Projective distance and $g$-measures

Abstract

We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to thed-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.

Constant-Length Random Substitutions and Gibbs Measures

“…Remark 1. In [23] we studied the salient features of the topology generated by the projective distance. There it was proved that M(A N ) is complete and non-separable with respect to ρ, so that the topology generated by ρ is strictly finer than the vague topology.…”