Erratum: Coding map for a contractive Markov system

TL;DR

This paper corrects a proof error in previous work on measures in contractive Markov systems, providing bounds and conditions under which certain measures are positive or well-defined, impacting the construction of equilibrium states.

Contribution

It identifies a proof error in earlier results and offers a corrected lower bound and generalized construction for measures in contractive Markov systems under specific conditions.

Findings

Explicit lower bounds for DDM in certain contractive systems

Conditions under which DDM is positive and well-defined

Generalized construction for measures in average contractive systems

Abstract

An error in the proof of Lemma 2 (ii) in [I. Werner, Math. Proc. Camb. Phil. Soc. 140(2) 333-347 (2006)], which claims the absolute continuity of dynamically defined measures (DDM), is identified. This undermines the assertion of the positivity of a DDM which provides a construction for equilibrium states in [I. Werner, J. Math. Phys. 52 122701 (2011)]. An explicit lower bound for the DDM appearing there is computed in the case when all maps of a contractive Markov system (CMS) are contractions, the probability functions are Dini-continuous and bounded away from zero, and there exists an equilibrium state of the CMS which is absolutely continuous with respect to the initial measure. In the case of the contraction only on average, a generalized construction is shown to provide a positive set function, but it is unknown whether it gives a measure on the Borel $\sigma$-algebra, and if it did, the measure would coincide with the original DDM.