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### Open Access Journals

DCDS

In this work we define a stochastic adding machine associated to a
quadratic base $(F_n)_{n \geq 0}$ formed by recurrent sequences of
order 2. We obtain a Markov chain with states in $\mathbb{Z}^+$ and we
prove that the spectrum of the transition operator associated to
this Markov chain is connected to the
filled Julia sets for a class of endomorphisms in
$\mathbb{C}^2$ of which we study topological properties.

DCDS

We consider in this paper a sequence of complex analytic functions constructed by the following procedure
$f_n(z)=f_{n-1}(z)f_{n-2}(z)+c$, where $c\in\mathbb{C}$ 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.

keywords:
holomorphic motion
,
endomorphisms of $\mathbb{C}^2$.
,
Julia sets
,
quasi-circle
,
complex dynamics

## Year of publication

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