Continuous mappings between spaces of arcs

TL;DR

This paper investigates the continuity properties of mappings between spaces of arcs induced by blow-analytic homeomorphisms, providing conditions for both continuity and discontinuity, and extends results to o-minimal structures.

Contribution

It offers new insights into the continuity of arc mappings under blow-analytic homeomorphisms and generalizes the results to definable arcs in o-minimal structures.

Findings

Positive continuity results under certain topologies

Negative results showing discontinuity in other topologies

Uniform continuity in o-minimal structures

Abstract

A blow-analytic homeomorphism is an arc-analytic subanalytic homeomorphism, and therefore it induces a bijective mapping between spaces of analytic arcs. We tackle the question of the continuity of this induced mapping between the spaces of arcs, giving a positive and a negative answer depending of the topology involved. We generalise the result to spaces of definable arcs in the context of o-minimal structures, obtaining notably a uniform continuity property.