Continuity of measure-dimension mappings

TL;DR

This paper investigates the continuity properties of various measure-dimension mappings under different topologies on Borel measures, providing examples of discontinuity and conditions for semi-continuity, with applications in dynamical systems.

Contribution

It offers a comprehensive analysis of the continuity and discontinuity of measure-dimension mappings under multiple topologies, including new semi-continuity results under specific conditions.

Findings

No general continuity under weak, setwise, or TV topology.

Semi-continuity results under setwise topology with restrictions.

Applications in determining measure dimensions in dynamical systems.

Abstract

We study continuity and discontinuity of the upper and lower (modified) box-counting, Hausdorff, packing, (modified) correlation measure-dimension mappings under the weak, setwise and TV topology on the space of Borel measures respectively in this work. We give various examples to show that no continuity can be guaranteed under the weak, setwise or TV topology on the space of Borel measures for any of these measure-dimension mappings. However, in some particular circumstances or by assuming some restrictions on the measures, we do have some (semi-)continuity results for some of these measure-dimension mappings under the setwise topology. In the end we point out some applications of our continuity results on deciding the dimensions of measures on ambient spaces with some dynamical structures.