$n$-supercyclic and strongly $n$-supercyclic operators in finite dimensions

We prove that on ${\mathbb{R}}^n$, there is no $N$-supercyclic operator with $1\le N<\lfloor \frac{n+1}{2}\rfloor$ i.e. if ${\mathbb{R}}^n$ has an $N$ dimensional subspace whose orbit under $T$ is dense in ${\mathbb{R}}^n$, then $N$ is greater than $\lfloor \frac{n+1}{2}\rfloor$. Moreover, this value is optimal. We then consider the case of strongly $N$-supercyclic operators. An operator $T$ is strongly $N$-supercyclic if ${\mathbb{R}}^n$ has an $N$-dimensional subspace whose orbit under $T$ is dense in ${\mathbb{P}}_N({\mathbb{R}}^n)$, the $N$-th Grassmannian. We prove that strong $N$-supercyclicity does not occur non-trivially in finite dimension.