Suppose $(X_n)_{n\in \mathbb N_0}$ is a discret-time Markov process on state space $(0,1]$ with transition kernel $\kappa(x,\cdot)$ possessing an absolute continuous (Lebesgue-)density for all $x\in (0,1]$ with support $(0,1]$. The transition density $p(x,y) $ may be written as continuous function of $(x,y)\in(0,1]^2$. Also, $\kappa(x,\cdot)$ converges weakly to $(\delta_0 + \delta_1)/2$ as $x\downarrow 0$.
Now, I would like to analyse, whether $(X_n)$ has an invariant measure, if it's unique and the law of $X_n$ converging to it for any initial distribution.
Are there references (text books etc.) for that context?
Thanks in advance :)