Let's say that we have a metric space $(M,d)$. Let's denote two subsets of $M$; $A$ and $B$ where $A \cap B=\emptyset$.
Essentially we divide our metric space into two non intersecting chunks. Now let's say that we have a function $f:M \rightarrow M$. The function $f$ has two fixed points that we will denote $a^\star$ and $b^\star$. Given any point in $A$ a repeated application of $f$ on the respective point would converge to $a^\star$. And the same thing with subset and point from $B$.
Another way to state this is that $f$ has a Lipschitz constant of less than 1 (contraction map), for any two points in A, and for any two points in B.
$$\forall a_1,a_2 \in A,d(f(a_1),f(a_2))\leq Ld(a_1,a_2) : L <1$$ $$\forall b_1,b_2 \in B,d(f(b_1),f(b_2))\leq Ld(b_1,b_2) : L <1$$
For this type of function to exist does $f$ have to be expansive ($L$ > 1) for any a point in $A$ and a point in $B$?
Can we state any other properties about $f$?
An example that might be helpful: $A = [-2,-1]$, $B = [1,2]$, with the usual metric of $\mathbb R$.
$$ f(x) = \cases{ \frac{1+x}{2} & $x \ge 1$\cr \frac{-1+x}{2} & $x \le -1$\cr} $$
Note that $d(f(x),f(y)) \le d(x,y)$ for all $x, y$. Of course equality holds in the case $x = a^* = -1, y=b^*=1$.