10:30 Jernej Činč (ICTP, Italy and University of Maribor, Slovenia)
Title: Lebesgue measure-preserving maps on one-dimensional manifolds
Abstract: In this talk I will survey recent advances in the study of Lebesgue measure-preserving maps on one-dimensional compact connected manifolds with particular emphasis on the circle case. If time permits I will also argue that there exists an open dense set of Lebesgue measure-preserving circle maps which satisfy a very strong topological expansion property. The talk is based on joint works with Jozef Bobok (CVUT Prague), Serge Troubetzkoy (Aix-Marseille) and Piotr Oprocha (AGH Krakow & IRAFM Ostrava).
11:30 Siniša Miličić (University of Pula, Croatia)
Title: Box dimensions of stuffed sets
Abstract: This seminar delves into the behaviour of box (Minkowski-Bouligand) dimensions of stuffed sets - sets with locally stable dimension and a compact core part with a dimensional jump (spirals, chirps etc.). We define stuffed sets and analyse how that property drives the box dimension, with the ultimate aim of justifying intuitive shortcuts in computing dimensions.
Afternoon (14:00 - 16:00) - Venue: Luigi Stasi seminar room (Leonardo Da Vinci Building)
14:00 Shaun Bullett (Queen Mary University of London, UK)
Title: Dynamics of holomorphic correspondences I: a guided tour of a family of examples
15:00 Luna Lomonaco (IMPA, Rio de Janeiro, Brazil)
Title: Dynamics of holomorphic correspondences II: matings between quadratic maps and the modular group
Joint Abstract for the above two talks:
A holomorphic correspondence on the Riemann sphere is a multivalued map z->w defined by a polynomial relation P(z,w)=0. This definition generalises those of a rational map and of a finitely generated Kleinian group, putting them into a common framework. An iterated correspondence may simultaneously exhibit the behaviour of a rational map of one part of the Riemann sphere and of a Kleinian group on another, in which case we describe it as a mating between the map and the group.
The first talk will describe the dynamical behaviour of the simplest 2-parameter family of holomorphic correspondences. With Christopher Penrose the speaker identified certain members of this family as matings between quadratic polynomials and the modular group, and in their 1994 Inventiones paper these authors made a series of conjectures concerning the family of such matings. The second speaker introduced a new tool to the problem: the theory of parabolic-like mappings. In her talk she will give an overview of how she and the first speaker have together resolved all the 1994 conjectures.