A local-global principle in the dynamics of quadratic polynomials

TL;DR

This paper investigates a local-global principle for the existence of periodic points in quadratic polynomials over number fields, establishing conditions under which local periodic points imply global ones, and identifying exceptions for periods 4 and 5.

Contribution

It proves a local-global principle for periods 1 to 3 and shows finiteness of counterexamples for periods 4 and 5 over number fields, extending previous results over the rationals.

Findings

If a quadratic polynomial has a period n point in every non-archimedean completion, it has such a point over the number field.

Finitely many conjugacy classes of quadratic polynomials over a number field can violate the principle for periods 4 and 5.

For every quadratic polynomial over Q, there are infinitely many primes p where no p-adic periodic point of period 4 exists.

Abstract

Let $K$ be a number field, $f\in K[x]$ a quadratic polynomial, and $n\in\{1,2,3\}$. We show that if $f$ has a point of period $n$ in every non-archimedean completion of $K$, then $f$ has a point of period $n$ in $K$. For $n\in\{4,5\}$ we show that there exist at most finitely many linear conjugacy classes of quadratic polynomials over $K$ for which this local-global principle fails. By considering a stronger form of this principle, we strengthen global results obtained by Morton and Flynn-Poonen-Schaefer in the case $K=\mathbf Q$. More precisely, we show that for every quadratic polynomial $f\in\mathbf Q[x]$ there exist infinitely many primes $p$ such that $f$ does not have a point of period 4 in the $p$-adic field $\mathbf Q_p$. Conditional on knowing all rational points on a particular curve of genus 11, the same result is proved for points of period 5.