On Absolutely Continuous Invariant Measures and Krieger-Type of Markov Subshifts

Abstract

It is shown that for a non-singular conservative shift on a topologically mixing Markov subshift with Doeblin Condition the only possible absolutely continuous shift-invariant measure is a Markov measure. Moreover, if it is not equivalent to a homogeneous Markov measure then the shift is of Krieger-type ${\mathrm{III}}_1$. A criterion for equivalence of Markov measures is included.

Incorporating referee's remark and notations as in the published version in Journal d'Analyse Mathématique

References in corpus (1)

• Ergodicity and type of nonsingular Bernoulli actions

Cited by in corpus (2)

• Some factors of nonsingular Bernoulli shifts
• Equivalence-Singularity Dichotomy in Markov Measures