Non-integrability of measure preserving maps via Lie symmetries

TL;DR

This paper develops a criterion linking Lie symmetries and the non-integrability of smooth measure preserving maps near elliptic fixed points, with applications to specific rational maps.

Contribution

It introduces a novel criterion connecting Lie symmetries and non-integrability, and applies it to prove non-integrability of certain rational maps.

Findings

Criterion relates non-integrability to Lie symmetries and periodic points.

Proves local non-integrability of Cohen map and other rational maps.

Studies regularity of the period function on period annuli.

Abstract

We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the local non-integrability of the Cohen map and of several rational maps coming from second order difference equations.