Let $f:M\rightarrow M$ be a homeomorphism. $f$ is called a $c$-expansive homeomorphism, whenever for every $x\neq y$, there is an integer $n$ with $d(f^n (x),f^n (y))>c$.
By double asymptotically points, we mean about two points $x$ and $y$ such that $d(f^n (x),f^n (y))$ tends to zero whenever $n$ tends to $\pm \infty$.
Question: Does there exist an example of an expansive homeomorphism $f$ in an infinite compact metric space $M$ without double asymptotically points?
Yes, there is an expansive homeomorphism on compact metric space without double asymptotically points. Consider for example the following link, page: 32
https://www.researchgate.net/publication/284900645_Expansive_Dynamical_Systems