共查询到20条相似文献,搜索用时 15 毫秒
1.
Yin Yongcheng 《数学学报(英文版)》1990,6(2):120-130
For any polynomialp(z)=a
0
z
n
+a|z
n–1++a
n
, a00, n2,F is the Julia set and
*
is the equilibrium distribution onF. Hans Brolin[1] proved that (F)>0, andS
*
=F. Up to now, we know nothing about rational functions. The aim of this paper is to discuss the case of rational functions.Project supported by the Science Fund of the Chinese Academy of Sciences. 相似文献
2.
Julia sets of rational semigroups 总被引:2,自引:0,他引:2
3.
Julia sets of rational semigroups 总被引:5,自引:0,他引:5
4.
In this paper, we prove that two rational maps with the Cantor Julia sets are quasicon- formally conjugate if they are topologically conjugate. 相似文献
5.
LetR be a rational function with nonempty set of normality that consists of basins of attraction only and let
相似文献
6.
Suppose a quadratic rational map has a Siegel disk and a parabolic fixed point. If the rotation number of the Siegel disk is an irrational of bounded type, then the Julia set of the map is shallow. This implies that its Hausdorff dimension is strictly less than two. 相似文献
7.
Atsushi Kameyama 《Advances in Mathematics》2006,200(1):217-244
Let be a subhyperbolic rational map of degree d. We construct a set of “proper” coding maps Cod°(f)={πr: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=(gi)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 πr is well-defined. Moreover, we see that the equivalence relation induced by πr is described by a finite directed graph, and give a necessary and sufficient condition for two coding maps πr and πr′ to be equal. 相似文献
8.
Leth be the Hausdorff dimension of the Julia setJ(R) of a Misiurewicz’s rational mapR :
(subexpanding case). We prove that theh-dimensional Hausdorff measure H
h
onJ(R) is finite, positive and the onlyh-conformal measure forR :
up to a multiplicative constant. Moreover, we show that there exists a uniqueR-invariant measure onJ(R) equivalent to H
h
. 相似文献
9.
Figen Çilingir Robert L. Devaney Elizabeth D. Russell 《Journal of Fixed Point Theory and Applications》2010,7(1):223-240
For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure
of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of
the form Fλ(z) = z
n
+ λ/z
n
where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend
these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann
sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the
external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology
of the Julia set. 相似文献
10.
Let T : J → J be an expanding rational map of the Riemann sphere acting on its Julia set J andf : J →R denote a Hölder continuous function satisfyingf(x)?log | T′(x vb for allx in J. Then for any pointz 0 in J define the set Dz 0(f) of “well-approximable” points to be the set of points in J which lie in the Euclidean ball $B(\gamma ,{\text{ exp(}} - \sum {_{i - 0}^{\mathfrak{n} - 1} } f(T^\ell x)))$ for infinitely many pairs (y, n) satisfying T n (y)=z0. We prove that the Hausdorff dimension of Dz 0(f) is the unique positive numbers(f) satisfying the equation P(T,?s(f).f)=0, where P is the pressure on the Julia set. This result is then shown to have consequences for the limsups of ergodic averages of Hölder continuous functions. We also obtain local counting results which are analogous to the orbital counting results in the theory of Kleinian groups. 相似文献
11.
Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy–Carleman–Ahlfors
theorem implies that if the set of all z for which |f(z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M(r, f) ≥ (N/2) log r − O(1), where M(r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Carleman and Tsuji related to the Denjoy–Carleman–Ahlfors
theorem. 相似文献
12.
Toshiyuki Sugawa 《Mathematische Zeitschrift》2001,238(2):317-333
A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate
C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né
and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and,
as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application,
we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of
quadratic polynomials in terms of the parameter c.
Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001 相似文献
13.
Huaibin Li 《印度理论与应用数学杂志》2013,44(6):849-863
In this paper, we consider a rational map f of degree at least two acting on Riemman sphere that is expanding away from critical points. Assuming that all critical points of f in the Julia set J(f) are reluctantly recurrent, we prove that the Hausdorff dimension of the Julia set J(f) is equal to the hyperbolic dimension, and the Lebesgue measure of Julia set is zero when the Julia set J(f) ≠ . 相似文献
14.
We discuss the dynamics as well as the structure of the parameterplane of certain families of rational maps with few criticalorbits. Our paradigm is the family Rt(z) = (1 + (4/27)z3/(1– z)), with dynamics governed by the behaviour of thepostcritical orbit (Rn())n. In particular, it is shown thatif escapes (that is, Rn() tends to infinity), then the Juliaset of R is a Cantor set, or a Sierpiski curve, or a curve withone or else infinitely many cut-points; each of these casesactually occurs. 相似文献
15.
Yuichi Kamiya 《Acta Mathematica Hungarica》2006,110(1-2):51-65
Summary A. Beurling introduced the concept of spectral sets of unbounded functions to study the possibility of the approximation of
those by trigonometric polynomials. We consider spectral sets of unbounded functions in a certain class which contains the
square of the Riemann zeta-function as a typical example. 相似文献
16.
Invariant sets under iteration of rational functions 总被引:11,自引:0,他引:11
Hans Brolin 《Arkiv f?r Matematik》1965,6(2):103-144
17.
We show that under the definition of computability which is natural from the point of view of applications, there exist non-computable quadratic Julia sets.
18.
19.
20.
Joachim Grispolakis John C. Mayer Lex G. Oversteegen 《Transactions of the American Mathematical Society》1999,351(3):1171-1201
We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.
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