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

TL;DR

This paper establishes bounds on the dimension of subspaces with dense orbits under linear operators in finite-dimensional real spaces, showing certain supercyclic behaviors are impossible or trivial.

Contribution

It proves the non-existence of certain supercyclic operators in finite dimensions and characterizes the triviality of strong supercyclicity.

Findings

No $N$-supercyclic operators exist for $N < loor{rac{n+1}{2}}$

The minimal $N$ for supercyclicity is optimal

Strong $N$-supercyclicity does not occur non-trivially in finite dimensions

Abstract

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.