A Simple Formula for the Hilbert Metric with Respect to a Sub-Gaussian Cone

Stéphane Chrétien,Juan-Pablo Ortega

doi:10.3390/math6030035

Abstract

The Hilbert metric is a widely used tool for analysing the convergence of Markov processes and the ergodic properties of deterministic dynamical systems. A useful representation formula for the Hilbert metric was given by Liverani. The goal of the present paper is to extend this formula to the non-compact and multidimensional setting with a different cone, taylored for sub-Gaussian tails.

Highlights

Let V be a topological vector space and C a closed convex cone inside V enjoying the property that C ∩ −C = ∅
We will focus on the case where V is the space of continuous integrable functions on Rn with Euclidean norm denoted by k · k
Letting δ tend towards zero and using continuity, we finaly conclude that inf e a k x − xσ(n) k g ( x σ (n ) ) − g ( x )

Summary

Introduction

The first application of the Hilbert metric to the field of dynamical systems is Birkhoff’s approach to the Perron-Frobenius theorem [9]; see [10]. Birkhoff’s theorem provides an elegant way to prove that certain maps between cones are contracts and obtain existence and uniqueness for certain problems such as in Perron-Frobenius theory for positive operators. The goal of this short note is to extend to the noncompact and multidimensional setting a useful formula for the Hilbert semi-metric which was previously given by Liverani [2] in the case in which V is the space L1 ([0, 1]) of integrable functions of the interval [0, 1]. E a| x −y| g (y ) − g ( x ) e a|u−v| f (v ) − f (u ) i e a| x −y| f (y ) − f ( x ) e a|u−v| g (v ) − g (u )

Main Result

Conclusions