Josias Reppekus

To understand the local behaviour of a dynamical system near a fixed point, a first step are local changes of coordinates that result in a simpler form of the system. If the system in question is an iterated holomorphic function, we can represent all of the involved mappings as convergent power series. Problems such as full linearisation or flattening of stable manifolds can then be split into two parts: 1. The formal solution of an equation of power series. 2. The convergence of the formal solution. The expression of such a formal solution naturally contains a series of divisors, that form an obstruction to general convergence when they are very small, making it a small divisor problem. These divisors depend only on the linear part of the map at the fixed point, hence one can formulate sufficient conditions for convergence purely in terms of this linear part. This talk will cover a series of simplification results under Brjuno type conditions on eigenvalues at fixed points, starting with the foundational linearisation results by Siegel and Brjuno, and continuing with more general simplification results by Pöschel and myself, that will enable the complete classifi- cation of the stable dynamics near certain so-called “one-resonant” fixed points under a Brjuno type condition.