A finiteness property for preperiodic points of Chebyshev polynomials

TL;DR

This paper proves a finiteness property for preperiodic points of Chebyshev polynomials over number fields, showing only finitely many S-integral preperiodic points relative to a non-preperiodic point.

Contribution

It establishes a new finiteness result for S-integral preperiodic points of Chebyshev polynomials in number fields.

Findings

Finiteness of S-integral preperiodic points relative to a non-preperiodic point

Extension of known results to Chebyshev polynomials over number fields

Provides a new perspective on the distribution of preperiodic points

Abstract

Let K be a number field with algebraic closure K-bar, let S be a finite set of places of K containing the archimedean places, and let f be a Chebyshev polynomial. We prove that if a in K-bar is not preperiodic, then there are only finitely many preperiodic points b in K-bar which are S-integral with respect to a.