%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%		ABSTRACTS FOR THE HOLOMORPHIC DYNAMICS WORKSHOP%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%------------------------------------------------------------------------------Avila, A: Hausdorff dimension of certain infinitely renormalizable Julia sets(joint with Mikhail Lyubich)We consider real quadratic polynomials which are infinitely renormalizable of constant type.  McMullen has asked if the corresponding Julia sets have always Hausdorff dimension two. We show that this is not the case and give examplesfor which the Julia set has Hausdorff dimension arbitrarily close to 1.------------------------------------------------------------------------------Brakalova, M: On the Measurable Riemann Mapping TheoremABSTRACT: I'll discuss some recent developments and extensions of David's results on the existence and uniqueness of solutions to the Beltrami Equation with ||\mu||_{\infty}=1.------------------------------------------------------------------------------Bullett, S: Dynamics of Hecke Groups, Chebyshev Polynomials and MatingsABSTRACT: We show how Chebyshev polynomials arise in two quite different ways in the study of matings between Hecke groups and polynomials (holomorphic correspondences which are conjugate to Hecke groups on one part of the Riemann sphere and conjugate to a polynomial on the complement). As a consequence we obtain a description of the connectivity locus of the one parameter family of scalar multiples of the nth Chebyshev polynomial, in the case that n is odd.------------------------------------------------------------------------------Buff, X: Siegel disks of Quadratic PolynomialsABSTRACT: We will present a technique introduced by Arnaud Cheritat which provides powerful results regarding Siegeldisks of quadratic polynomials. In particular, one can provethe existence of quadratic polynomials having a Siegel disk with smooth boundary. One can also obtain good estimates for the conformal radius of quadratic Siegel disks. ------------------------------------------------------------------------------Deroin, B: Levi-flat hypersurfaces in complex surfaces of positive curvatureABSTRACT: Given a harmonic measure on a Riemann surface foliation of a closed 3-dimensional manifold, we define its "normal class",which measures how two leaves of the foliation are converging to each other, using the brownian motion along the leaves. We bound this normal class and we give applications to Levi-flat hypersurfaces in complex surfaces of positive curvature.  -----------------------------------------------------------------------------De Marco, L: Stability, Lyapunov exponents and metrics on the sphereABSTRACT: In any holomorphic family of rational maps, the Liapounoff exponentsas a function of parameters is shown to characterize stability. We give a potential-theoretic formula for the Liapounoff exponent, and show howthe homogeneous capacity in C^2 is related to the study of conformal metricsin the Riemann spehere. ----------------------------------------------------------------------------Dominguez, P: Dynamics of the sine family(joint with Guillermo Sienra)-----------------------------------------------------------------------------Earle, C: Holomorphic contractibility of the normalized symmetric homeomorphisms of the circleABSTRACT: Gardiner and Sullivan showed that the space of normalized symmetric homeomorphisms of an oriented circle has a natural complex structure making it a complex Banach manifold. We explain how to contract it to a point in such a way that for each t in [0,1] the map f_t of the space into itself is holomorphic. Our construction does not generalize to the space of normalized quasisymmetric homeomorphisms.-----------------------------------------------------------------------------Epstein, A: Degenerate parabolic points and parameter spacediscontinuitiesABSTRACT: A common strategy in holomorphic dynamics is to attempt todecompose a complicated dynamical system into several simpler maps,typically of lower degree. In certain cases, the inverse procedure may beimplemented by a surgical construction: for example, intertwining a pairof quadratic polynomials to obtain a cubic polynomial, or mating them toobtain a quadratic rational map. Any such construction yields a mapbetween subsets of parameter spaces. These maps are holomorphic away from bifurcation loci. For one-parameter constructions, there areknown results assertion the global continuity of such maps. Onthe other hand, we have learned to expect discontinuity inmulti-parameter settings. We present several mechanisms for producingsuch discontinuities. These mechanisms are quite different, but they allarise in connection with degenerate parabolic cycles. The existence ofsuitably degenerate degenerate parabolics is in turn a property ofnatural multi-parameter families.-----------------------------------------------------------------------------Geyer, L: Linearizability of irrationally indifferent fixed points  ABSTRACT: We present partial results supporting the conjecture that there are no "exotic" Siegel discs, i.e. with non-Brjuno rotation numbers, for polynomials and rational functions. Special attention will be given to the case of cubic polynomials, where the critical part of parameter space is amenable to (very limited) computer experiments.------------------------------------------------------------------------------Gomez-Mont, X: On the geometry and dynamics of holomorphic flows(joint with  E. Ghys, J. Saludes; Ch. Bonatti, M. Viana and R. Vila)ABSTRACT: We will explain how to apply Teichmuller Theory (quasiconformalmaps, Ahlfors-Bers Theory and the solution of the d-bar problem) to obtaina dynamic division of a holomorphic foliation into 2 pieces: One withautomorphisms (Fatou component) and another one without symmetries (Juliacomponent). Then we will classify the Fatou components and give anergodicity result for the Julia set.Restricting to a special class of polynomial differential equations(Riccati equations) we will show that under a mild condition thestatistics of all leaves is the same. We use the Poincare metric on theleaves to parametrize the leaf and we use the foliated geodesic andhorocyclic flows.------------------------------------------------------------------------------Haissinsky: Tuning by surgery   ABSTRACT: In this talk, I will provide a constructive proof of thetuning theorem which asserts the presence of small copies of theMandelbrot set in itself.------------------------------------------------------------------------------Hruska, S: Hyperbolicity in the complex Henon familyABSTRACT: The Henon map, H_{a,c}(x,y)=(x^2+c-ay,x) is a widely studiedfamily of maps with complicated dynamical behavior. Here weregard H as a holomorphic diffeomorphism of C^2 and allow a,c to becomplex. Foundational work on the complex Henon family has been done byHubbard, Bedford and Smillie, Fornaess and Sibony, and others; however,basic questions remain unanswered. A good first step would be to understand hyperbolic Henon maps, which area class of maps which exhibit the simplest type of chaotic dynamics. Acomplex Henon map is called hyperbolic if it is hyperbolic over its Juliaset, or equivalently, its chain recurrent set.In this talk, we will describe the algorithm and results of a rigorouscomputer program for testing whether for a given a,c, the complexHenon map H_{a,c} is hyperbolic.  Time permitting, we will alsodiscuss a (non-rigorous) program of Papadontanakis and Hubbard, whichdraws pictures illuminating the rich and subtle dynamics of the complexHenon family.------------------------------------------------------------------------------Kameyama, A: Coding and tiling of Julia sets for subhyperbolic rational maps------------------------------------------------------------------------------Kotus, J: Geometry and ergodic theory of nonrecurrent elliptic functionsABSTRACT: We explore the class of elliptic functions whose critical pointsin the Julia set are all nonrecurrent and whose omega-limit sets arecompact subsets of the complex plane. In particular, this class containshyperbolic, subhyperbolic and parabolic elliptic functions. Let h denotethe Hausdorff dimension of the Julia set of such a function. We constructan atomless h-conformal measure m and we show that the h-dimensionalHausdorff measure of the Julia set vanishes except when the Julia set isthe entire complex plane. The h-dimensional packing measure is alwayspositive and it is finite if and only if there are no rationallyindifferent periodic points. Furthemore, we prove the existence (anduniqueness, up to a multiplicative constant) of a sigma-finitef-invariant measure mu equivalent to m. This measure is then proved to beergodic and conservative. We identify the set of those points whose openneighorhoods have infinite mu-measure, and show that infinity is not inthat set. ------------------------------------------------------------------------------Lei, T: Cui's extension of Thurston's theoremABSTRACT: This is to characterize topologically geometricallyfinite rational maps. It is a powerful tool to construct access to toperturbe maps with parabolic points.------------------------------------------------------------------------------Levin, G: Universality and dynamics of unimodal maps with infinite criticality(joint with Greg Swiatek)ABSTRACT: The universality in one-dimensional dynamics is described by fixed points of renormalization operators. We study the limitingbehavior of these fixed-point maps as the order of the critical point increases to infinity. Itis shown that a limiting dynamics exists, with a critical point that isflat, but still having a well-behaved analytic continuation to aneighborhood of the real interval pinched at the critical point. We studythe dynamics of limiting maps and prove their rigidity. In particular, thesequence of fixed points of renormalization converges, uniformly on thereal domain, to a mapping of the limiting type, as the criticalitytends to infinity along the reals. (This generalizes our result announcedon the December workshop.) We prove also a straightening theorem for thelimiting maps. -----------------------------------------------------------------------------Makienko, P: Poincare series and Fatou conjectureABSTRACT: Let R be a rational map. A critical point c is called summableif the series sum_n 1/(R^n)'(R(c)) is absolutelyconvergent. Under certain topological conditions on the postcritical setwe prove that R can not be structurally stable if it has a summablecritical point c in J(R).  ------------------------------------------------------------------------------Markovic, V: Isomorphisms of Teichmueller spaces and isometries of L^ptype spacesABSTRACT: We show that every biholomorphic map between Teichmuller spaces(of Riemann surfaces which are of non exceptional type) must be ageometric isomorphism. In particular, the group of automorphisms of theTeichmuller space of a surface of non-exceptional type agrees with themodular group of the surface. ------------------------------------------------------------------------------Okuyama, Y: The Siegel-Cremer problem from the Nevanlinna theoretical viewpointABSTRACT: We study irrationally indifferent cycles of points or Jordancurves for a rational function f - such a cycle is Siegel or Cremer, bydefinition. We present a new argument from the viewpoint of Nevanlinnatheory. Using this argument, we give a clear interpretation of aDiophantine quantity associated with an irrationally indifferent cycle.This quantity turns out to be Nevanlinna-theoretical. As a consequence, weshow that an irrationally indifferent cycle is Cremer if thisNevanlinna-theoretical quantitiy does not vanish.------------------------------------------------------------------------------Oudkerk, R: The parabolic implosion and convergence to Lavaurs maps ABSTRACT: We are interested in a convergent sequence of rational maps f_n->f_0 where f_0 has a parabolic cycle.  It can be shown that either there is no "parabolic implosion" or else we can pass to a subsequence such that we have f_n -> (f_0,g) for some Lavaurs map g of f_0. This means that in some way the "liminf" of the dynamical systems is a semi-group whose generators include both f_0 and g, and that        "liminf J(f_n) \supseteq J(g) \supsetneq J(f_0)"------------------------------------------------------------------------------Penrose, C: Regular and limit sets for holomorphic correspondences------------------------------------------------------------------------------Perez, R: Geometry and combinatorics of Lyubich's principal nestABSTRACT: We present a description of admissible combinatorics for theprincipal nest of a quadratic polynomial; this information helps forinstance, in the computation of exact moduli growth rates. As examples, wecharacterize complex quadratic Fibonacci maps, construct complexrotation-like maps and present a new dense autosimilarity result on theMandelbrot set.------------------------------------------------------------------------------Rebelo, J: Dynamics of meromorphic vector fields and the geometry of complex surfacesABSTRACT: Relations between the dynamics of certain meromorphic vector fields and the geometry of complex surfaces.------------------------------------------------------------------------------Rempe, L: Topology of Julia sets of Exponential MapsABSTRACT:  We present a universal model for the dynamics of an exponential mapon its set of escaping points, which is a complete topological modelfor the case of attracting or parabolic parameters. In fact, we show thattopologically the principle of renormalization is valid for these parameters.We also remark on some results of rigidity of escaping dynamics and existenceof non-landing dynamic rays in the case where the Julia set is C. ------------------------------------------------------------------------------Rees, M: Views of the space of quadratic rational mapsABSTRACT: The parameter space of quadratic rational maps is essentially a space of two complex dimensions. There are many natural subspaces ofone complex dimension to consider. These subspaces tend to havenontrivial topology, and even in a topological sennse, there is more thanone natural path from one rational map to another. When considering  rational maps as dynamical systems, there is, of course, much moreconsider than just topological structure. I shall talk about how oneviews one rational map in terms of another, probably with particularreference to matings of polynomials.------------------------------------------------------------------------------Rippon, P: On a question of Fatou(joint with Gwyneth Stallard) ABSTRACT: Let f be a transcendental entire function and let I(f) be the set of points whose iterates tend to infinity. We show that I(f) has atleast one unbounded component. In the case that f has a Baker wandering domain, we show that I(f) is a connected unbounded set.------------------------------------------------------------------------------Shcherbakov, A: Generic properties of foliations determined by algebraicdifferential equations on C^2.A differential equation dw/dz=P_n(z,w)/Q_n(z,w) (where P_n and Q_n arepolynomials of degree at most n) can be extended to CP^2 as a holomorphicfoliation with singularities. The class of such equations is denoted byA_n. A generic foliation from A_n has the line at infinity as a leaf. Thisleaf has a non-trivial fundamental group. The corresponding holonomytransformation group consists of germs of conforal mappings (C,0)->(C,0).The orbit of a point z under the action of this holonomy group is the setof images of z under the representatives of germs from this group, for allgerm having representatives defined at z. If a group of germs of conformal mappings (C,0)->(C,0) is nonsolvablethen:1) Its orbits are dense in sectors. That is, there is a finite set ofreal analytic curves passing through 0 such that orbits are dense indomains bounded by these curves.2) The group is topologically rigid. That is, any homeomorphism whichconjugates it with another such group is holomorphic or antiholomorphic. 3) There is a countable set of germs whose representatives have isolatedfixed points away from 0.Foliations in A_n have corresponding properties. More precisely, thereexists a real algebraic subset of the space A_n such that for any equationfrom the complement:1) Any leaf, other than the leaf at infinity, is dense in C^n.2) The foliation is absolutely rigid. That is, if it is topologicallyconjugate to some other foliation by a homeomorphism sufficiently close tothe identity then it is affinely equivalent to this foliation.3) There exists a countable set of homologically independent complex limitcycles.There are other generic properties: a generic equation from A_n has nocycle on the infinity leaf with identity holonomy map, and any leaf for ageneric equation is hyperbolic.     ------------------------------------------------------------------------------Shen, W: Density of Axiom A in the space of real polynomialsABSTRACT: In this joint work with Sebastian van Strien and Oleg Kozlovski, we prove that for any d>=2, Axiom A maps are dense in the space of real polynomials with real critical points of degree d, through arigidity approach.  ------------------------------------------------------------------------------Singh, A: Transcendental entire functions whose Julia set is the complexplaneABSTRACT: We give a proceedure for constructing a new class of entirefunction whose Julia set is the whole complex plane. We utilize theproperties of Schwarzian derivative and consider the critical andasymptotic values of the function. These functions can be constructed tobe composite and non-periodic in nature.------------------------------------------------------------------------------Stallard, G: Dimensions of Julia setsABSTRACT: We give examples of hyperbolic meromorphic and entire functions for which the Julia sets have different Hausdorff and box dimensions.------------------------------------------------------------------------------Urbanski, M: Fractal properties and ergodic theory of elliptic functionsABSTRACT: Let q be the maximal order of all poles of an ellipticfunction f. We will discuss the following results:  1. HD(J(f)) is greater than 2q/(q+1).  2. Hausdorff dimension of points escaping to infinity is at most      2q/(q+1).  We now consider the function f as mapping the torus T-f^{-1}(\infty) onto T. Given a potential \phi, Holder continuous far from poles and of the form u(z)+gamma\log|z-b|, gamma>2, u Holder continuous, near a pole b, we define a pointwise pressure P(phi) and the corresponding transfer operator L. Assuming that P(phi)> sup(phi) we will discuss the following results:  1. P(phi)=log lambda\$, where lambda is a positive eigenvalue of L.  2. There exists a unique real parameter c and a unique probability measure m on J(f) such that (dm\circ f)/dm=exp(c-phi). This constant c is equal to P(phi).  3. There is a unique f-invariant probability measure mu absolutely continuous with respect to m. The dynamical system (f,mu) is metrically exact, in particular mixing of all orders.  4. The transfer operator L acting on C(T) is almost periodic.----------------------------------------------------------------------------Zdunik, A: Conformal measures and Hausdorff measures for the exponentialfamily(joint with Mariusz Urbanski)ABSTRACT: For a large class  of maps a*exp(z) we study the dimension ofsome natural, dynamically defined essential subset of a Julia set (It canbe understood as an analogue of a conical limit set). A conformal measuresupported on this special set is built and its ergodic properties arestudied. We also introduce a corresponding thermodynamical formalism.Using this tool, we study the dependence of the above dimension on theparameter a.------------------------------------------------------------------------------