The concept of a complex box mapping (or puzzle mapping) is a generalization of the classical notion of polynomial-like map to the case when one allows for countably many components in the domain and finitely many components in the range of the mapping. In one-dimensional dynamics, box mappings appear naturally as first return maps to certain nice sets, and hence one arrives at a notion of box renormalization. We say that a rational map is box renormalizable if the first return map to a well-chosen neighborhood of the set of critical points (intersecting the Julia set) has a structure of a box mapping. In our talk, we will discuss various features of general box mappings, as well as so-called dynamically natural box mappings, focusing on their rigidity properties. We will then show how these results can be used almost as 'black boxes' to conclude similar rigidity properties for box renormalizable rational maps. We will give several examples to illustrate this procedure, these examples include, most prominently, complex polynomials of arbitrary degree and their Newton maps. (The talk is based on joint work with Trevor Clark, Oleg Kozlovski, Dierk Schleicher and Sebastian van Strien.)[-]

In a recently completed paper Pascale Roesch and I have given a complete proof that the connectedness locus $M_{1}$ in the space moduli space of quadratic rational maps with a parabolic fixed point of multiplier 1 is homeomorphic to the Mandelbrot set. In this talk I will outline and discus the proof, which in an essential way involves puzzles and a theorem on local connectivity of $M_{1}$ at any parameter which is neither renormalizable nor has all fixed points non-repelling similar to Yoccoz celebrated theorem for local connectivity of $M$ at corresponding parameters.[-]

In this talk we present bifurcation phenomena in natural families of rational or (transcendental) meromorphic functions of finite type $\left\{f_{\lambda}:=\varphi_{\lambda} \circ f_{\lambda_{0}} \circ \psi_{\lambda}^{-1}\right\}_{\lambda \in M}$, where $M$ is a complex connected manifold, $\lambda_{0} \in M, f_{\lambda_{0}}$ is a meromorphic map and $\varphi_{\lambda}$ and $\psi_{\lambda}$ are families of quasiconformal homeomorphisms depending holomorphically on $\lambda$ and with $\psi_{\lambda}(\infty)=\infty$. There are fundamental differences compared to the rational or entire setting due to the presence of poles and therefore of parameters for which singular values are eventually mapped to infinity (singular parameters). Under mild conditions we show that singular (asymptotic) parameters are the endpoint of a curve of parameters for which an attracting cycle progressively exits the domain, while its multiplier tends to zero, proving a conjecture from [Fagella, Keen, 2019]. We also present the connections between cycles exiting the domain, singular parameters, activity of singular orbits and $\mathcal{J}$-unstability, converging to a theorem in the spirit of Mañé-Sad-Sullivan's celebrated result.[-]