On the Collection of Integers that Index the Fixed Points of Maps on the Space of Rational Functions

TL;DR

This paper investigates the set of integers that index fixed points of certain maps on rational functions, showing they form a union of arithmetic progressions and exploring their algebraic structure.

Contribution

It characterizes the set of distinguished integers for maps on rational functions, revealing their union of arithmetic progressions and relating new generating sets to previous bases.

Findings

The set of distinguished integers can be expressed as a union of infinitely many arithmetic progressions.

A new generating set for fixed rational functions is constructed and related to existing bases.

The properties of these integers depend on the parameters s and t, with implications for the structure of fixed points.

Abstract

Given integers s and t, define a function phi_{s,t} on the space of all formal complex series expansions by phi_{s,t} (sum a_n x^n) = sum a_{sn+t} x^n. We define an integer r to be distinguished with respect to (s,t) if r and s are relatively prime and and r divides t (1 + s + ... s^{ord_r(s)-1}). The vector space consisting of all rational functions whose Taylor expansions at zero are fixed by phi_{s,t} was previously classified by constructing a basis that is partially indexed by integers that are distinguished with respect to the pair (s,t). In this paper, we study the properties of the set of distinguished integers with respect to (s,t). In particular, we demonstrate that the set of distinguished integers with respect to (s,t) can be written as a union of infinitely many arithmetic progressions. In addition, we construct another generating set for the collection of rational functions that are fixed by phi_{s,t} and discuss the relationship between this generating set and the basis that was generated previously.