Topological transitivity is a property of dynamical systems.
My question is: Does there exist a topological transitive dynamical system in the usual plane or the usual space that diverges to infinity for all initial conditions. This means that the image of any point by the successive compositions of the map goes to infinity.
No. For otherwise, by definition, there is some $x_0$ s.t. $\{x_0,T(x_0),T^2(x_0),\dots\}$ is dense in $X$, but this easily contradicts $T^n(x_0) \to \infty$.