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