Abstract: In 2009, H. Bruin and S. van Strien proved that the level sets of constant topological entropy of real polynomial maps of given degree are connected, solving a 20 year old conjecture of Milnor. Their proof is quite complicated. I will try to show how to prove Milnor's conjecture in a much simpler way.
Weixiao Shen: On Viana maps based on quadratic polynomials
Davoud Cheraghi: Universality of the Mandelbrot set
Abstract: We discuss some recent results on the analytic properties of a renormalization operator on an infinite dimensional space of holomorphic maps. We show how the small scale geometry of the Mandelbrot set emerges out of the dynamics of this operator. In particular, we establish the local connectivity of the Mandelbrot set at certain parameters.
Lasse Rempe-Gillen: Topology of Julia sets of hyperbolic entire functions
Abstract: For hyperbolic rational maps (i.e., those rational maps with the simplest type of dynamical behaviour), the Julia set is always locally connected, and its structure is essentially well-understood. On the other hand, in the transcendental entire case, even the Julia sets of hyperbolic functions can have very complicated topological structure. Indeed, with Rottenfußer, R¨uckert and Schleicher, we proved that there is a hyperbolic entire function with connected Fatou set whose Julia set contains no arcs, answering questions of Fatou from 1926 and Eremenko from 1989 in the negative. I will describe more recent work that gives a much more precise description of the possible topology of the connected components of Julia sets of hyperbolic entire functions with connected Fatou set. In particular, there exists such a function for which the Julia set is an uncountable union of pseudo-arcs (a certain type of hereditarily indecomposable plane continuum).