Abstracts of the talks at the

Ph.d. course and workshop Holomorphic Dynamics - Geometry of Julia sets in one and more complex variables

The focus of the Ph.d. course and workshop is on the geometry of Julia sets in one and more complex variables and the structure of Fatou components in two or more complex variables.

The core of the workshop wil be a mini-course by Eric Bedford and Han Peters on the structure of Fatou-components for Henon-maps.

Title : Mini-course Fatou Dynamics of Complex Hénon maps

Speakers : Eric Bedford and Han Peters
Abstract : The mini-course consists of 6 lectures. See Mini-course plan for titles and abstracts.

Title : : Infinitely many sinks for residual sets of automorphisms of low degree in $\C^{3}$
Speaker : Sébastien Biebler

Abstract : Hyperbolic systems such as Smale's horseshoe were originally supposed to be dense in the set of parameters in the 1960's. The discovery in the seventies of the so-called Newhouse's phenomenon, i.e. the existence of residual sets of diffemorphisms of compact surfaces with infinitely many sinks ( periodic attractors ) showed it was false for diffeomorphisms of surfaces. The technical core of the proof is the reduction to a line of tangency between the stable and unstable foliations where two Cantor sets must have persistent intersection.

In the complex setting, this reduction is not possible anymore and to get persistent homoclinic tangencies and then residual sets of diffeomorphisms displaying infinitely many sinks, we have to intersect in fact two Cantor sets in the plane. This was done by Buzzard in 1994 who proved there exists d > 0 such that Newhouse's phenomenon happens in the space of polynomial automorphisms of degree greater than d.

Bonatti and Diaz introduced a type of horseshoes verifying several properties they called blender horseshoe. The important property of such hyperbolic sets lies in the fractal configuration of one of their stable/unstable manifold which implies persistent intersection between any well oriented graph and this foliation. They used them to get robust homoclinic tangencies for diffeomorphisms of $\R^{3}$.

Title : Singular perturbations of Blaschke products and connectivity of Fatou components.
Speaker : Jordi Canella
Abstract : The study of singular perturbations  is a very active research field in  holomorphic dynamics. They were used by C. McMullen to show the existence of buried Julia components for rational maps. In this work a family of singular perturbations of degree 4 Blaschke products is considered.  The  free critical points of the Blaschke products lead to the emergence of new dynamic phenomena after the singular perturbation. We prove that, under certain conditions, all Fatou components of the singularly perturbed Blaschke product have finite connectivity but there are components of arbitrarily large connectivity within its dynamical plane. Under the same conditions we prove that the Julia set is the union of countably many Cantor sets of quasicircles and uncountably many point components.

Title : Periodic Points in non-Autonomous Iteration.
Speaker : Mark Comerford
Abstract : The concept of periodicity does not exist in non-autonomous iteration where one considers compositions arising from a sequence of mappings which in general is allowed to vary. Nevertheless, indifferent periodic points are extremely important for constructing examples of polynomial sequences with interesting properties. Among other examples, we describe a sequence of polynomials where all the critical points escape but which possesses bounded Fatou components and a polynomial sequence which possesses an invariant sequence of measurable line fields on the iterated Julia sets.

Title : TBA
Speaker : Tanya Firsova
Abstract : TBA

Title : Convergence properties of the Thurston algorithm for quadratic matings.
Speaker : Wolf Jung
Abstract : Mating is an operation due to Douady and Hubbard, which combines the dynamics of two complex quadratic polynomials. It gives a rational map,  whose Julia set is obtained by gluing together the polynomial Julia sets. The standard construction uses the Thurston pullback map in Teichmüller space, which has an attracting fixed point in good cases.The algorithm was  implemented by Buff and Cheritat to obtain movies of slow mating. When post-critical points of the two polynomials need to be identified during mating,  the algorithm is known to diverge in Teichmüller space, but it is shown to converge in a generalized sense. In the talk,  basic definitions of mating and (augmented) Teichmüller space will be reviewed and illustrated with a few examples. The idea of the convergence proof is explained, which is using Thurston obstructions as a tool to control the position of postcritical points.

Time permitting, the divergence for Lattes maps and/or convergence of equipotential gluing is discussed as well.The latter is an alternative definition of mating,  which is based on a holomorphic motion of polynomial Julia sets.

Title : An implosion arising from saddle connection in 2D complex dynamics.
Speaker : Shizuo Nakane
Abstract : : If a polynomial skew product on C^2 has a connection between two saddles,

numerical pictures show that fiber Julia sets oscillate and behave discontinuously.

This phenomenon quite resembles the parabolic implosion, and we explain such behaviors

by an analogous argument as parabolic implosion. This is a joint work with Hiroyuki Inou.

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