Equivalence-Singularity Dichotomy in Markov Measures

TL;DR

This paper introduces a new dichotomy for one-dimensional Markov measures, distinguishing when they are equivalent or singular, using a novel approach that handles both one-sided and two-sided chains without relying on existing 0-1 laws.

Contribution

It presents a novel equivalence-singularity dichotomy for Markov measures, including a new 0-1 law derived from this dichotomy, applicable to both one-sided and two-sided chains.

Findings

Established a dichotomy for Markov measures

Developed a new approach handling one-sided and two-sided chains

Derived a new 0-1 law from the dichotomy

Abstract

We establish an equivalence-singularity dichotomy for a large class of one-dimensional Markov measures. Our approach is new in that we deal with one-sided and two-sided chains simultaneously, and in that we do not appeal to any 0-1 law. In fact we deduce a new 0-1 law from the dichotomy.