Fedor Pakovich
Ben Gurion University, Beer Sheva, Israel

Let $A$ be a rational function of degree at least two on $\mathbb C\mathbb P^1$. For a point $z_1\in \mathbb C\mathbb P^1$ we denote by $O_A(z_1)$ the forward orbit of $A$, that is, the set $$\{z_1,A(z_1),A^{\circ 2}(z_1),\dots \}.$$ In the talk, we address the following problem: given two rational functions $A$ and $B$ of degree at least two, under what conditions do there exist orbits $O_A(z_1)$ and $O_B(z_2)$ having an infinite intersection?

We show that under a mild restriction on $A$ and $B$ this happens if and only if $A$ and $B$ have an iterate in common, that is, if and only if $A^{\circ k}=B^{\circ l}$ for some $k,l\geq 1.$ Put another way, unless rational functions $A$ and $B$ have the same global dynamics, an orbit of $A$ may intersect an orbit of $B$ at most at finitely many places.