Iterated monodromy groups of rational functions and periodic points over finite fields

TL;DR

This paper investigates the behavior of periodic points of rational functions over finite fields, revealing new results for classes beyond power maps and Chebyshev polynomials, and connecting the problem to iterated monodromy groups and geometric methods.

Contribution

It provides the first results on the asymptotic proportion of periodic points for a broader class of rational functions over finite fields, using monodromy groups and orbifold metrics.

Findings

For quadratic functions over odd finite fields, the proportion of periodic points has lim inf 0.

For certain quadratic polynomials over odd square finite fields, the proportion of periodic points tends to 0.

The methods connect periodic point counts to properties of iterated monodromy groups and geometric expansions.

Abstract

Let $q$ be a prime power and $\phi$ a rational function with coefficients in a finite field $\mathbb{F}_q$. For $n \geq 1$, each element of $\mathbb{P}^1(\F_{q^n})$ is either periodic or strictly preperiodic under iteration of $\phi$. Denote by $a_n$ the proportion of periodic elements. Little is known about how $a_n$ changes as $n$ grows, unless $\phi$ is a power map or Chebyshev polynomial. We give the first results on this question for a wider class of rational functions: $a_n$ has lim inf $0$ when $q$ is odd and $\phi$ is quadratic and neither Latt\`es nor conjugate to a one-parameter family of exceptional maps. We also show that $a_n$ has limit $0$ when $\phi$ is a non-Chebyshev quadratic polynomial with strictly preperiodic finite critical point and $q$ is an odd square. Our methods yield additional results on periodic points for reductions of post-critically finite (PCF) rational functions defined over number fields. The difficulty of understanding $a_n$ in general is that $\mathbb{P}^1(\F_{q^n})$ is a finite set with no ambient geometry. In fact, $\phi$ can be lifted to a PCF rational map on the Riemann sphere, where we show that $a_n$ is given by counting elements of the iterated monodromy group (IMG) that act with fixed points at all levels of the tree of preimages. Using a martingale convergence theorem, we translate the problem to determining whether certain IMG elements exist. This in turn can be decisively addressed using the expansion of PCF rational maps in the orbifold metric.