I was reading this post and wondered. Does there exist a topologically transitive (TT) map $f:\mathbb{R}^n\to\mathbb{R}^n$ when $n\geq 2$? I know that post asks for compactness and topological mixing but if we relax the requirement to only TT is it possible?

Note: If $\mathbb{R}^n$ is replaced by an infinite-dimensional Hilbert space, then the Ansari-Bernal theorem guarantees such a map exists; moreover it can be linear... So maybe it can exist in the finite-dimensional case?