%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % 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 examples for which the Julia set has Hausdorff dimension arbitrarily close to 1. ------------------------------------------------------------------------------ Brakalova, M: On the Measurable Riemann Mapping Theorem ABSTRACT: 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 Matings ABSTRACT: 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 Polynomials ABSTRACT: We will present a technique introduced by Arnaud Cheritat which provides powerful results regarding Siegel disks of quadratic polynomials. In particular, one can prove the 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 curvature ABSTRACT: 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 sphere ABSTRACT: In any holomorphic family of rational maps, the Liapounoff exponents as a function of parameters is shown to characterize stability. We give a potential-theoretic formula for the Liapounoff exponent, and show how the homogeneous capacity in C^2 is related to the study of conformal metrics in 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 circle ABSTRACT: 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 space discontinuities ABSTRACT: A common strategy in holomorphic dynamics is to attempt to decompose a complicated dynamical system into several simpler maps, typically of lower degree. In certain cases, the inverse procedure may be implemented by a surgical construction: for example, intertwining a pair of quadratic polynomials to obtain a cubic polynomial, or mating them to obtain a quadratic rational map. Any such construction yields a map between subsets of parameter spaces. These maps are holomorphic away from bifurcation loci. For one-parameter constructions, there are known results assertion the global continuity of such maps. On the other hand, we have learned to expect discontinuity in multi-parameter settings. We present several mechanisms for producing such discontinuities. These mechanisms are quite different, but they all arise in connection with degenerate parabolic cycles. The existence of suitably degenerate degenerate parabolics is in turn a property of natural 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 (quasiconformal maps, Ahlfors-Bers Theory and the solution of the d-bar problem) to obtain a dynamic division of a holomorphic foliation into 2 pieces: One with automorphisms (Fatou component) and another one without symmetries (Julia component). Then we will classify the Fatou components and give an ergodicity result for the Julia set. Restricting to a special class of polynomial differential equations (Riccati equations) we will show that under a mild condition the statistics of all leaves is the same. We use the Poincare metric on the leaves to parametrize the leaf and we use the foliated geodesic and horocyclic flows. ------------------------------------------------------------------------------ Haissinsky: Tuning by surgery ABSTRACT: In this talk, I will provide a constructive proof of the tuning theorem which asserts the presence of small copies of the Mandelbrot set in itself. ------------------------------------------------------------------------------ Hruska, S: Hyperbolicity in the complex Henon family ABSTRACT: The Henon map, H_{a,c}(x,y)=(x^2+c-ay,x) is a widely studied family of maps with complicated dynamical behavior. Here we regard H as a holomorphic diffeomorphism of C^2 and allow a,c to be complex. Foundational work on the complex Henon family has been done by Hubbard, 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 are a class of maps which exhibit the simplest type of chaotic dynamics. A complex Henon map is called hyperbolic if it is hyperbolic over its Julia set, or equivalently, its chain recurrent set. In this talk, we will describe the algorithm and results of a rigorous computer program for testing whether for a given a,c, the complex Henon map H_{a,c} is hyperbolic. Time permitting, we will also discuss a (non-rigorous) program of Papadontanakis and Hubbard, which draws pictures illuminating the rich and subtle dynamics of the complex Henon family. ------------------------------------------------------------------------------ Kameyama, A: Coding and tiling of Julia sets for subhyperbolic rational maps ------------------------------------------------------------------------------ Kotus, J: Geometry and ergodic theory of nonrecurrent elliptic functions ABSTRACT: We explore the class of elliptic functions whose critical points in the Julia set are all nonrecurrent and whose omega-limit sets are compact subsets of the complex plane. In particular, this class contains hyperbolic, subhyperbolic and parabolic elliptic functions. Let h denote the Hausdorff dimension of the Julia set of such a function. We construct an atomless h-conformal measure m and we show that the h-dimensional Hausdorff measure of the Julia set vanishes except when the Julia set is the entire complex plane. The h-dimensional packing measure is always positive and it is finite if and only if there are no rationally indifferent periodic points. Furthemore, we prove the existence (and uniqueness, up to a multiplicative constant) of a sigma-finite f-invariant measure mu equivalent to m. This measure is then proved to be ergodic and conservative. We identify the set of those points whose open neighorhoods have infinite mu-measure, and show that infinity is not in that set. ------------------------------------------------------------------------------ Lei, T: Cui's extension of Thurston's theorem ABSTRACT: This is to characterize topologically geometrically finite rational maps. It is a powerful tool to construct access to to perturbe 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 limiting behavior of these fixed-point maps as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization converges, uniformly on the real domain, to a mapping of the limiting type, as the criticality tends to infinity along the reals. (This generalizes our result announced on the December workshop.) We prove also a straightening theorem for the limiting maps. ----------------------------------------------------------------------------- Makienko, P: Poincare series and Fatou conjecture ABSTRACT: Let R be a rational map. A critical point c is called summable if the series sum_n 1/(R^n)'(R(c)) is absolutely convergent. Under certain topological conditions on the postcritical set we prove that R can not be structurally stable if it has a summable critical point c in J(R). ------------------------------------------------------------------------------ Markovic, V: Isomorphisms of Teichmueller spaces and isometries of L^p type spaces ABSTRACT: We show that every biholomorphic map between Teichmuller spaces (of Riemann surfaces which are of non exceptional type) must be a geometric isomorphism. In particular, the group of automorphisms of the Teichmuller space of a surface of non-exceptional type agrees with the modular group of the surface. ------------------------------------------------------------------------------ Okuyama, Y: The Siegel-Cremer problem from the Nevanlinna theoretical viewpoint ABSTRACT: We study irrationally indifferent cycles of points or Jordan curves for a rational function f - such a cycle is Siegel or Cremer, by definition. We present a new argument from the viewpoint of Nevanlinna theory. Using this argument, we give a clear interpretation of a Diophantine quantity associated with an irrationally indifferent cycle. This quantity turns out to be Nevanlinna-theoretical. As a consequence, we show that an irrationally indifferent cycle is Cremer if this Nevanlinna-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 nest ABSTRACT: We present a description of admissible combinatorics for the principal nest of a quadratic polynomial; this information helps for instance, in the computation of exact moduli growth rates. As examples, we characterize complex quadratic Fibonacci maps, construct complex rotation-like maps and present a new dense autosimilarity result on the Mandelbrot set. ------------------------------------------------------------------------------ Rebelo, J: Dynamics of meromorphic vector fields and the geometry of complex surfaces ABSTRACT: Relations between the dynamics of certain meromorphic vector fields and the geometry of complex surfaces. ------------------------------------------------------------------------------ Rempe, L: Topology of Julia sets of Exponential Maps ABSTRACT: We present a universal model for the dynamics of an exponential map on its set of escaping points, which is a complete topological model for the case of attracting or parabolic parameters. In fact, we show that topologically the principle of renormalization is valid for these parameters. We also remark on some results of rigidity of escaping dynamics and existence of non-landing dynamic rays in the case where the Julia set is C. ------------------------------------------------------------------------------ Rees, M: Views of the space of quadratic rational maps ABSTRACT: The parameter space of quadratic rational maps is essentially a space of two complex dimensions. There are many natural subspaces of one complex dimension to consider. These subspaces tend to have nontrivial topology, and even in a topological sennse, there is more than one natural path from one rational map to another. When considering rational maps as dynamical systems, there is, of course, much more consider than just topological structure. I shall talk about how one views one rational map in terms of another, probably with particular reference 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 at least 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 algebraic differential equations on C^2. A differential equation dw/dz=P_n(z,w)/Q_n(z,w) (where P_n and Q_n are polynomials of degree at most n) can be extended to CP^2 as a holomorphic foliation with singularities. The class of such equations is denoted by A_n. A generic foliation from A_n has the line at infinity as a leaf. This leaf has a non-trivial fundamental group. The corresponding holonomy transformation 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 set of images of z under the representatives of germs from this group, for all germ having representatives defined at z. If a group of germs of conformal mappings (C,0)->(C,0) is nonsolvable then: 1) Its orbits are dense in sectors. That is, there is a finite set of real analytic curves passing through 0 such that orbits are dense in domains bounded by these curves. 2) The group is topologically rigid. That is, any homeomorphism which conjugates it with another such group is holomorphic or antiholomorphic. 3) There is a countable set of germs whose representatives have isolated fixed points away from 0. Foliations in A_n have corresponding properties. More precisely, there exists a real algebraic subset of the space A_n such that for any equation from 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 topologically conjugate to some other foliation by a homeomorphism sufficiently close to the identity then it is affinely equivalent to this foliation. 3) There exists a countable set of homologically independent complex limit cycles. There are other generic properties: a generic equation from A_n has no cycle on the infinity leaf with identity holonomy map, and any leaf for a generic equation is hyperbolic. ------------------------------------------------------------------------------ Shen, W: Density of Axiom A in the space of real polynomials ABSTRACT: 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 a rigidity approach. ------------------------------------------------------------------------------ Singh, A: Transcendental entire functions whose Julia set is the complex plane ABSTRACT: We give a proceedure for constructing a new class of entire function whose Julia set is the whole complex plane. We utilize the properties of Schwarzian derivative and consider the critical and asymptotic values of the function. These functions can be constructed to be composite and non-periodic in nature. ------------------------------------------------------------------------------ Stallard, G: Dimensions of Julia sets ABSTRACT: 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 functions ABSTRACT: Let q be the maximal order of all poles of an elliptic function 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 exponential family (joint with Mariusz Urbanski) ABSTRACT: For a large class of maps a*exp(z) we study the dimension of some natural, dynamically defined essential subset of a Julia set (It can be understood as an analogue of a conical limit set). A conformal measure supported on this special set is built and its ergodic properties are studied. We also introduce a corresponding thermodynamical formalism. Using this tool, we study the dependence of the above dimension on the parameter a. ------------------------------------------------------------------------------