Let $T: X \to X$ be a contraction of a compact metric space. How many recurrent points does it have? Can it be mixing?
Being contraction of a compact metric space means that $d(T(x),T(y)))\le kd(x,y)$ for some $0\le k<1$. We are looking for points fulfilling condition $d(T^n(x),x)<\epsilon$ for $n$ greater than some $N$. My intuition is that we should look for the recurrent points only among fixed points.