Iterations and groups of formal transformations

TL;DR

This paper investigates formal iteration in transformation groups, providing a counterexample to Moser's theorem and examples of groups where iteration solutions always exist, advancing understanding of formal transformations.

Contribution

It constructs a counterexample to Moser's existence theorem and presents examples of transformation groups with universally solvable iteration problems.

Findings

Constructed an area-preserving map without a square root.

Provided a counterexample to Moser's theorem.

Showed existence of transformation groups with solutions for all elements.

Abstract

In this paper, we consider the problem of formal iteration. We construct an area preserving mapping which does not have any square root. This leads to a counterexample to Moser's existence theorem for an interpolation problem. We give examples of formal transformation groups such that the iteration problem has a solution for every element of the groups.