Långsam rekurrens och svagt expanderande funktioner

Given a function f defined on the complex plane (or the Riemann sphere or other complex manifolds) we say that the nth iterate of f is the new function formed by composing f with itself n times.

A fundamental question in dynamical systems is to understand the behaviour of the iterates of f as the number of iterates goes to infinity. Also, one is interested in what kind of dynamics is ´´generic´´ in some parameter space. What happens with the dynamics if one perturbs f to a nearby function g?

A famous conjecture, called the Fatou conjecture, states that there is an open and dense set of functions that are ´´nice´´ in the sense that they have uniformly expanding properties, meaning roughly that nearby points repel each other under iteration. Moreover, maps where the critical set is recurrent are generic.

Conjecturally, in many spaces of maps considered, for almost all maps, the critical set is even slowly recurrent. Hence expanding and critically recurrent maps are very interesting and will be a main study object of this project.

Basically we want to understand the perturbation properties of these maps, where we change the rate of recurrence and expansion.

If we let the these dynamical constants to approach certain critical values, we reach a map with entropy zero, addressing the famous Fatou conjecture.

We also aim to prove that ´´typical´´ parameters (slightly stronger than the slow recurrence condition) are abundant.