Julia集和等势线上的Chebyshev多项式

用P表示一个度为d的首一多项式,J_P表示它的Julia集.本文得到Julia集J_P和其等势线Γ_P(R)上的d~n-阶Chebyshev多项式,并举例说明二者并不总是相等.     

The Julia sets of quadratic Cremer polynomials

We study the topology of the Julia set of a quadratic Cremer polynomial P. Our main tool is the following topological result. Let be a homeomorphism of a plane domain U and let T⊂U be a non-degenerate invariant non-separating continuum. If T contains a topologically repelling fixed point x with an invariant external ray landing at x, then T contains a non-repelling fixed point. Given P, two angles θ,γ are K-equivalent if for some angles x₀=θ,…,xn=γ the impressions of xi−1 and xi are non-disjoint, 1?i?n; a class of K-equivalence is called a K-class. We prove that the following facts are equivalent: (1) there is an impression not containing the Cremer point; (2) there is a degenerate impression; (3) there is a full Lebesgue measure dense Gδ-set of angles each of which is a K-class and has a degenerate impression; (4) there exists a point at which the Julia set is connected im kleinen; (5) not all angles are K-equivalent.     

Locally connected models for Julia sets

Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map φ onto a locally connected continuumJP_∼, i.e. a monotone map φ:J→JP_∼ such that for any other monotone map ψ:J→J^′ there exists a monotone map h with ψ=h°φ. Then we extend φ onto the complex plane C (keeping the same notation) and show that φ monotonically semiconjugates PC_| to a topological polynomialg:C→C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map φ not to collapse all of J into a point.     

Symmetries of the Julia sets of Newton’s method for multiple root

The symmetries of Julia sets of Newton’s method is investigated in this paper. It is shown that the group of symmetries of Julia set of polynomial is a subgroup of that of the corresponding standard, multiple and relax Newton’s method when a nonlinear polynomial is in normal form and the Julia set has finite group of symmetries. A necessary and sufficient condition for Julia sets of standard, multiple and relax Newton’s method to be horizontal line is obtained.     

Non-uniform hyperbolicity in complex dynamics

We say that a rational function F satisfies the summability condition with exponent α if for every critical point c which belongs to the Julia set J there exists a positive integer n _c so that \(\sum_{n=1}^{\infty} |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles. Let μ _max be the maximal multiplicity of the critical points.The objective is to study the Poincaré series for a large class of rational maps and establish ergodic and regularity properties of conformal measures. If F is summable with exponent \(\alpha<\frac{\delta_{\textit{Poin}}(J)}{\delta_{\textit{Poin}}(J)+\mu_{\textit{max}}}\) where δ _Poin (J) is the Poincaré exponent of the Julia set then there exists a unique, ergodic, and non-atomic conformal measure ν with exponent δ _Poin (J)=HDim(J). If F is polynomially summable with the exponent α, \(\sum_{n=1}^{\infty}n |(F^{n})^{\prime}(F^{n_{c}}(c))|^{-\alpha}<\infty\) and F has no parabolic periodic cycles, then F has an absolutely continuous invariant measure with respect to ν. This leads also to a new result about the existence of absolutely continuous invariant measures for multimodal maps of the interval.We prove that if F is summable with an exponent \(\alpha< \frac{2}{2+\mu_{\textit{max}}}\) then the Minkowski dimension of J is strictly less than 2 if \(J\neq\hat{\mathbb{C}}\) and F is unstable. If F is a polynomial or Blaschke product then J is conformally removable. If F is summable with \(\alpha<\frac{1}{1+\mu_{\textit{max}}}\) then connected components of the boundary of every invariant Fatou component are locally connected. To study continuity of Hausdorff dimension of Julia sets, we introduce the concept of the uniform summability.Finally, we derive a conformal analogue of Jakobson’s (Benedicks–Carleson’s) theorem and prove the external continuity of the Hausdorff dimension of Julia sets for almost all points c from the Mandelbrot set with respect to the harmonic measure.     

Function weighted measures and orthogonal polynomials on Julia sets

LetT(z) be a monic polynomial of degreed ?2 chosen so that its Julia setJ is real. A class of invariant measures supported onJ is constructed and discussed. We then construct the Jacobi matrices associated with these measures and show that they satisfy a renormalization group equation, a special case of which was discovered by Bellissard. Finally, we examine the asymptotic behavior of the orthogonal polynomials associated with these operators. We note that the operators have singular continuous spectra.     

Attracting cycles for the relaxed Newton’s method

We study the relaxed Newton’s method applied to polynomials. In particular, we give a technique such that for any n≥2, we may construct a polynomial so that when the method is applied to a polynomial, the resulting rational function has an attracting cycle of period n. We show that when we use the method to extract radicals, the set consisting of the points at which the method fails to converge to the roots of the polynomial p(z)=z^m−c (this set includes the Julia set) has zero Lebesgue measure. Consequently, iterate sequences under the relaxed Newton’s method converge to the roots of the preceding polynomial with probability one.     

Specializations of indecomposable polynomials

We address some questions concerning indecomposable polynomials and their behaviour under specialization. For instance we give a bound on a prime p for the reduction modulo p of an indecomposable polynomial ${P(x)\in {\mathbb{Z}}[x]}$ to remain indecomposable. We also obtain a Hilbert like result for indecomposability: if f(t ₁, . . . , t _r , x) is an indecomposable polynomial in several variables with coefficients in a field of characteristic p?=?0 or p?>?deg(f), then the one variable specialized polynomial ${f(t_1^\ast+\alpha_1^\ast x,\ldots,t_r^\ast+\alpha_r^\ast x,x)}$ is indecomposable for all ${(t_1^\ast, \ldots, t_r^\ast, \alpha_1^\ast, \ldots,\alpha_r^\ast)\in \overline k^{2r}}$ outside a proper Zariski closed subset.     

Topologie locale des méthodes de Newton cubiques: plan dynamique

For a cubic Newton map N, we obtain the following theorems: 1) The boundary of the immediate basin of each fixed critical point is locally connected. 2) The Julia set J(N) is locally connected provided either N has no irrational indifferent periodic point or N has no Siegel disc and the orbit of the non-fixed critical point doesn 't accumulate on the boundary of the fixed immediate basins. In particular, in contrast with Julia sets of polynomials, J(N) can be locally connected even if N has a periodic Cremer point.The proofs rely on the construction of articulated rays which are very special simple arcs landing on J(N).     

Attractors and recurrence for dendrite-critical polynomials

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.     

On Smital properties

Let A be an algebra and J⊂A an ideal of subsets of a group 〈X,+〉 with an invariant topology τ. We say that a triple 〈A,J,τ〉 has the Smital property if, for any set A∈A?J and any set D dense in τ, the set c(A+D) belongs to J. In the paper we compare this property and similar ones with the well-known Steinhaus type properties. We consider several weak and strong versions of Smital properties.     

A complete unitary similarity invariant for unicellular matrices

We present necessary and sufficient conditions for an n×n complex matrix B to be unitarily similar to a fixed unicellular (i.e., indecomposable by similarity) n×n complex matrix A.     

Inverse limits of postcritically finite polynomials

We examine inverse limits of postcritically finite polynomials restricted to their Julia sets. We define the “trunk” of a Julia set, a forward-invariant set related to the Hubbard tree, and use it to show that the inverse limit always contains at least one indecomposable subcontinuum. We characterize when the inverse limit is indecomposable and also examine how the trunk behaves in the inverse limit.     

The arithmetic of distributions in free probability theory

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I ₀, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I ₀ consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.     

On linear equations with general polynomial solutions

We provide necessary and sufficient conditions for which an nth-order linear differential equation has a general polynomial solution. We also give necessary conditions that can directly be ascertained from the coefficient functions of the equation.     

Coding and tiling of Julia sets for subhyperbolic rational maps

Let be a subhyperbolic rational map of degree d. We construct a set of “proper” coding maps Cod^°(f)={π_r:Σ→J}_r of the Julia set J by geometric coding trees, where the parameter r ranges over mappings from a certain tree to the Riemann sphere. Using the universal covering space for the corresponding orbifold, we lift the inverse of f to an iterated function system I=(g_i)_i=1,2,…,d. For the purpose of studying the structure of Cod^°(f), we generalize Kenyon and Lagarias-Wang's results : If the attractor K of I has positive measure, then K tiles φ^-1(J), and the multiplicity of π_r is well-defined. Moreover, we see that the equivalence relation induced by π_r is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps π_r and π_r^′ to be equal.     

Harmonic measure on the Julia set for polynomial-like maps

For a generalized polynomial-like mapping we prove the existence of an invariant ergodic measure equivalent to the harmonic measure on the Julia set J( f). We also prove that for polynomial-like mappings the harmonic measure is equivalent to the maximal entropy measure iff f is conformally equivalent to a polynomial. Next, we show that the Hausdorff dimension of harmonic measure on the Julia set of a generalized polynomial-like map is strictly smaller than 1 unless the Julia set is connected. Oblatum 24-IV-1995 & 22-VII-1996     

Polyhedral results for the bipartite induced subgraph problem

Given a graph G=(V,E) with node weights, the Bipartite Induced Subgraph Problem (BISP) is to find a maximum weight subset of nodes V^′ of G such that the subgraph induced by V^′ is bipartite. In this paper we study the facial structure of the polytope associated with that problem. We describe two classes of valid inequalities for this polytope and give necessary and sufficient conditions for these inequalities to be facet defining. For one of these classes, induced by the so-called wheels of order q, we give a polynomial time separation algorithm. We also describe some lifting procedures and discuss separation heuristics. We finally describe a Branch-and-Cut algorithm based on these results and present some computational results.     

Indecomposable modules over one-dimensional Noetherian rings

In this article we consider finitely generated torsion-free modules over certain one-dimensional commutative Noetherian rings R. We assume there exists a positive integer N_R such that, for every indecomposable R-module M and for every minimal prime ideal P of R, the dimension of M_P, as a vector space over the field R_P, is less than or equal to N_R. If a nonzero indecomposable R-module M is such that all the localizations M_P as vector spaces over the fields R_P have the same dimension r, for every minimal prime P of R, then r=1,2,3,4 or 6. Let n be an integer ≥8. We show that if M is an R-module such that the vector space dimensions of the M_P are between n and 2n−8, then M decomposes non-trivially. For each n≥8, we exhibit a semilocal ring and an indecomposable module for which the relevant dimensions range from n to 2n−7. These results require a mild equicharacteristic assumption; we also discuss bounds in the non-equicharacteristic case.     

Misiurewicz points on the Mandelbrot-like set concerning renormalization transformation

We study Misiurewicz points on the parameter space about a family of rational maps Tλ concerning renormalization transformation in statistical mechanic. We determine the intersection points of the Julia set J(Tλ) and the positive real axis R+and discuss the continuity of the Hausdorff dimension HD(J(f)) about real parameter λ.