Let $S=\{1,2\}$, $\omega_1, \dots, \omega_n, \dots$ are i.i.d discrete random variable taking values in $S$, consider $f_1, f_2 :\mathbb R_{+}\to \mathbb R_{+}$ as
$f_1(x)=\begin{cases} \frac{3x}{2} & x\leq 1 \\ \frac{1}{2}+\sqrt{x} & x>1 \\ \end{cases} $
$f_2(x)=\begin{cases} \frac{x}{4}+\frac{3}{4} & x\leq 1 \\ \sqrt{x} & x>1 \\ \end{cases} $
$p_k(1)=\frac{1}{2}+\frac{1}{2^k}, p_k(2)=\frac{1}{2}-\frac{1}{2^k}$
I start from some $X(0)=x(0)>0$ and choose $\omega_1$ according to probability distribution $p_1$ defined above and hence immediately get $f_{\omega_1}$.
Define $X(1)=f_{\omega_1}(X(0))$, repeat this 9 times and when chose $\omega_{10}$ according to probability distribution $p_2$ and define $X(10) =f_{\omega_{10}}(X(9))$ and repeat this iteration in this way and change the probability distribution again at $20^{th}, 30^{th}, \dots, 10k^{th}$ step $k=1,2,\dots$ according to probability dist $p_k$, continue this process.
And note that, from step 0 to step 9, we have $\mathbb P( \omega_i= j)= p_1(j), 1\le i\le 9, 1\le j\le 2$, similarly from step 10 to step 19, $\mathbb P( \omega_i= j)= p_2(j), 10\le i\le 19, 1\le j\le 2$, and so on.
(1) Could any-one tell me whether this is a Markov chain( time-inhomogeneous)? I understand that until the 49th iteration it is a Markov chain but I am confused as at the $50^{th}$ iteration probability distribution changes, does this means this forms an inhomogeneous Markov chain over-all?
(2) Any idea of how to conclude if there exists a stationary probability distribution $\pi$ such that $\mathbb P(X(n)\in \mathcal A)\rightarrow \pi(\mathcal A)$ as $n\to \infty$ for any Borel subset $\mathcal A$. Thanks.
(3) What if $f_1, f_2 :\mathbb R\to \mathbb R$ both are Lipschitz function with Lipschitz constant $<1$? Thanks