On the asymptotics of the scenery flow

TL;DR

This paper investigates the asymptotic behavior of scenery flows, which are ways of 'zooming in' on measures, and explores how these flows behave under local diffeomorphisms, providing conditions for their convergence.

Contribution

It establishes necessary and sufficient conditions for the asymptotic convergence of scenery flows under transformations, advancing understanding of measure zooming behaviors.

Findings

Identifies conditions for scenery flow asymptotics

Provides examples illustrating the theory

Clarifies the impact of local diffeomorphisms on measures

Abstract

Various notions of "zooming in" on measures exist in the literature and the scenery flow is one of them. It is of interest to describe the joint asymptotics of the scenery flows generated by a measure and the measure transported by a local diffeomorphism. We give both sufficient and necessary conditions for the scenery distributions to be asymptotic and provide some examples.