Starting from a presentation of the many recent applications of Galois theory of functional equations to enumerative combinatorics, we will introduce the Galois theory of (different kinds) of difference equations. We will focus on the point of view of the applications, hence with little emphasis on the technicalities of the domain, but I'm willing to do an hour of « exercises » (i.e. to go a little deeper into the proofs), if a part of the audience is interested.[-]

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.[-]