Projective distance and $g$-measures

TL;DR

This paper introduces a new projective distance for probability measures on symbolic spaces, compares it with existing metrics, and explores its implications for the continuity of entropy and ergodic properties in $g$-measures.

Contribution

It proposes a novel projective distance inspired by Hilbert's metric and applies it to analyze $g$-measures, revealing new insights into entropy continuity and ergodic property preservation.

Findings

The projective distance is generally not comparable to the $ar{d}$-distance.

It helps assess entropy continuity at $g$-measures with uniqueness.

It relates convergence speed and regularity of $g$-functions to ergodic property preservation.

Abstract

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective distance is inspired on Hilbert's projective metric, and in the framework of $g$-measures, it allows to assess the continuity of the entropy at $g$-measures satisfying uniqueness. It also allows to relate the speed of convergence and the regularity of sequences of locally finite $g$-functions, to the preservation at the limit, of certain ergodic properties for the associate $g$-measures.