Spectra related to the length spectrum

Abstract

We show how to extend the Covering Spectrum (CS) of Sormani-Wei to two spectra, called the Extended Covering Spectrum (ECS) and Entourage Spectrum (ES) that are new for Riemannian manifolds but defined with useful properties on any metric on a Peano continuum. We do so by measuring in two different ways the size of a topological generalization of the $\delta$-covers of Sormani-Wei called covers. For Riemannian manifolds $M$ of dimension at least 3, we characterize entourage covers as those covers corresponding to the normal closures of finite subsets of $\pi_{1}(M)$. We show that CS$\subset$ES$\subset$MLS and that for Riemannian manifolds these inclusions may be strict, where MLS is the set of lengths of curves that are shortest in their free homotopy classes. We give equivalent definitions for all of these spectra that do not actually involve lengths of curves. Of particular interest are resistance metrics on fractals for which there are no non-constant rectifiable curves, but where there is a reasonable notion of Laplace Spectrum (LaS). The paper opens new fronts for questions about the relationship between LaS and subsets of the length spectrum for a range of spaces from Riemannian manifolds to resistance metric spaces.