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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 $f$?