Integer Points in Backward Orbits

TL;DR

This paper explores the distribution of $S$-integer points in backward orbits of rational maps, proving a conjecture for specific maps like $z^d$ and Chebyshev polynomials, and relating it to Galois orbit bounds.

Contribution

It formulates a conjecture on backward orbits and proves it for certain maps by bounding Galois orbits, extending Silverman's results to backward dynamics.

Findings

Proved the conjecture for $ ext{z}^d$ and Chebyshev polynomials.

Bounded the number of Galois orbits for $z^n-eta$ when $eta$ is a non-root of unity.

Connected the conjecture's validity to uniform bounds on Galois orbits.

Abstract

A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.