Dimensions of Julia Sets of Hyperbolic Entire Functions

It is known that, if f is a hyperbolic rational function, thenthe Hausdorff, packing and box dimensions of the Julia set J(f)are equal. It is also known that there is a family of hyperbolictranscendental meromorphic functions with infinitely many polesfor which this result fails to be true. In this paper, new methodsare used to show that there is a family of hyperbolic transcendentalentire functions f_K, K N, such that the box and packing dimensionsof Jf_K are equal to two, even though as K the Hausdorff dimensionof Jf_K tends to one, the lowest possible value for the Hausdorffdimension of the Julia set of a transcendental entire function.2000 Mathematics Subject Classification 30D05, 37F10, 37F15,37F35, 37F50.     

On Sets Where Iterates of a Meromorphic Function Zip Towards Infinity

For a transcendental meromorphic function f, various propertiesof the set [formula] were obtained in [8] and [9]. Here we establish analogous propertiesfor the smaller sets [formula] introduced in [5], and [formula] We deduce a symmetry result for Julia sets J(f), and also indicatesome techniques for showing that certain invariant curves liein I'(f), Z(f) and J(f). 2000 Mathematics Subject Classification30D05, 37F10, 37F50.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

Dynamics of meromorphic functions with direct or logarithmic singularities

Let f be a transcendental meromorphic function and denote byJ(f) the Julia set and by I(f) the escaping set. We show thatif f has a direct singularity over infinity, then I(f) has anunbounded component and I(f)J(f) contains continua. Moreover,under this hypothesis I(f)J(f) has an unbounded component ifand only if f has no Baker wandering domain. If f has a logarithmicsingularity over infinity, then the upper box dimension of I(f)J(f)is 2 and the Hausdorff dimension of J(f) is strictly greaterthan 1. The above theorems are deduced from more general resultsconcerning functions which have ‘direct or logarithmictracts’, but which need not be meromorphic in the plane.These results are obtained by using a generalization of Wiman–Valirontheory. This method is also applied to complex differentialequations.     

The Hausdorff Dimension of Julia Sets of Entire Functions IV

It is known that, for a transcendental entire function f, theHausdorff dimension of J(f) satisfies 1 dimJ(f) 2. For eachd (1, 2), an example of a transcendental entire function fwith dimJ(f) = d is given. It is then indicated how this functioncan be modified to produce a transcendental meromorphic functionF with one pole with dimJ(F) = d. These appear to be the firstexamples of Julia sets with non-integer dimensions whose dimensionshave been calculated exactly.     

Julia Sets of Polynomials are Uniformly Perfect

Let f be a polynomial of degree at least two. We shall showthat the Julia set J(f) of f is uniformly perfect. This meansthat there is a constant c(0,1) depending on f only such thatwhenever zJ(f) and 0 < r < diam J(f) then J(f) intersectsthe annulus {w:cr |w — z| r}.     

On Uniformly Perfect Boundaries of Stable Domains in Iteration of Meromorphic Functions

Let f:C be a function which is either transcendental meromorphicor rational with degree at least 2. We discuss the uniform perfectnessof the attracting or parabolic cycle of stable domains of f(z),and include a proof that the Julia set of a meromorphic functionof finite type is uniformly perfect. 2000 Mathematics SubjectClassification 37F10, 37F50, 30D05.     

Packing Measure Analysis of Harmonic Measure

We show that, for any Jordan domain J in R², harmonic measureis supported by a Borel set of packing dimension 1. We alsoobtain incomplete analogs to the results of Makarov, which connectthe almost everywhere behavior of the derivative near the boundaryfor the conformal mapping function from the unit disk J withthe Hausdorff measure properties of sets supporting the harmonicmeasure.     

A note on a quadratic rational map with two Siegel disks

Suppose f（z） is a quadratic rational map with two Siegel disks. If the rotation numbers of the Siegel disks are both of bounded type, the Hausdorff dimension of the Julia set satisfies Dim （J（f））〈2.     

Hausdorff and Packing Measures of the Level Sets of Iterated Brownian Motion

Burdzy and Khoshnevisan⁽⁹⁾ have shown that the Hausdorff dimension of the level sets of an iterated Brownian motion (IBM) is equal to 3/4. In this paper, the exact Hausdorff measure function and the packing measure of the levels set of IBM are given. Our approach relies on some accurate analysis on the local asymptotic of local times.     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

The Hausdorff Dimension of Julia Sets of Meromorphic Functions II

A family of transcendental meromorphic functions, f_p(z), p N is considered. It is shown that, if p 6, then the Hausdorffdimension of the Julia set of f_p satisfies dim J(f_p) 1/p, for0 < < 1/6^p, and dim J(f_p) 1–(30 ln ln p/ln p),for p^4p–1/10⁵ ln p < < p^4p–1/10⁴ ln p. Theseresults are used elsewhere to show that, for each d (0, 1),there exists a transcendental meromorphic function for whichdim J(f) = d.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.     

Concerning the Problem of Subsets of Finite Positive Packing Measure

For every Hausdorff function h, the paper gives a constructionof a compact metric space of infinite diameter-based h-packingmeasure which has no subsets of positive finite measure. Itthen indicates how such a construction can be modified to providethe same result for a radius-based packing measure in the caseof certain Hausdorff functions which do not satisfy a doublingcondition. This is in surprising contrast to another resultwhich shows that for a particular radius-based definition ofpacking measure no such example can be found.     

Best Approximation in an Asymmetrically Weighted L1 Measure

Let f(x) be a given, real-valued, continuous function definedon an interval [a,b]of the real line. Given a set of m real-valued,continuous functions _j(x) defined on [a,b], a linear approximatingfunction can be formed with any real setA = {a₁, a₂,..., a_m}. We present results for determining A sothat F(A, x) is a best approximation to(x) when the measureof goodness of approximation is a weighted sum of |F(A, x)–f(x)|,the weights being positive constants, w, when F(A, x) f(x)and w2 otherwise (when w, = w2 = 1, the measure is the L₁, norm).The results are derived from a linear programming formulationof the problem. In particular, we give a theorem which shows when such bestapproximations interpolate the function at fixed ordinates whichare independent of f(x). We show how the fixed points can becalculated and we present numerical results to indicate thatthe theorem is quite robust.     

On Maximal Regularity and Semivariation of Cosine Operator Functions

It is proved that a cosine operator function C(·), withgenerator A, is locally of bounded semivariation if and onlyif u'(t) = Au(t)+f(t), t>0, u(0), u'(0)D(A), has a strongsolution for every continuous function f, if and only if thefunction , is twice continuously differentiable for every continuous function f, that is, C(·)has the maximal regularity property if and only if A is a boundedoperator. Some other characterisations of bounded generatorsof cosine operator functions are also established in terms oftheir local semivariations.     

Dimensions of subsets of Moran fractals related to frequencies of their codings

The Moran fractal considered in this paper is an extension of the self-similar sets satisfying the open set condition. We consider those subsets of the Moran fractal that are the union of an uncountable number of sets each of which consists of the points with their location codes having prescribed mixed group frequencies. It is proved that the Hausdorff and packing dimensions of each of these subsets coincide and are equal to the supremum of the Hausdorff (or packing) dimensions of the sets in the union. An approach is given to calculate their Hausdorff and packing dimensions. The main advantage of our approach is that we treat these subsets in a unified manner. Another advantage of this approach is that the values of the Hausdorff and packing dimensions do not need to be guessed a priori.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

Tjurina and Milnor Numbers of Matrix Singularities

To gain understanding of the deformations of determinants andPfaffians resulting from deformations of matrices, the deformationtheory of composites f F with isolated singularities is studied,where f : YC is a function with (possibly non-isolated) singularityand F : XY is a map into the domain of f, and F only is deformed.The corresponding T¹(F) is identified as (something like) thecohomology of a derived functor, and a canonical long exactsequence is constructed from which it follows that = µ(f F) – ß₀ + ß₁, where is the length of T¹(F) and ß_i is the lengthof Tor_i^O_Y(O_Y/J_f, O_X). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulaysingular locus (for example when f is the determinant function),relations between and the rank of the vanishing homology ofthe zero locus of f F are obtained.     

Finitary Power Semigroups of Infinite Groups are not Finitely Generated

For a semigroup S, the finitary power semigroup of S, denotedP_f(S), consists of all finite subsets of S under the usual multiplication.The main result of this paper asserts that P_f(G) is not finitelygenerated for any infinite group G. 2000 Mathematics SubjectClassification 20M05 (primary), 20M30, 20F99 (secondary).