Let $f:\mathbb{R}^2\to \mathbb{R}^2$ be defined by \begin{equation*} f(x)= \left(% \begin{array}{cc} \lambda & 0 \\ 0 & n\lambda \\ \end{array}% \right) x, \end{equation*} where $\lambda>0$ and $n\in\mathbb{N}$
It is easy to see that $f$ is expansive, this means that there is $c>0$ such that if $d(f^n(x), f^n(y))<c$ for all $n\in\mathbb{Z}$, then $x=y$.
What can say about homeomorphisms that are closed to $f$? i.e. is there is $\delta>0$ such that if $g:\mathbb{R}^2\to \mathbb{R}^2$ is a homeomorphism with $d(f(x), g(x))<\delta$, for all $x\in\mathbb{R}$, then $g$ is an expansive homeomorphism?