Projective distance and $g$-measures

Liliana  Trejo-Valencia; Edgardo  Ugalde

doi:10.3934/dcdsb.2015.20.3565

Article Contents

2015, Volume 20, Issue 10: 3565-3579. Doi: 10.3934/dcdsb.2015.20.3565

This issue Previous Article Topological mixing, knot points and bounds of topological entropy Next Article

Projective distance and $g$-measures

Liliana Trejo-Valencia^1, and
Edgardo Ugalde^1,

1.
Instituto de Física, Universidad Autónoma de San Luis Potosí, Avenida Manuel Nava 6, Zona Universitaria, 78290 San Luis Potosí

Received: January 2015

Revised: March 2015

Published: December 2015

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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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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