On the Fine Structure of Stationary Measures in Systems Which Contract-on-Average

Suppose {f ₁,...,f _m } is a set of Lipschitz maps of ^d . We form the iterated function system (IFS) by independently choosing the maps so that the map f _i is chosen with probability p _i ( ^m _i=1 p _i =1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ^d and as a corollary show that the measure will be singular if the modulus of the entropy _i p _i log p _i is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannon's Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of .     

岩石类材料裂纹分叉非规则性几何的分形效应

本文考察了岩石裂纹(从晶粒尺寸的微裂纹到几千米的断层构造)的分叉在几何上存在的非规则性,在文献[10,11]的基础上,应用分形(fractal)几何建立了裂纹分叉的分形模型,分析了裂纹分叉的分形效应,从几何的角度研究了裂纹分叉使断裂韧性提高的物理力学现象,导出了一个比smith公式更合理的断裂韧性计算式。     

The Dimension of Interior Levels of the Boolean Lattice,II

Extending an old lemma by Dushnik, we establish the dimension d(3, k; n) of the containment order generated by the 3-element and k-element subsets of an n-element set for most k between and n.     

Feller Processes Generated by Pseudo-Differential Operators: On the Hausdorff Dimension of Their Sample Paths

Let {X _t} _t0 be a Feller process generated by a pseudo-differential operator whose symbol satisfiesÇⁿ|q(Ç,)|c(1=)()) for some fixed continuous negative definite function (). The Hausdorff dimension of the set {X _t:tE}, E [0, 1] is any analytic set, is a.s. bounded above by dim E. is the Blumenthal–Getoor upper index of the Levy Process associated with ().     

A continuity property for the inverse of mañé's projection

Let X be a compact subset of a separable Hilbert space H with finite fractal dimension d _F(X), and P ₀an orthogonal projection in H of rank greater than or equal to 2 d _F (X) + 1. For every > 0, there exists an orthogonal projection P in H of the same rank as P ₀, which is injective when restricted to X and such that P – P ₀ < This result follows from Mañé's paper. Thus the inverse (P|_X)^–1 of the restricted mapping P|_X: X PX is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé's projection (P| _X )^–1. It is known that when H is finite dimensional then (P| _X )^–1 is Hölder continuous. In this paper we shall prove that if X is a global attractor of an infinite dimensional dissipative evolutionary equation then in some cases (e.g. two-dimensional Navier-Stokes equations with homogeneous Dirichlet boundary conditions) for every x, y X such that , where is a positive constant.     

The Solution Set to BVP for Some Functional Differential Inclusions

We consider a multivalued BVP x'(t) A(t)x((t))+ F(t,x((t))), Lx= . Under appropriate assumptions on A, L and F, we prove that for sufficiently small the set of solutions to this problem is a nonempty infinite dimensional AR-space (Theorem 4).     

Lipschitz images with fractal boundaries and their small surface wrapping

Assume and is a Lipschitz -mapping; and denote the volume and the surface area of . We verify that there exists a figure with , and, of course, , where depends only on the dimension and on . We also give an example when is a square and ; in fact, the boundary of can contain a fractal of Hausdorff dimension exceeding one.

     

Integration over a fractal curve and the jump problem

A definition of integration, i.e., a generalization of a functional of the form

A dimension formula for Bernoulli convolutions

We present a dynamical approach to the study of the singularity of infinitely convolved Bernoulli measuresv , for the golden section. We introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square, the fat baker's transformation. Our approach strongly relies on the Markov structure of the underlying dynamical system. Indeed, if =golden mean, the fat baker's transformation has a very simple Markov coding. The ambiguity (of order two) of this coding, which appears when projecting on the line, due to passages for the central, overlapping zone, can be expressed by means of products of matrices (of order two). This product has a Markov distribution inherited by the Markov structure of the map. The dimension of the projected measure is therefore associated to the growth of this product; our dimension formula appears in a natural way as a version of the Furstenberg-Guivarch formula. Our technique provides an explicit dimension formula and, most important, provides a formalism well suited for the multifractal analysis of this measure, as we will show in a forthcoming paper.     

三分子反应在—维介质上的动力学行为

提出了一种简明的理论分析方法，求出了三分子反应Ａ＋２Ｘ　３Ｘ在一维介质上的动力学行为的解析解，并成功地解决了该反应动力学行为对初始条件的依赖性问题。还通过数值模拟证实了提出的理论方法的正确性。