Realizability of integer sequences as differences of fixed point count sequences

TL;DR

This paper extends the criteria for realizing integer sequences as differences of fixed point count sequences in dynamical systems, providing a new characterization and structural insights.

Contribution

It introduces a new realizability criterion for integer sequences as differences of fixed point counts from a system and its factor, expanding previous understanding.

Findings

Provides a criterion for realizability as differences of fixed point sequences.

Characterizes the structure of the set of such integer sequences.

Extends the concept of exact realizability to a broader class.

Abstract

A sequence of non-negative integers is exactly realizable as the fixed point counts sequence of a dynamical system if and only if it gives rise to a sequence of non-negative orbit counts. This provides a simple realizability criterion based on the transformation between fixed point and orbit counts. Here, we extend the concept of exact realizability to realizability of integer sequences as differences of the two fixed point counts sequences originating from a dynamical system and a topological factor. A criterion analogous to the one for exact realizability is given and the structure of the resulting set of integer sequences is outlined.