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

We prove that on $\mathbb{R}^n$, there is no $N$-supercyclic operator with $1\leq N< \lfloor \frac{n+1}{2}\rfloor$ 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.