The stochastic adding machine (SAM) was defined by Killeen and Taylor in [KT], as follows: let
$N\in\mathbb{N}=\{0,1,2,\ldots\}$. By using the greedy algorithm we may write N as
$N=\sum_{i=0}^{k(N)}\varepsilon_i(N)2^i$, in a unique
way where $\varepsilon_i(N)\in\{0,1\}$, for all $i\in\{0,\ldots,k(N)\}$. So, the representation of N in base 2
is given by $N=\varepsilon_{k(N)}(N)\ldots\varepsilon_0(N)$. They
defined a systems of evolving equation that calculates the digits of
$N+1$ in base 2, introducing an auxiliary variable "carry 1",
$c_i(N)$, for each digit $\varepsilon_i(N)$, as follows:

Define $c_{-1}(N+1)=1$ and for all $i\geq 0$, do
\begin{eqnarray}
\varepsilon_i(N+1)=(\varepsilon_{i}(N)+c_{i-1}(N+1))\mod 2;\\
c_i(N+1)=\left[\frac{\varepsilon_i(N)+c_{i-1}(N+1)}{2}\right],\nonumber
\end{eqnarray}
where $[x]$ is the integer part of $x\in\mathbb{R}^+$.

Killeen and Taylor [KT] defined a SAM considering a family of independent, identically distributed randon
variables $\{e_i(n):i\geq 0, \textrm{ } n\in\mathbb{N}\}$,
parametrized by nonnegative integers i and n, where each
$e_i(n)$ takes the value 0 with probability $1-p$ the value 1
with probability $p$. More precisely, let N be a nonnegative integer and
consider the sequences $(\varepsilon(N+1))_{i\geq 0}$ and
$(c_i(N+1))_{i\geq -1}$ defined by $c_{-1}(N+1)=1$ and for all
$i\geq 0$
\begin{eqnarray}
\varepsilon_i(N+1)=(\varepsilon_{i}(N)+e_i(N)c_{i-1}(N+1))\mod 2;\label{2}\\
c_i(N+1)=\left[\frac{\varepsilon_i(N)+e_i(N)c_{i-1}(N+1)}{2}\right].
\nonumber
\end{eqnarray}
Killeen and Taylor [KT] studied the spectrum of the transition
operator $S$ associated to the SAM in base 2, acting in $l^\infty$, and they proved that the spectrum of $S$ is equal to the filled Julia set of the quadratic map $f:\mathbb{C}\longrightarrow\mathbb{C}$ defined by $f(z)=\left(\frac{z-(1-p)}{p}\right)^2$.

In [MSV], the authors considered the SAM taking a probabilities sequence
$(p_i)_{i\geq 1}$, where the probability change in each
state, i.e. on the description (\ref{2}) we have $e_i(N)=1$ with
probability $p_{i+1}$ and $e_i(N)=0$ with probability $1-p_{i+1}$,
for all $i\geq 0$, and they constructed the transition operator
$S$ related to this probabilities sequence. In particular, they proved that the spectrum of $S$ acting in $l^\infty$, is equal to the fibered Julia set $E:=\{z\in\mathbb{C}: (\tilde{f}_j(z)) \textrm{ is bounded}\}$,
where $\tilde{f}_j:=f_j\circ\ldots\circ f_1$ and $f_j:\mathbb{C}\longrightarrow\mathbb{C}$ are maps defined by $f_j(z)=\left(\frac{z-(1-p_j)}{p_j}\right)^2$, for all $j\geq 1$.

In this work, instead of base 2, we will consider the Fibonacci
base $(F_n)_{n\geq 0}$ defined by $F_n=F_{n-1}+F_{n-2}$, for all
$n\geq 2$, where $F_0=1$ and $F_1=2$. Also, we will consider a
probabilities sequence $(p_i)_{i\geq 1}$, instead of an
unique probability $p$ (as was done in [MS] and [UM]). Thenceforth, we will define
the Fibonacci SAM and considering the transition operator $S$, we will prove that the Markov chain is transient if only if $\prod_{i=1}^{\infty}p_i>0$. Otherwise, if $\sum_{i=1}^{+\infty}p_i=+\infty$, then the Markov chains is null recurrent and if $\sum_{i=2}^{+\infty}p_iF_{2(i-1)}<+\infty$, then
the Markov chain is recurrent positive.

We will compute the point spectrum and prove that it is connected to the fibered Julia sets for a class of endomorphisms in $\mathbb{C}^2$. Precisely $\sigma_{pt}(S)\subset E \subset \sigma(S)$ where $E=\{z\in\mathbb{C}:(g_n\circ\ldots\circ g_0(z,z))_{n\geq 1}\textrm{ is bounded}\}$ and $g_n:\mathbb{C}^2\longrightarrow\mathbb{C}^2$ are maps defined by $g_0(x,y)=\left(\frac{x-(1-p_1)}{p_1},\frac{y-(1-p_1)}{p_1}\right)$ and $g_n(x,y)=\left(\frac{1}{r_n}xy-\left(\frac{1}{r_n}-1\right),x\right)$ for all $n\geq 1$, where $r_n=p_{\left[\frac{n+1}{2}\right]+1}$. Moreover, if $\liminf_{i\to+\infty}p_i>0$ then $E$ is compact and $\mathbb{C}\setminus E$ is connected.

REFERENCES

[C] D. A. Caprio, A class of adding machine and Julia sets, arXiv:1508.05062 [math.DS].

[KT] P.R. Killeen, T.J. Taylor, A stochastic adding machine and complex dynamics, Nonlinearity 13 (2000) 1889-1903.

[MSV] A. Messaoudi, O. Sester, G. Valle, Spectrum of stochastic adding machines and fibered Julia sets, Stochastics and Dynamics, 13(3), 26 pp, 2013.

[MS] A. Messaoudi, D. Smania, Eigenvalues of stochastic adding machine, Stochastics and Dynamics, Vol.10. N0. 2 (2010) 291-313.

[MU] A. Messaoudi, R.M.A. Uceda, Stochastic adding machine and 2-dimensional Julia sets, Discrete Contin. Dyn. Syst. 34, No. 12, 5247-5269, 2014.