Let \(\phi: \mathbb{P}^1(\mathbb{C}) \rightarrow \mathbb{P}^1(\mathbb{C})\) be a rational map of degree \(d \geq 2\). Fix two points \(A, P \in \mathbb{P}^1(\mathbb{C})\), and consider the forward orbit

of \(P\) under \(\phi\). An interesting question is how close can \(\phi^{n_k}(P)\) be to the point \(A\) along a subsequence \(n_k\). In some cases the answer is easier, such as when \(A\) is an attracting periodic point and \(P\) is in its attracting basin. However, in general it is not so clear, such as when both \(A, P\) are in the Julia set \(J_\phi\) and when \(P\) is not periodic. Specifically we would like to know whether it is possible for \(\phi^{n_k}(P)\) to be arbitrarily close to \(A\) along some subsequence \(n_k\).

It turns out that if the coefficients of \(\phi\) and the points \(A, B\) are defined over some number field \(K\), then under some mild assumptions the answer to above is no. This was proved by J. Silverman in 1993 and the proof is based on the arithmetic information provided by Roth’s Theorem.

In this talk I will give an overview of the proof of this famous Silverman’s Theorem E.