite Iterated Function Systems

We develop the thermodynamic formalism for equilibrium states of strongly Hölder families of functions. These equilibrium states are supported on the limit set generated by iterating a system of infinitely many contractions. The theory of these systems was laid out in an earlier paper of the last two authors. The first five sections of this paper except Section 3 are devoted to developing the thermodynamic formalism for equilibrium states of Hölder families of functions. The first three sections provide us with the tools needed to carry out the multifractal analysis for the equilibrium states mentioned above assuming that the limit set is generated by conformal contractions. The theory of infinite systems of conformal contractions is laid out in [13]. The multifractal analysis is then given in Section 7. In Section 8 we apply this theory to some examples from continued fraction systems and Apollonian packing.     

Conjugacies for linear and nonlinear perturbations of nonuniform behavior

We construct topological conjugacies between linear and nonlinear evolution operators that admit either a nonuniform exponential contraction or a nonuniform exponential dichotomy. We consider evolution operators defined by nonautonomous differential equations x^′=A(t)x+f(t,x) in a Banach space. The conjugacies are obtained by first considering sufficiently small linear and nonlinear perturbations of linear equations x^′=A(t)x. In the case of linear perturbations, we construct in a more or less explicit manner topological conjugacies between the two linear flows. In the case of nonlinear perturbations, we obtain a version of the Grobman-Hartman theorem for nonuniformly hyperbolic dynamics. Furthermore, all the conjugacies that we construct are locally Hölder continuous provided that the vectors fields are of class C¹. As a byproduct of our approach, we give conditions for the robustness of strong nonuniform exponential behavior, in the sense that under sufficiently small perturbations the structure determined by the stable and unstable bundles persists up to small variations. We also show that the constants determining the nonuniform exponential contraction or nonuniform exponential dichotomy vary continuously with the perturbation. All the results are obtained in Banach spaces.     

华沙圈上连续映射的distality与等度连续性

设W为华沙圈,f:W→W为连续映射.本文得到了f为distal的一个刻画并且讨论了f的distality与等度连续性的关系.证明了:(i)f是distal的当且仅当f为恒等映射.(ii)如果f为满射,则f是distal的当且仅当f为等度连续的.     

圆周自映射的一些动力系统性质及其等价条件

本文介绍了近年来关于圆周连续自映射的动力系统性质的一些研究工作，并补充了一些新的结果，从链回归点的角度对圆周连续自映射作了新的探讨。     

Krylov and Safonov estimates for degenerate quasilinear elliptic PDEs

We here establish an a priori Hölder estimate of Krylov and Safonov type for the viscosity solutions of a degenerate quasilinear elliptic PDE of non-divergence form. The diffusion matrix may degenerate when the norm of the gradient of the solution is small: the exhibited Hölder exponent and Hölder constant only depend on the growth of the source term and on the bounds of the spectrum of the diffusion matrix for large values of the gradient. In particular, the given estimate is independent of the regularity of the coefficients. As in the original paper by Krylov and Safonov, the proof relies on a probabilistic interpretation of the equation.     

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on S ¹ . In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group exists if and only if the quotient group does not contain a free subgroup with two generators (here is some specific subgroup of G defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups of homeomorphisms of the line.     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Small ball estimates for Brownian motion and the Brownian sheet

Small ball estimates are obtained for Brownian motion and the Brownian sheet when balls are given by certain Hölder norms. As an application of these results we include a functional form of Chung's LIL in this setting.Both authors were supported in part by NSF Grant Number DMS-9024961.     

Multifractal Formalism for Self-Similar Functions Under the Action of Nonlinear Dynamical Systems

We study functions which are self-similar under the action of some nonlinear dynamical systems. We compute the exact pointwise H{?}lder regularity, then we determine the spectrum of singularities and the Besov ``smoothness' index, and finally we prove the multifractal formalism. The main tool in our computation is the wavelet analysis. October 1, 1996. Date revised: May 13, 1997. Date re-revised: January 10, 1998. Date accepted: February 27, 1998.     

The Multifractal Formalism for Measures,Review and Extension to Mixed Cases

The multifractal formalism for single measure is reviewed.Next,a mixed generalized multifractal formalism is introduced which extends the multifractal formalism of a single measure based on generalizations of the Hausdorff and packing measures to a vector of simultaneously many measures.Borel-Cantelli and Large deviations Theorems are extended to higher orders and thus applied for the validity of the new variant of the multifractal formalism for some special cases of multi-doubling type measures.     

A dynamical property for planar homeomorphisms and an application to the problem of canonical position around an isolated fixed point

The first theorem in this paper gives a criterion for detecting a fixed point of a planar homeomorphism into a disc. We then use this result to re-prove the path-connectedness of the space of all planar homeomorphisms fixing only one point, with a given Lefschetz index.     

Existence and uniqueness for non-linear singular integral equations used in fluid mechanics

Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further studied for non-linear singular integral equations over a finite number of arbitrarily ordered arcs. An application to fluid mechanics theory is finally given for the determination of the form of the profiles of a turbomachine in two-dimensional flow of an incompressible fluid.     

Cesàro Asymptotics for Orthogonal Polynomials on the Unit Circle and Classes of Measures

The convergence in L²( ) of the even approximants of the Wall continued fractions is extended to the Cesàro–Nevai class CN, which is defined as the class of probability measures σ with lim_n→∞ ∑ⁿ⁻¹_k=0 |a_k|=0, {a_n}_n0 being the Geronimus parameters of σ. We show that CN contains universal measures, that is, probability measures for which the sequence {|_n|² dσ}_n0 is dense in the set of all probability measures equipped with the weak-* topology. We also consider the “opposite” Szeg class which consists of measures with ∑^∞_n=0 (1−|a_n|²)^1/2<∞ and describe it in terms of Hessenberg matrices.     

一类五次多项式系统的奇点量与极限环分支

该文研究一类五次多项式微分系统在高次奇点与无穷远点的极限环分支问题. 该系统的原点是高次奇点, 赤道环上没有实奇点. 首先推导出计算高次奇点与无穷远点奇点量的代数递推公式,并用之计算系统原点、无穷远点的奇点量，然后分别讨论了系统原点、无穷远点中心判据. 给出了多项式系统在高次奇点分支出5个极限环同时在无穷远点分支出2个极限环的实例. 这是首次在同步扰动的条件下讨论高次奇点与无穷远点分支出极限环的问题.     

Some Exact Penalty Results for Nonlinear Programs and Mathematical Programs with Equilibrium Constraints

Recently, some exact penalty results for nonlinear programs and mathematical programs with equilibrium constraints were proved by Luo, Pang, and Ralph (Ref. 1). In this paper, we show that those results remain valid under some other mild conditions. One of these conditions, called strong convexity with order , is discussed in detail.     

一类具临界指数和等值面边值条件椭圆型方程解的存在性

本文利用临界点理论给出了RN(N≥3)中有界光滑区域上的拟线性椭圆型方程-△pU=|u|p*-2u a(x)|u|p-2u f(x,u),X∈Ω(P*=Np/(N-p),1相似文献   

四阶奇异边值问题正解的多重性与无解性

应用不动点指数理论和上下解的方法,研究了一类非线性四阶微分方程组奇异边值问题,给出了其正解存在性与无解性定理.     

Existence and Multiplicity of Positive Solutions for a Singular Nonlinear High Order Fractional Differential Problem with Multi-point Boundary Conditions

In this paper, a singular nonlinear high order fractional differential problem involving multi-point boundary conditions is solved by means of the fixed point index theory. Some properties of the first eigenvalue corresponding to relevant operator and some new height functions are also used to prove the existence and multiplicity of positive solutions. The nonlinearity depends on arbitrary fractional derivative.     

Defect indices and definiteness conditions for a class of discrete linear Hamiltonian systems

This paper is concerned with a class of discrete linear Hamiltonian systems in finite or infinite intervals. A definiteness condition and its equivalent statements are discussed and three sufficient conditions for the definiteness condition are given. A precise relationship between the defect index of the minimal subspace generated by the system and the number of linearly independent square summable solutions of the system is established. In particular, they are equal if and only if the definiteness condition is satisfied. Finally, two criteria for the limit point case and one criterion for the limit circle case are obtained.