Herman Rings and Arnold Disks

For (,a) C* x C, let f_,a be the rational map defined by f_,a(z)= z² (az+1)/(z+a). If R/Z is a Brjuno number, we let D bethe set of parameters (,a) such that f_,a has a fixed Hermanring with rotation number (we consider that (e²ⁱ,0) D). Resultsobtained by McMullen and Sullivan imply that, for any g D, theconnected component of D(C* x (C/{0,1})) that contains g isisomorphic to a punctured disk. We show that there is a holomorphic injection F:DD such thatF(0) = (e²ⁱ ,0) and , where r is the conformal radius at 0 of the Siegel disk of the quadraticpolynomial z e²ⁱ z(1+z). As a consequence, we show that for a (0,1/3), if f_l,a has afixed Herman ring with rotation number and if m_a is the modulusof the Herman ring, then, as a0, we have e m_a=(r/a) + O(a). We finally explain how to adapt the results to the complex standardfamily z e^(a/2)(z-1/z).     

A Mean Value Property of Poly-Temperatures on a Strip Domain

We consider the iterates of the heat operator on Rⁿ⁺¹={(X, t); X=(x₁, x₂, ..., x_n)Rⁿ, tR}. Let Rⁿ⁺¹ be a domain,and let m1 be an integer. A lower semi-continuous and locallyintegrable function u on is called a poly-supertemperatureof degree m if (–H)^mu0 on (in the sense of distribution). If u and –u are both poly-supertemperatures of degreem, then u is called a poly-temperature of degree m. Since His hypoelliptic, every poly-temperature belongs to C(), andhence (–H)^m u(X, t)=0 (X, t). For the case m=1, we simply call the functions the supertemperatureand the temperature. In this paper, we characterise a poly-temperature and a poly-supertemperatureon a strip D={(X, t);XRⁿ, 0<t<T} by an integral mean on a hyperplane. To state our result precisely,we define a mean A[·, ·]. This plays an essentialrole in our argument.     

Sierpinski and non-Sierpinski curve Julia sets in families of rational maps

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 R_t(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.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Real Interpolation and Two Variants of Gehring's Lemma

Let be a fixed open cube in Rⁿ. For r[1, ) and [0, ) we define where Q is a cube in Rⁿ (with sides parallel to the coordinateaxes) and _Q stands for the characteristic function of the cubeQ. A well-known result of Gehring [5] states that if (1.1) for some p(1, ) and c(0, ), then there exist q(p, ) and C=C(p,q, n, c)(0, ) such that for all cubes Q, where |Q| denotes the n-dimensional Lebesguemeasure of Q. In particular, a function fL¹() satisfying (1.1)belongs to L^q(). In [9] it was shown that Gehring's result is a particular caseof a more general principle from the real method of interpolation.Roughly speaking, this principle states that if a certain reversedinequality between K-functionals holds at one point of an interpolationscale, then it holds at other nearby points of this scale. Usingan extension of Holmstedt's reiteration formulae of [4] andresults of [8] on weighted inequalities for monotone functions,we prove here two variants of this principle involving extrapolationspaces of an ordered pair of (quasi-) Banach spaces. As an applicationwe prove the following Gehring-type lemmas.     

A Strong Notion of Universal Taylor Series

For a holomorphic function f in the open unit disc D, the Nthpartial sum of its Taylor series with center D is denotedby S_N(f,)(z)= . Generically, all functions f in H(D) satisfy the following. For every compactset K C with KD=Ø and K^c connected and every polynomialh, there exists a sequence of positive integers such that, for every 1 {0,1,2,...},      

An Access Theorem for Continuous Functions

Let f be a continuous function on an open subset of R² suchthat for every x there exists a continuous map : [–1,1] with (0) = x and f increasing on [–1, 1]. Thenfor every there exists a continuous map : [0, 1) suchthat (0) = y, f is increasing on [0; 1), and for every compactsubset K of , max{t : (t) K} < 1. This result gives an answerto a question posed by M. Ortel. Furthermore, an example showsthat this result is not valid in higher dimensions.     

Skew Derivations with Central Invariants

If is an automorphism and is a -derivation of a ring R, thenthe subring of invariants is the set R⁽⁾={rR|(r)=0}. The mainresult of this paper is ‘let R be a semiprime ring withan algebraic -derivation such that R⁽⁾ is central; then R iscommutative’. This theorem generalizes results on theinvariants of automorphisms and derivations and is proved byreducing down to the special cases of automorphisms and derivations.     

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.     

Asymptotics for Fractional Nonlinear Heat Equations

The Cauchy problem is studied for the nonlinear equations withfractional power of the negative Laplacian where (0,2), with critical = /n and sub-critical (0,/n)powers of the nonlinearity. Let u₀ L^1,a L C, u₀(x) 0 in Rⁿ, = . The case of not small initial data is of interest. It is proved that the Cauchy problemhas a unique global solution u C([0,); L L^1,a C) and the largetime asymptotics are obtained.     

Weighted Bloch Spaces, Hp, and BMOA

It is determined which Bloch-type conditions on a function f _0<p<H^p ensure that f BMOA.     

Antisupercyclic Operators and Orbits of the Volterra Operator

We say that a bounded linear operator T acting on a Banach spaceB is antisupercyclic if for any x B either Tⁿx = 0 for somepositive integer n or the sequence {Tⁿx/||Tⁿx||} weakly convergesto zero in B. Antisupercyclicity of T means that the angle criterionof supercyclicity is not satisfied for T in the strongest possibleway. Normal antisupercyclic operators and antisupercyclic bilateralweighted shifts are characterized. As for the Volterra operator V, it is proved that if 1 p and any f L_p [0,1] then the limit lim_n (n!||Vⁿf||_p)^1/n doesexist and equals 1 – inf supp (f). Upon using this asymptoticformula it is proved that the operator V acting on the Banachspace L_p[0,1] is antisupercyclic for any p (1,). The same statementfor p = 1 or p = is false. The analogous results are provedfor operators when the real part of z C is positive.     

On Artin's Braid Group And Polyconvexity In The Calculus Of Variations

Let R² be a bounded Lipschitz domain and let be a Carathèodory integrand such that F(x,·) is polyconvex for L²-a.e. x . Moreover assume thatF is bounded from below and satisfies the condition as det for L²-a.e. x . The paper describes the effect of domain topologyon the existence and multiplicity of strong local minimizersof the functional wherethe map u lies in the Sobolev space W_id^1,p (, R²) with p 2and satisfies the pointwise condition u(x) >0 for L²-a.e.x . The question is settled by establishing that F[·]admits a set of strong local minimizers on that can be indexed by the group P_n Zⁿ, the directsum of Artin's pure braid group on n strings and n copies ofthe infinite cyclic group. The dependence on the domain topologyis through the number of holes n in and the different mechanismsthat give rise to such local minimizers are fully exploitedby this particular representation.     

Ergodicity of a Class of Cocycles Over Irrational Rotations

It is proved that if is irrational and L²(S¹) with o(l/n)then for each mZ\{0} the corresponding skew product is ergodic. The rigidity of specialflows over irrational rotations with roof functions whose Fouriercoefficients are in o(l/n) is also shown.     

Necessary and Sufficient Conditions for Exponential Stability and Ultimate Boundedness of Systems Governed by Stochastic Partial Differential Equations

Consider the following infinite dimensional stochastic evolutionequation over some Hilbert space H with norm |·|: It is proved that under certain mild assumptions, the strongsolution X_t(x₀)VHV*, t 0, is mean square exponentially stableif and only if there exists a Lyapunov functional (·,·):HxR₊R¹ which satisfies the following conditions: (i)c₁|x|²–k₁e^–µ₁t(x,t)c₂|x|²+k₂+k₂e^–µ₂t; (ii) L(x,t)–c₃(x,t)+k₃e^–µ₃t, xV, t0; where L is the infinitesimal generator of the Markov processX_t and c_i, k_i, µ_i, i = 1, 2, 3, are positive constants.As a by-product, the characterization of exponential ultimateboundedness of the strong solution is established as the nulldecay rates (that is, µ_i = 0) are considered.     

Identity Theorems for Functions of Bounded Characteristic

Suppose that f(z) is a meromorphic function of bounded characteristicin the unit disk :|z|<1. Then we shall say that f(z)N. Itfollows (for example from [3, Lemma 6.7, p. 174 and the following])that where h₁(z), h₂(z) are holomorphic in and have positive realpart there, while ₁(z), ₂(z) are Blaschke products, that is, where p is a positive integer or zero, 0<|a_j|<1, c isa constant and (1–|a_j|)<. We note in particular that, if c0, so that f(z)0, (1.1) so that f(z)=0 only at the points a_j. Suppose now that z_j isa sequence of distinct points in such that |z_j|1 as j and (1–|z_j|)=. (1.2) If f(z_j)=0 for each j and fN, then f(z)0. N. Danikas [1] has shown that the same conclusion obtains iff(z_j)0 sufficiently rapidly as j. Let _j, _j be sequences of positivenumbers such that _j< and _j as j. Danikas then defines and proves Theorem A.     

Geometry of Critical Loci

Let :(Z,z)(U,0) be the germ of a finite (that is, proper with finite fibres)complex analytic morphism from a complex analytic normal surfaceonto an open neighbourhood U of the origin 0 in the complexplane C². Let u and v be coordinates of C² defined on U. Weshall call the triple (, u, v) the initial data. Let stand for the discriminant locus of the germ , that is,the image by of the critical locus of . Let ()_A be the branches of the discriminant locus at O whichare not the coordinate axes. For each A, we define a rational number d by where I(–, –) denotes the intersection number at0 of complex analytic curves in C². The set of rational numbersd, for A, is a finite subset D of the set of rational numbersQ. We shall call D the set of discriminantal ratios of the initialdata (, u, v). The interesting situation is when one of thetwo coordinates (u, v) is tangent to some branch of , otherwiseD = {1}. The definition of D depends not only on the choiceof the two coordinates, but also on their ordering. In this paper we prove that the set D is a topological invariantof the initial data (, u, v) (in a sense that we shall definebelow) and we give several ways to compute it. These resultsare first steps in the understanding of the geometry of thediscriminant locus. We shall also see the relation with thegeometry of the critical locus.     

Livsic Regularity Theorems for Twisted Cocycle Equations Over Hyperbolic Systems

Let be a hyperbolic map. Cocycle equations of the form f =u·g·u^–1 are considered, with f, g, u takingvalues in a compact connected Lie group G, being an automorphismof G and f, g being Hölder continuous. When the eigenvaluesof the derivative of have modulus 1, it is proved that anymeasurable solution u has a Hölder continuous version.This condition on is optimal. When f, g are C^k then u may betaken to be C^k–1+ for any (0, 1).     

On Borel Sets in Function Spaces with the Weak Topology

It is proved that the duality map ,:(, weak)x(()*, weak*)R isnot Borel. More generally, the evaluation e:(C)(K),x KR, e(f,x) = f(x), is not Borel for any function space C(K) on a compactF-space. It is also shown that a non-coincidence of norm-Boreland weak-Borel sets in a function space does not imply thatthe duality map is non-Borel.     

Compact Embeddings of Besov Spaces in Exponential Orlicz Spaces

Let 1 < p < , 0 < v < p', let be a bounded domainin Rⁿ, and denote by id the limiting compact embedding of theBesov space (Rⁿ) into the exponentialOrlicz space L_exp(t^v)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.