Invariant measures as obstructions to attractors in dynamical systems and their role in nonholonomic mechanics

TL;DR

This paper investigates how the absence of certain invariant measures can prevent attractors in dynamical systems, especially in nonholonomic mechanics, highlighting the need to consider measures beyond positive smooth densities.

Contribution

It provides rigorous results on invariant measure obstructions and extends the analysis of nonholonomic systems beyond classical measure classes.

Findings

Invariant measures can obstruct attractor existence in dynamical systems.

Classical invariant measures with positive densities are insufficient for nonholonomic systems.

Extended measure classes are necessary to understand dynamical obstructions.

Abstract

We present some rigorous results on the absence of a wide class of invariant measures for dynamical systems possessing attractors. We then consider a generalization of the classical nonholonomic Suslov problem which shows how previous investigations of existence of invariant measures for nonholonomic systems should necessarily be extended beyond the class of measures with strictly positive $C^1$ densities if one wishes to determine dynamical obstructions to the presence of attractors.