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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 $N$-dimensionalsubspace 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 occurnon-trivially in finite dimension.

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