  In this work we define a stochastic adding machine associated to the
Fibonacci base and to a probabilities sequence $\overline{p}=(p_i)_{i\geq 1}$.
We obtain a Markov chain whose states are the set of nonnegative integers. We
study probabilistic properties of this chain, such as transience and
recurrence. We also prove that the spectrum associated to this Markov chain is
connected to the fibered Julia sets for a class of endomorphisms in
$\mathbb{C}^2$.
