Concerning a stochastic dynamical system

We study the discrete-time dynamical system
$$ X_{n+1}=2\sigma\cos(2\pi\theta_n)g(X_n),\qquad n\in\mathbb Z, $$
Where $\theta_n$ is an ergodic stationary process whose univariate distribution is uniform on the interval $[0,1]$, the function $g(x)$ is odd, bounded, increasing, and continuous, and $\mathbb Z$ is the ring of integers. It is proved that under certain conditions there exists a unique stationary process that is a solution of the above equation and this process has a continuous purely singular spectrum.