Consider a circle $C$ with radius of $r$, we place $m$ balls(treated as point) randomly on it, and each ball $i$ has the mass $m_i$. We define a function $\varphi:C\rightarrow C$ which maps $x\in C$ to the centroid of those balls with distance less than $l<2\pi r$ to $x$(work on arcs).

My question is that, for any point $x\in C$ whether the sequence $\{\varphi(x),\varphi\circ\varphi(x),\cdots,\varphi^n(x)\}$ terminated at some point $x_0$?(no cycles).

If the circle is not discrete but continuous with a density distribution function, what would be the answer?

(I'm not a English native speaker, so sorry for my poor English)