TITLE: Dynamical properties of injective continuous functions on $\mathbb{R}^d$
QUESTION [28 upvotes]: Let $\varphi:\mathbb{R}^d\to\mathbb{R}^d$ be an injective continuous function. 
Denote by $\varphi_n$ the $n$-th iterate of $\varphi$, i.e. 
$\varphi_n(x)=\varphi_{n-1}(\varphi(x))$ for all $x\in\mathbb{R}^d$.
Consider two properties of $\varphi$:
1) For every compact set $K\subset\mathbb{R^d}$ there exists $n\in\mathbb{N}$ such that
$\varphi_n(K)\cap K=\emptyset$.
2) For every compact set $K\subset\mathbb{R^d}$ there exists $n_0\in\mathbb{N}$ such that
for every $n\geq n_0$ we have
$\varphi_n(K)\cap K=\emptyset$.
The question is the following: is 1 equivalent to 2?
Remark 1. It is obvious that 2 implies 1.
Remark 2. If $d=1$, then the answer is YES and it is a simple consequence of the fact 
that an injective function must be monotonic.

REPLY [2 votes]: Martin's negative answer leaves open the case of $n=2$, and it seems this is for good reason.
Indeed, I think that, for a homeomorphism $\phi$ of $\mathbb{R}^2$, the two properties are indeed equivalent.
Firstly, note that $\phi$ can be assumed to be orientation-preserving (otherwise, replace $\phi$ by its second iterate).
Let $\phi$ be an orientation-preserving homeomorphism of $\mathbb{R}^2$ without fixed points. The Brouwer plane translation theorem states that every point is contained in a domain of translation (which is disjoint from both its image and preimage). See
http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2143652
So you can cover your compact set $K$ with finitely many such translation domains. It should follow (though I have not checked the details) that $K$ is disjoint from all of its sufficiently large iterates.
(I noticed that you did not say that $\phi$ should be a homeomorphism, but only injective and continuous. It appears plausible that any counterexample in this setting would also give rise to a homeomorphic one, but one should always be careful with such topological questions - as indeed Brouwer himself discovered in the course of the proof of his theorem, I believe.)