Could anyone give me some references for the convergence rate of Markov chain arising from the random iteration of Lipschitz functions which moves as follows:

$$ X_{n+1}= f_{\omega_n}(X_n)$$ where $f_1,\dots, f_s$ are Lipschitz functions (with Lipschitz constants $L_i$ and $\sum_{k=1}^{s} p_kL_k <1$ )on $\mathcal X$, a metric space with a metric $d$ it is complete and separable say. $\omega_0,\omega_1,\dots, \omega_n,\dots$ are i.i.d discrete random variable taking values in $\{1,2,\dots,s<\infty\}$. $p_k=\mathbb P(\omega_i=k)$.

I found Diaconis and Freedman's Paper where they didn't talk about the finite number of functions though and convergence rate is exponential in Prohorov's metric.

While talking about the convergence rate of probability measures towards the invariant measure, we need to talk about some metric on the space of probability measures, which also a question, which metric to use in this case to find the rate. Thanks.

$\mu_n(A)=\mathbb P(X_n\in A)$ i.e $X_n\sim \mu_n$,

I am looking for some reference on the Wasserstein metric. Thanks for helping!