Corbes invariants en sistemes forçats quasi-periòdicament

Abstract

[en] In this work, we prove the inverse function and implicit function theorems in functions between Banach spaces, firstly defining the concepts needed to prove the theorems. This allows us to study the continuity w.r.t. parameters of the invariant curves in a particular type of dynamical system. A quasi-periodically forced one-dimensional dynamical system is a system that has neither fixed points nor periodic points. The simplest invariant objects that can be found are curves. We prove that a sufficient condition for the continuation of an invariant curve w.r.t. a parameter is the attraction of this curve. Attraction condition is deduced by the negative sign of the Lyapunov exponent of the curve, which provides us with information about the transfer operator spectrum. We also study the linear behavior of the curves and the conditions needed to simplify their behavior.