Sistemi dinamici discreti olomorfi locali

Abate, Marco

La Matematica nella società e nella cultura, Série 1, Tome 1 (2008) no. 3, pp. 409-441.

Riassunto (VO)

La teoria dei sistemi dinamici si distingue da altri settori della matematica non per gli oggetti che studia ma per le domande che si pone su di loro. Per esempio, un sistema dinamico discreto è semplicemente un'applicazione (misurabile, continua, differenziable, olomorfa...) di uno spazio in sé. Studiare un'applicazione $f$ dal punto di vista dinamico significa allora studiare il comportamento qualitativo delle iterate $f^{k} = f \circ f \circ \cdots \circ f$ al tendere di $k$ all'infinito. In questo articolo vogliamo dare un'idea del tipo di questioni che si affrontano in dinamica restringendoci a un argomento limitato ma importante, la dinamica discreta olomorfa locale, che studia il comportamento dinamico di applicazioni olomorfe definite nell'intorno di un punto fisso. Nata alla fine dell'ottocento, più o meno in contemporanea con l'intero campo dei sistemi dinamici, ha avuto un grosso sviluppo negli ultimi trent'anni, con la dimostrazione di importanti risultati e lo sviluppo di nuove significative tematiche e naturali problemi aperti. Ne presenteremo le problematiche di base e i principali risultati ottenuti, evidenziando le idee più significative, almeno nel caso unidimensionale.

Abstract

The difference between the theory of dynamical systems and other branches of Mathematics is not in the objects of study, but in the questions asked about them. For instance, a discrete dynamical system simply is a (measurable, continuous, differentiable, holomorphic...) self-map of a space. Studying a map $f$ from a dynamical point of view then means studying the qualitative behavior of the iterates $f^{k} = f \circ f \circ \cdots \circ f$ as $k$ goes to infinity. In this paper we would like to give an idea of the kind of arguments the theory of dynamical systems deals with, concentrating our attention to a limited but important subject, the local discrete holomorphic dynamics, that is the study of the dynamical behaviour of holomorphic maps defined in a neighbourhood of a fixed point. Born toward the end of the nineteenth century, more or less in the same years the general theory of dynamical systems was born, local discrete holomorphic dynamics have seen major developments in the last thirty years, when several important results have been proved, and new significants areas have started to be explored, providing a wealth of natural open problems. We shall describe the basic themes and main results of the theory, stressing the more significant ideas, at least in the one-dimensional case.

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