On deformations of the spectrum of a Finsler--Laplacian that preserve the length spectrum

TL;DR

This paper demonstrates that a Finsler--Laplacian can detect metric deformations invisible to the length spectrum and constructs examples of non-reversible Finsler metrics with specific spectral and entropy properties.

Contribution

It introduces a Finsler--Laplacian capable of detecting metric changes beyond the length spectrum and provides explicit examples of non-reversible Finsler metrics with unique spectral characteristics.

Findings

Finsler--Laplacian detects metric deformations invisible to the length spectrum

Constructs non-reversible Finsler metrics with $4\\lambda_1 > h^2$ in negative curvature

Shows the Finsler--Laplacian's sensitivity surpasses traditional spectral invariants

Abstract

In this article, we show that a Finsler--Laplacian introduced previously can detect changes in the Finsler metric that the marked length spectrum cannot. We also construct examples of non-reversible Finsler metrics in negative curvature such that $4\lambda_1 > h^2$, where $\lambda_1$ is the bottom of the $L^2$-spectrum and $h$ the topological entropy of the flow.