arXiv subjects
Publications and source records attributed to Alexandre DeZotti.
We survey some of the connections linking complex dynamics to other fields of mathematics and science. We hope to show that complex dynamics is not just interesting on its own but also has value as an applicable theory.
We prove that for an infinite dimensional class of holomorphic maps with an elliptic fixed point, the post-critical set has Hausdorff dimension two, provided the rotation number is non-Herman of sufficiently high type. We identify classes of Brjuno but not Herman, and non-Brjuno, numbers such that the post-critical set satisfies the Karpinska's dimension paradox. That is, the set of end points of the post-critical set has dimension two, but without those end points, the dimension drops to one.
Let $ P \colon \mathbb{C} \to \mathbb{C} $ be an entire function. A Poincar\'e function $ L \colon \mathbb{C} \to \mathbb{C} $ of $ P $ is the entire extension of a linearising coordinate near a repelling fixed point of $ P $. We propose such Poincar\'e functions as a rich and natural class of dynamical systems from the point of view of measurable dynamics, showing that the measurable dynamics of $ P $ influences that of $ L $. More precisely, the hyperbolic dimension of $ P $ is a lower bound for the hyperbolic dimension of $ L $. Our results allow us to describe a large collection of hyperbolic entire functions having full hyperbolic dimension, and hence no natural invariant measures. (The existence of such examples was only recently established, using very different and much less direct methods.) We also give a negative answer to a natural question concerning the behaviour of eventual dimensions under quasiconformal equivalence.