I know that $S^1$ and $S^2$ admit no expansive homeomorphism. But I don't know for higher dimension $S^n$ for some $n\ge3$.
Definition: Let $X$ be locally compact metric space and $T$ is homeomorphism map on $X$ to itself. Then $T$ is said to be expansive if there exists positive real number $C$ and for any two distinct point $x$,$y\in X$ there exists integer number $n$ such that:
$$d(f^n(x), f^n(y)) > C$$
Edit: That $S^2$ admits no expansive self-homeomorphism is a result of K. Hiraide (1987 PNAS note, 1990 Osaka J Math Paper).