I was wondering if anyone knows a good paper/survey/treatise on the dynamics of self-maps of $\mathbb{R}^2$? My advisor and I are trying to answer some questions, and we were wondering if there are any results we could use. Here's one question to give you an idea of what we're looking for.
Let $\psi: \mathbb{R}^2 \to \mathbb{R}^2$ be injective and continuous. Let $x \in \mathbb{R}^2$ and define $\{\psi^{n}(x)\}$ to be the orbit of x under $\psi$ where $\psi^{n} = \psi \circ \psi \circ ... \circ \psi$. So this is a countably infinite set. Assume that for every $x \in \mathbb{R}^2$, the orbit of x is unbounded. Does this imply that for every $x \in \mathbb{R}^2$, the orbit of x tends to $\infty$?
The former states that there is a subsequence $\{\psi^{n_k}(x)\}$ where $\lvert \psi^{n_k}(x) \rvert \to \infty$ (limit taken on k) while the latter states that for the whole sequence $\{\psi^{n}(x)\}$, $\lvert \psi^{n}(x) \rvert \to \infty$.
Thanks in advance.