I've casually proved, as application of some ideas that I am developing, a result that might be of interest in itself. I am completely new in this field and then I would like to ask your help to understand: 1) might it be of interest? 2) Is it trivial, in the sense that it can be proved directly? 3) is it well-known?

Let me fix a *planar* and *regular* setting, even if I could state the result in more general settings: let $n\geq1$ be a fixed integer and $P=[-n,n]^2\subseteq\mathbb Z^2$. Let me fix the following notation: given $(x,y)\in P$, I will denote by $A(x,y)$ the set formed by the following *at most* five points: $(x-1,y),(x,y),(x+1,y),(x,y-1),(x,y+1)$, where *at most* means that if one of those points does not belong to $P$, then I will not consider it.

Now, the situation is the following: for any point $p\in P$, let $\gamma_p$ a walk starting on $p$ and ending on $p^+$. I suppose that: if a walk hits the boundary, then it ends. In particular, if $p\in\partial P$, then $p^+=p$.

**Definition:** A flow is a family of walks $\gamma_p$, one for each $p\in P$, such that: for all $p\in P$, whenever $q\in A(p)$, then $q^+\in A(p^+)$.

My result would be: Given a flow of walks, there is at least one walk which does not hit the boundary.

I was thinking that it might be useful to prove that some walks are bounded, but I repeat that I am really new in this field.

Every comment is welcome and also references are appreciated.

Thanks in advance,

Valerio