On the Fibonacci complex dynamical systems

Abstract

We consider in this paper a sequence of complex analytic functions constructed by the following procedure fn(z) = f n−1 (z)f n−2 (z) + c, where c ∈ C is a parameter. Our aim is to give a thorough dynamical study of this family, in particular we are able to extend the familiar notions of Julia sets and Green function and to analyze their properties. As a consequence, we extend some well-known results. Finally we study in detail the case where c is small.

Julia sets for Fibonacci endomorphisms of ℝ2

“…Although the examples given in the figure 4 show horizontal slices of the set K + that are bounded, the entire set K + itself is never bounded. The following proposition is a consequence of the results developed in [EABMS16]:…”

“…In particular, it was shown that the spectrum of the transfer operator associated with a stochastic adding machine in an exotic base (given by Fibonacci numbers) is related to the set K + (f c ), for a real value of c (a result inspired by [KT00]). Also, in [EABMS16], various topological properties of certain slices of the sets K + (f c ) were discussed. Here K + (f c ) is the forward filled Julia set of f c , made of all the points whose forward orbits are bounded:…”

Filled Julia set of some class of Hénon-like map

“…In [6], the authors proved that the spectrum of theses operators in other Banach spaces contains the set {z ∈ C 2 : (z, z) ∈ K + (f α )}. In [5], the authors studied many topological properties of the set K + (f α ) ∩ G(h) where h is a non null polynomial function and G(h) is the graph of h in C 2 . In particular, they proved that K + (f α ) ∩ G(h) is a quasi-disk, if |α| is small and h(z) = z.…”