Sets of Fibre Homotopy Classes¶and Twisted Order Parameter Spaces

We propose a machinery to calculate the set of global defect indices of a regularly defected order taking values in arbitrary parameter bundles. Received: 26 January 1999/ Accepted: 7 December 1999     

Constructing Locally Connected Non-Computable Julia Sets

A locally connected quadratic Siegel Julia set has a simple explicit topological model. Such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution. We constructively produce parameter values for Siegel quadratics for which the Julia sets are non-computable, yet locally connected. This research was partially conducted during the period the first author was employed by the Clay Mathematics Institute as a Liftoff Fellow. The second author’s research is supported by NSERC operating grant.     

On Computational Complexity of Siegel Julia Sets

It has been previously shown by two of the authors that some polynomial Julia sets are algorithmically impossible to draw with arbitrary magnification. On the other hand, for a large class of examples the problem of drawing a picture has polynomial complexity. In this paper we demonstrate the existence of computable quadratic Julia sets whose computational complexity is arbitrarily high. The first and third authors are partially supported by NSERC Discovery grants. The second author is partially supported by NSERC Postgraduate Scholarship     

广义Julia 集的参数辨识

本文研究了一类相当广泛的复系统Julia 集的参数辨识问题.基于非线性反馈控制器和差分方程稳定性理论, 设计了普遍适用的自适应同步控制器和参数自适应律的解析表达式.理论证明设计的控制器可使得此类广义复系统Julia 集达到同步,并且可以辨识广义Julia 集的未知参数.通过仿真实例验证了该方法的有效性.另外, 本文特别地讨论了最基本的Julia 集的参数辨识问题. 关键词： Julia集 参数辨识 同步     

On Connectivity of Julia Sets of Yang--Lee Zeros

We prove that the Julia sets of the Yang–Lee zeros of the Potts model on the diamond hierarchical lattice are connected sets on the complex plane. Received: 23 May 2000 / Accepted: 9 May 2001     

Reproducing Kernels and Coherent States on Julia Sets

We construct classes of coherent states on domains arising from dynamical systems. An orthonormal family of vectors associated to the generating transformation of a Julia set is found as a family of square integrable vectors, and, thereby, reproducing kernels and reproducing kernel Hilbert spaces are associated to Julia sets. We also present analogous results on domains arising from iterated function systems. The research of the first two authors was supported by Natural Sciences and Engineering Research Council of Canada.     

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.     

Rigidity and Non-local Connectivity of Julia Sets of Some Quadratic Polynomials

For an infinitely renormalizable quadratic map \({f_c: z\mapsto z^2+c}\) with the sequence of renormalization periods {k _m } and rotation numbers {t _m  = p _m /q _m }, we prove that if \({\limsup k_m^{-1} \log |p_m| >0 }\), then the Mandelbrot set is locally connected at c. We prove also that if \({\limsup |t_{m+1}|^{1/q_m} <1 }\) and q _m → ∞, then the Julia set of f _c is not locally connected and the Mandelbrot set is locally connected at c provided that all the renormalizations are non-primitive (satellite). This quantifies a construction of A. Douady and J. Hubbard, and weakens a condition proposed by J. Milnor.     

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.     

Addendum to: Rigidity and Non-Local Connectivity of Julia Sets of Some Quadratic Polynomials

We improve one of the main results of Levin (Commun Math Phys 304:295–328, 2011).     

Metrics of Homogeneous Spaces Admitting (3.1)-Type Complete Sets

A classification has been obtained for spaces which admit complete sets of motion integrals of the (3.1)-type and are spatially homogeneous. The metrics and Killing vectors have been found in explicit form for all types of these spaces. A classification by Bianchi is given for the spaces obtained.     

Considerable Sets of Linear Operators in Hilbert Spaces as Operator Generalized Effect Algebras

We show that considerable sets of positive linear operators namely their extensions as closures, adjoints or Friedrichs positive self-adjoint extensions form operator (generalized) effect algebras. Moreover, in these cases the partial effect algebraic operation of two operators coincides with usual sum of operators in complex Hilbert spaces whenever it is defined. These sets include also unbounded operators which play important role of observables (e.g., momentum and position) in the mathematical formulation of quantum mechanics.     

Relativistic Correction on Neutrino Emission from Neutron Stars in Various Parameter Sets

In the relativistic mean field theory and cooling theories, relativistic correction on neutrino emission from neutron stars in four typical nuclear parameter sets, GM1, GL85, GPS250 and GPS300 is studied. Results show that relativistic effect makes the neutrino emissivity, neutrino luminosity and cooling rate lower, compared with the non-relativistic case. And the influence of relativistic effect grows with the mass of the neutron star. GPS300 set leads to the biggest fall in neutrino emissivity, whereas GM1 set leads to the largest disparity in cooling rate caused by relativistic effect.     

Metrics of Homogeneous Spaces Admitting Complete Sets of the (2.1) type

A classification of spaces admitting complete sets of integrals of motion of the (2.1) type and being spatially homogeneous has been obtained. The explicit form of metrics and Killing vectors have been found for all types of such spaces. The Bianchi classification of the spaces obtained is given.     

Two-magnetic-monopole Julia sets

We study the global behavior of two interacting magnetic monopoles in the non-relativistic limit. We identify what appear to be regions in phase space of regular and stochastic motions. We also construct Julia sets, and approximate stable and unstable sets. We predict imprisoned motion for some initial conditions with infinite monopole separation, and note that the set of bounded initial conditions can sometimes separate regular motion from stochastic motion.     

Percolation and Julia sets

The types of Julia sets for the renormalization group (RG) transformation for percolation on the hierarchial model are derived. The RG transformation for the concentration for two-dimensional triangular lattice site percolation problems and two-dimensional square lattice bond percolation problems are generalized to the complex plane. Julia sets for both cases are found.     

Stable Sets and \epsilon-Stable Sets in Positive-Entropy Systems

In this paper, the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated. It is shown that in any positive-entropy system, there is a measure-theoretically “rather big” set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set. Moreover, the Bowen entropy of these weak mixing sets are also estimated. At the same time, it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre-images of -stable sets, which answers an open question posed by D. Fiebig, U.R. Fiebig and Z.H. Nitecki (Ergod. Th & Dynam. Sys. 23, 1785-1806 (2003)). The author is supported by NSFC, 973 project and FANEDD (200520).     

基于Julia分形的多涡卷忆阻混沌系统

忆阻器作为一种非线性电子元件,能用作混沌系统中的非线性项,从而提高系统的复杂度.分形与混沌是密切相连的,分别对两者的研究都已成熟,却鲜有将分形过程应用到混沌系统中,以产生丰富的混沌吸引子.为了探索将分形与混沌系统相结合的可能性,本文首先提出了一个新的忆阻混沌系统,并从对称性、耗散性、平衡点稳定性、功率谱、Lyapunov指数和分数维等方面探讨了系统的动力学特性;紧接着,把经典的Julia分形过程应用到该忆阻混沌系统中,产生了新的混沌吸引子,并将几种由Julia分形衍生的变形Julia分形过程应用于文中提出的忆阻混沌系统,获得了丰富的混沌吸引子;最后,讨论了分形过程中的复常数对系统的影响.从仿真结果可以看出,分形过程与混沌系统的结合能产生丰富的多涡卷混沌吸引子.这不仅为产生多涡卷混沌吸引子提供了一种新方法,还弥补了使用功能函数方法造成混沌系统不光滑的不足.