Spectral geometries on a compact metric space

TL;DR

This paper introduces spectral geometries as a way to discretely approximate compact metric spaces, inspired by non-commutative geometry, and explores their properties and relationships to traditional dimensions.

Contribution

It defines axioms for spectral geometries, studies their deformations, and links their dimension to classical metric space dimensions.

Findings

Spectral geometries can be characterized by a specific set of axioms.

Bounded deformations of spectral geometries are analyzed.

A relationship between spectral geometry dimension and traditional metric dimensions is established.

Abstract

The notion of a spectral geometry on a compact metric space X is introduced. This notion serves as a discrete approximation of X motivated by the notion of a spectral triple from non-commutative geometry. A set of axioms charaterising spectral geometries is given. Bounded deformations of spectral geometries are studied and the relationship between the dimension of a spectral geometry and more traditional dimensions of metric spaces is investigated.