Let $(X,d)$ is a metric space and $X$ has no isolated points, $T:X\rightarrow X$ is a continuous self-map.
$T$ is sensitive if there exist $\epsilon>0$, for any $x\in X$ and any positive number $\delta>0$, we can find a point $y\in B(x,\delta)$ and $n\in\mathbb{N}$, such that $d(T^nx,T^ny)>\epsilon$.
$T$ is expansive if there exist $\varepsilon>0$, for any $x,y\in X$, $x\neq y$, we can find a number $n\in\mathbb{N}$, such that $d(T^nx,T^ny)>\varepsilon$.
If $T$ is sensitive, then for any $k\in\mathbb{N}$, $T^k$ is sensitive?
If $T$ is expansive, then for any $k\in\mathbb{N}$, $T^k$ is expansive?
If $T$ is an injection and $T$ is expansive, then for any $k\in\mathbb{N}$, $T^k$ is expansive?
Thanks a lot.