Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Classification of quadratic systems admitting the existence of an algebraic limit cycle

In the paper we find a set of necessary conditions that must be satisfied by a quadratic system in order to have an algebraic limit cycle. We find a countable set of ?5 parameter families of quadratic systems such that every quadratic system with an algebraic limit cycle must, after a change of variables, belong to one of those families. We provide a classification of all the quadratic systems which can have an algebraic limit cycle based on geometrical properties of the embedding of the system in the Poincaré compactification of R². We propose names for all the classes we distinguish and we classify all known examples of quadratic systems with algebraic limit cycle. We also prove the integrability of certain classes of quadratic systems.     

Fractal Measures for Parabolic IFS

Let h be the Hausdorff dimension of the limit set of a conformal parabolic iterated function system in dimension d?2. In case the system of maps is finite, we provide necessary and sufficient conditions for the h-dimensional Hausdorff measure to be positive and finite and also, assuming the strong open set condition holds, characterize when the h-dimensional packing measure of the limit set is positive and finite. We also prove that the upper ball (box)-counting dimension and the Hausdorff dimension of this limit set coincide. As a byproduct we include a compact analysis of the behaviour of parabolic conformal diffeomorphisms in dimension 2 and separately in any dimension greater than or equal to 3.     

Transitive Limit Closures of Convex Dynamical Systems

Convex dynamical systems are iterated set-valued maps with convex graphs. The closed union of all finite powers of a given convex relation will be called its limit closure. We address the question of transitivity of limit closures and establish a sufficient condition for such transitivity (limit transitivity). We also present examples showing that the limit closure of a general compact convex system is not necessarily transitive. limit closure can be intransitive as well. It is also shown that the restriction of a linear single-valued map to a convex set containing an open neighborhood of the origin is always limit transitive.     

Dimension sets for infinite IFSs: the Texan Conjecture

We consider the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system and refer to it as the restricted dimension set. The corresponding set for all subsystems will be referred to as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both general results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval.     

共轭可交换性及其在集合形式单调类定理中的应用

引入了集列的极限运算与二元集运算之间的共轭可交换性的概念,讨论了常见的和重要的集列的极限运算与集代数运算之间所具有的共轭可交换性质,在一般的情形下证明了共轭可交换性是集类的极限运算生成类对集代数运算具有封闭性的一个充分条件,使得集合形式的单调类定理的理论得到了统一的解释,并且证明更简单,思路更清晰,最后还给出了一个新的集合形式的单调类定理.     

渐近时间周期系统

本文证明了渐近时间周期系统的Ｐｏｉｎｃａｒｅ映射的极限集是极限系统的Ｐｏｉｎｃａｒｅ映射的不变集。     

VIRTUAL SYSTEM METHOD FOR SYSTEM TREE

ＶＩＲＴＵＡＬＳＹＳＴＥＭＭＥＴＨＯＤＦＯＲＳＹＳＴＥＭＴＲＥＥ￥ＺＨＥＮＧＺＨＯＮＧＧＵＯ（ＤｅｐａｒｔｍｅｎｔｏｆＰｒｏｂａｂｉｌｉｔｙａｎｄＳｔａｔｉｓｔｉｃｓ，ＢｅｉｊｉｎｇＵｎｉｖｅｒｓｉｔｙ）Ｂｅ１Ｊｉｎｇ１００８７１，Ｃｈｉｎａ．）（Ｐ...     

The limit functions of a random iteration system

This paper discusses the limit functions of a random iteration system formed by finitely many rational functions. Applying these results we prove that a hyperbolic iteration system has no wandering domain and that its limit functions are constant. Finally the continuity on its Julia set is considered.     

The Shunted Homopolar Dynamo—An Analytic Approach to a Poincaré Map

We study the set of ordinary differential equations governing the homopolar disk dynamo. It is found that this set, which is a modification of the Lorenz system, has strange attractors of Lorenz type when R (which corresponds to the Rayleigh number of the Lorenz system) tends to infinity. A central aspect of this study is that the Poincare map for this limit can be obtained through Melnikov's perturbation method, in contrast to the usual dependence on numerical computation.     

On Necessary Conditions of Probability Limit Theorems in Finite Algebras

Doklady Mathematics - We consider the conditions for a finite set with a given system of operations (a finite algebra) to be subject to a probability limit theorem, i.e., arbitrary computations...     

On the dynamics and recursive properties of multidimensional symbolic systems

We study the (sub)dynamics of multidimensional shifts of finite type and sofic shifts, and the action of cellular automata on their limit sets. Such a subaction is always an effective dynamical system: i.e. it is isomorphic to a subshift over the Cantor set the complement of which can be written as the union of a recursive sequence of basic sets. Our main result is that, to varying degrees, this recursive-theoretic condition is also sufficient. We show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems, and that a general effective system can be realized, modulo a small extension, as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set (after removing a dynamically trivial set). As applications, we characterize, in terms of their computational properties, the numbers which can occur as the entropy of cellular automata, and construct SFTs and CAs with various interesting properties.     

Boundary behaviour of inner functions and holomorphic mappings

Let be a holomorphic function in the unit disk omitting a set of values of the complex plane. If has positive logarithmic capacity, R. Nevanlinna proved that has a radial limit at almost every point of the unit circle. If is any infinite set, we show that has a radial limit at every point of a set of Hausdorff dimension 1. A localization technique reduces this result to the following theorem on inner functions. If is an inner function omitting a set of values in the unit disk, then for any accumulation point of in the disk, there exists a set of Hausdorff dimension 1 of points in the circle where has radial limit . Received: 13 February 1997     

钟摆模型极限环的存在性研究

钟摆系统是一类典型的分段光滑系统,结合Filippov系统刻画语言,解释了当钟摆无能量补充时,钟摆最终会停止在滑动集上的原因.利用数值模拟的方法,给出钟摆系统在有能量补充时,存在极限环的条件.最后,结合环域定理证明了一般的钟摆模型存在唯一稳定的极限环.     

On topological entropy of billiard tables with small inner scatterers

We present in this paper an approach to studying the topological entropy of a class of billiard systems. In this class, any billiard table consists of strictly convex domain in the plane and strictly convex inner scatterers. Combining the concept of anti-integrable limit with the theory of Lyusternik-Shnirel'man, we show that a billiard system in this class generically admits a set of non-degenerate anti-integrable orbits which corresponds bijectively to a topological Markov chain of arbitrarily large topological entropy. The anti-integrable limit is the singular limit when scatterers shrink to points. In order to get around the singular limit and so as to apply the implicit function theorem, on auxiliary circles encircling these scatterers we define a length functional whose critical points are well-defined at the anti-integrable limit and give rise to billiard orbits when the scatterers are not points. Consequently, we prove the topological entropy of the first return map to the scatterers can be made arbitrarily large provided the inner scatterers are sufficiently small.     

The String Method as a Dynamical System

This paper is devoted to the theoretical analysis of the zero-temperature string method, a scheme for identifying minimum energy paths (MEPs) on a given energy landscape. By definition, MEPs are curves connecting critical points on the energy landscape which are everywhere tangent to the gradient of the potential except possibly at critical points. In practice, MEPs are mountain pass curves that play a special role, e.g., in the context of rare reactive events that occur when one considers a steepest descent dynamics on the potential perturbed by a small random noise. The string method aims to identify MEPs by moving each point of the curve by steepest descent on the energy landscape. Here we address the question of whether such a curve evolution necessarily converges to an MEP. Surprisingly, the answer is no, for an interesting reason: MEPs may not be isolated, in the sense that there may be families of them that can be continuously deformed into one another. This degeneracy is related to the presence of critical points of Morse index 2 or higher along the MEP. In this paper, we elucidate this issue and completely characterize the limit set of a curve evolving by the string method. We establish rigorously that the limit set of such a curve is again a curve when the MEPs are isolated. We also show under the same hypothesis that the string evolution converges to an MEP. However, we identify and classify situations where the limit set is not a curve and may contain higher dimensional parts. We present a collection of examples where the limit set of a path contains a 2D region, a 2D surface, or a region of an arbitrary dimension up to the dimension of the space. In some of our examples the evolving path wanders around without converging to its limit set. In other examples it fills a region, converging to its limit set, which is not an MEP.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

一个A（f）≠M（f）的紧致系统（X，f）

本文证明，存在紧致系统，其几乎周期点集闭包不等于其测度中心．藉此，我们否定地回答了两个未解决的问题．     

Continuous time trading of a small investor in a limit order market

We provide a mathematical framework to model continuous time trading of a small investor in limit order markets. We show how elementary strategies can be extended in a suitable way to general continuous time strategies containing orders with infinitely many different limit prices. The general limit buy order strategies are predictable processes with values in the set of nonincreasing demand functions. It turns out that our strategy set of limit and market orders is closed, but limit orders can turn into market orders when passing to the limit, and any element can be approximated by a sequence of elementary strategies.     

Convergence of stochastic search algorithms to finite size pareto set approximations

In this work we investigate the convergence of stochastic search algorithms toward the Pareto set of continuous multi-objective optimization problems. The focus is on obtaining a finite approximation that should capture the entire solution set in a suitable sense, which will be defined using the concept of ε-dominance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and analyse two different archiving strategies which lead to a different limit behavior of the algorithms, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting Pareto set approximation.