p-dimensional cones and applications

TL;DR

This paper introduces p-dimensional cones and gauges, generalizing Birkhoff cones, and proves spectral gap and regularity results for operators acting on these cones, with applications to random products of cone-contracting operators.

Contribution

It develops a new framework of p-dimensional cones and gauges, extending classical cone theory to higher dimensions, and establishes spectral and regularity results using Grassmannian and exterior algebra tools.

Findings

Spectral gap for the p largest eigenvalues of cone-contracting operators.

Regularity of characteristic exponents for random products of such operators.

Generalization of contraction principles to p-dimensional cones.

Abstract

We introduce a notion of p-dimensional cones made of $p$-dimensional subspaces and gauges on these cones, giving rise to a contraction principle which generalizes the one for Birkhoff cones. Using tools on the grassmannian and the exterior algebra, we prove a spectral gap result for the p largest eigenvalues of an operator and a regularity result for the characteristic exponents of a random product of cone-contracting operators.