Markov chains and mappings of distributions on compact spaces

Consider a compact metric space $S$ and a pair $(j,k)$ with $k\ge 2$ and $1\le j\le k$. For any probability distribution $\theta \in P(S)$, define a Markov chain on $S$ by: from state $s$, take $k$ i.i.d. ($\theta$) samples, and jump to the $j$’th closest. Such a chain converges in distribution to a unique stationary distribution, say ${\pi }_{j,k}(\theta )$. So this defines a mapping ${\pi }_{j,k}:P(S)\to P(S)$. What happens when we iterate this mapping? In particular, what are the fixed points of this mapping? We present a few rigorous results, to complement our extensive simulation study elsewhere.