# Does there exist a topological transitive dynamical system that diverges to infinity for all initial conditions?

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.

• wut does 'diverges to infinity' mean – mathworker21 Dec 17 '18 at 9:12
• @mathworker21: This means that the image of any point by the successive compositions of the map goes to infinity. – China Dec 17 '18 at 9:14
• what is 'infinity' for an arbitrary metric space – mathworker21 Dec 17 '18 at 9:16
• @mathworker21: The same as the case of the real line. – China Dec 17 '18 at 9:19
• if you want to talk about $\mathbb{R}^n$, just say so. also, the answer in $\mathbb{R}^n$ is obviously "no", by the definition of topological transitivity. if a point goes off to infinity, then it can't visit every open set. – mathworker21 Dec 17 '18 at 9:25

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 $$T^n(x_0) \to \infty$$.