Mixing in the Absence of the Shrinking Target Property

Using reparametrizations of linear flows, we show that thereexist area-preserving real analytic maps of the three-dimensionaltorus that are ‘mixing of all orders’ and do notenjoy the monotone shrinking target property. Prior to that,we give a short proof of a result of Kurzweil from 1955: namely,that a translation T of the torus T^d has the monotone shrinkingtarget property if and only if the vector is badly approximable(that is, of constant type). 2000 Mathematics Subject Classification37E45, 37A25, 11J13. 2000 Mathematics Subject Classification37E45, 37A25, 11J13.     

Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.     

Recurrence, Dimension and Entropy

Let (_A, T) be a topologically mixing subshift of finite typeon an alphabet consisting of m symbols and let :_A R^d be a continuousfunction. Denote by (x) the ergodic limit when the limit exists. Possible ergodic limits arejust mean values dµ for all T-invariant measures. Forany possible ergodic limit , the following variational formulais proved: where h_µ denotes the entropy of µ and h_top denotestopological entropy. It is also proved that unless all pointshave the same ergodic limit, then the set of points whose ergodiclimit does not exist has the same topological entropy as thewhole space _A     

DIFFERENCES OF ERGODIC AVERAGES FOR CESARO BOUNDED OPERATORS

We prove that the weighted differences of ergodic averages,induced by a Cesàro bounded, strongly continuous, one-parametergroup of positive, invertible, linear operators on L^p, 1 <p < , converge almost every where and in the L^p-norm. Weobtain first the boundedness of the ergodic maximal operatorand the convergence of the averages.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

Harmonic morphisms on heaven spaces

We prove that any (real or complex) analytic horizontally conformalsubmersion from a three-dimensional conformal manifold (M³,c_M) to a two-dimensional conformal manifold (N², c_N) can be,locally, ‘extended’ to a unique harmonic morphismfrom the (eaven)-space (H⁴, g) of (M³, c_N) to (N², c_N). Moreover,any positive harmonic morphism with two-dimensional fibres from(H⁴, g) is obtained in this way.     

Singularities and Limit Functions in Iteration of Meromorphic Functions

Let f(z) be a transcendental meromorphic function. The paperinvestigates, using the hyperbolic metric, the relation betweenthe forward orbit P(f) of singularities of f^–1 and limitfunctions of iterations of f in its Fatou components. It ismainly proved, among other things, that for a wandering domainU, all the limit functions of {fⁿ|_U} lie in the derived setof P(f) and that if f^np|_V q(n +) for a Fatou component V, theneither q is in the derived set of S_p (f) or f^p(q) = q. As applicationsof main theorems, some sufficient conditions of the non-existenceof wandering domains and Baker domains are given.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient

We analyse approximate solutions generated by an upwind differencescheme (of Engquist–Osher type) for nonlinear degenerateparabolic convection–diffusion equations where the nonlinearconvective flux function has a discontinuous coefficient (x)and the diffusion function A(u) is allowed to be strongly degenerate(the pure hyperbolic case is included in our setup). The mainproblem is obtaining a uniform bound on the total variationof the difference approximation u, which is a manifestationof resonance. To circumvent this analytical problem, we constructa singular mapping (, ·) such that the total variationof the transformed variable z = (, u) can be bounded uniformlyin . This establishes strong L¹ compactness of z and, since(, ·) is invertible, also u. Our singular mapping isnovel in that it incorporates a contribution from the diffusionfunction A(u). We then show that the limit of a converging sequenceof difference approximations is a weak solution as well as satisfyinga Krukov-type entropy inequality. We prove that the diffusionfunction A(u) is Hölder continuous, implying that the constructedweak solution u is continuous in those regions where the diffusionis nondegenerate. Finally, some numerical experiments are presentedand discussed.     

Simplices of Maximal Volume or Minimal Total Edge Length in Hyperbolic Space

In this paper, we are mainly concerned with n-dimensional simplicesin hyperbolic space Hⁿ. We will also consider simplices withideal vertices, and we suggest that the reader keeps the Poincaréunit ball model of hyperbolic space in mind, in which the sphereat infinity Hⁿ() corresponds to the bounding sphere of radius1. It is known that all hyperbolic simplices (even the idealones) have finite volume. However, explicit calculation of theirvolume is generally a very difficult problem (see, for example,[1] or [16]). Our first theorem states that, amongst all simplicesin a closed geodesic ball, the simplex of maximal volume isregular. We call a simplex regular if every permutation of itsvertices can be realized by an isometry of Hⁿ. A correspondingresult for simplices in the sphere has been proved by Böröczky[4].     

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.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

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.     

Dimension theory of iterated function systems

Let {S_i} be an iterated function system (IFS) on ?^d with attractor K. Let (Σ, σ) denote the one‐sided full shift over the alphabet {1, …, ??}. We define the projection entropy function h_π on the space of invariant measures on Σ associated with the coding map π : Σ → K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under π is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures. © 2008 Wiley Periodicals, Inc.     

A Remark About the Lie Algebra of Infinitesimal Conformal Transformations of the Euclidean Space

Infinitesimal conformal transformations of Rⁿ are always polynomialand finitely generated when n > 2. Here we prove that theLie algebra of infinitesimal conformal polynomial transformationsover Rⁿ, n 2, is maximal in the Lie algebra of polynomial vectorfields. When n is greater than 2 and p, q are such that p +q = n, this implies the maximality of an embedding of so(p +1, q + 1, R) into polynomial vector fields that was revisitedin recent works about equivariant quantizations. It also refinesa similar but weaker theorem by V. I. Ogievetsky. 1991 MathematicsSubject Classification 17B66, 53A30.     

Statistical properties of topological Collet-Eckmann maps

We study geometric and statistical properties of complex rational maps satisfying a non-uniform hyperbolicity condition called “Topological Collet-Eckmann”. This condition is weaker than the “Collet-Eckmann” condition. We show that every such map possesses a unique conformal probability measure of minimal exponent, and that this measure is non-atomic, ergodic, and that its Hausdorff dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we show that there is a unique invariant probability measure that is absolutely continuous with respect to this conformal measure, and that this invariant measure is exponentially mixing (it has exponential decay of correlations) and satisfies the Central Limit Theorem.We also show that for a complex rational map the existence of such invariant measure characterizes the Topological Collet-Eckmann condition: a rational map satisfies the Topological Collet-Eckmann condition if, and only if, it possesses an exponentially mixing invariant measure that is absolutely continuous with respect to some conformal measure, and whose topological support contains at least 2 points.     

On full groups of measure-preserving and ergodic transformations with uncountable cofinalities

The group of all measure-preserving permutations of the unitinterval and the full group of an ergodic transformation ofthe unit interval are shown to have uncountable cofinality andthe Bergman property. Here, a group G is said to have the Bergmanproperty if, for any generating subset E of G, some boundedpower of EE^–1{1} already covers G. This property arosein a recent interesting paper of Bergman, where it was derivedfor the infinite symmetric groups. We give a general sufficientcriterion for groups G to have the Bergman property. We showthat the criterion applies to a range of other groups, includingsufficiently transitive groups of measure-preserving, non-singular,or ergodic transformations of the reals; it also applies tolarge groups of homeomorphisms of the rationals, the irrationals,or the Cantor set.     

The fcc lattice and the cusped hyperbolic 4-orbifold of minimal volume: In memoriam H. S. M. Coxeter

The 1-cusped hyperbolic coset space of H⁴ by the Coxeter group[4, 3^2,1] of volume ²/1440 is the unique minimal volume orbifoldamong all non-compact complete hyperbolic 4-orbifolds. Our proofis geometric and based on horoball geometry combined with Gauss'scharacterization of the face centered cubic lattice packingas the densest one in euclidean 3-space.     

Invariant Subsets and Invariant Measures for Irreducible Actions on Zero-Dimensional Groups

Algebraic higher-rank actions on connected groups are oftenremarkably rigid in their topological and measurable structure.In contrast to this, the author of this paper constructs uncountablymany closed invariant sets and uncountably many invariant measureswith positive entropy for irreducible algebraic Z^d-actions onzero-dimensional groups. 2000 Mathematics Subject Classification37A45 (primary), 37B50, 37A35, 37B40 (secondary).