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

TL;DR

This paper investigates the conditions under which absolutely continuous invariant measures exist for certain Markov subshifts, establishing their Markovian nature and linking measure equivalence to Krieger-type classifications.

Contribution

It proves that for non-singular conservative shifts on topologically mixing Markov subshifts with Doeblin condition, the only absolutely continuous invariant measure is Markov, and relates measure equivalence to Krieger-type III₁.

Findings

Only Markov measures are absolutely continuous invariant measures.

Non-equivalence to homogeneous Markov measures implies Krieger-type III₁.

Provides a criterion for measure equivalence.

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.