$X$ be finite dimensional real NLS , let $x \in X$ , does there exist $T \in \mathcal B(X)$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$?

Question

Let $X$ be a finite dimensional real normed linear space , let $x \in X$ , then does there exist a continuous linear transformation $T:X \to X$ such that $\{T^n(x):n \in \mathbb N\}$ is dense in $X$ ?

Answer 1

Your set is countable. Thus the answer is no, at least when $X$ is not separable, e.g. the space of bounded real sequences w.r.t. the infinity norm.

Answer 2

No, there are no hypercyclic operators on finite dimensional spaces. See e.g the article of Große-Erdmann in the Transactions of the AMS.