Consider deterministic discrete dynamical system defined as $$x_{n+1}=f(x_n).$$
Then Lyapunov exponent is defined as
$$\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{i=0}^{n-1}\ln(f'(x_i)).$$
I think that I understand main idea why the Lyapunov exponent is defined in this form.
My problem is that I have not deterministic discrete dynamical system but I have random discrete dynamical system.
Consisder very simple random dynamical system of the form $$X_{n+1}= \begin{cases} f(X_n) \text{ with prob. } p,\\ g(X_n) \text{ with prob. } 1-p. \end{cases} $$
The question is how the Lyapunov exponent can be define for this random dynamical system (or for random dynamical systems in general). I have been reading somtehing about Lyapunov exponet for sequence of random matrices but I am not sure if it is usable for my problem.
I am studing these things by myself and I am litle bit lost in it. Any help will be apreciated.