[Paper Review] On Absolutely Continuous Invariant Measures and Krieger-Types of Markov Subshifts

Nachi Avraham-Re’em|arXiv (Cornell University)|Apr 13, 2020

Mathematical Dynamics and Fractals34 references4 citations

TL;DR

This paper establishes that for non-singular conservative shifts on topologically mixing Markov subshifts satisfying the Doeblin condition, any absolutely continuous shift-invariant measure must be a Markov measure. Furthermore, if such a measure is not equivalent to a homogeneous Markov measure, the system is of Krieger type III₁, providing a complete classification of invariant measures under these conditions and a criterion for equivalence of Markov measures.

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.

Motivation & Objective

• To classify absolutely continuous shift-invariant measures on non-singular conservative Markov subshifts under the Doeblin condition.
• To determine the Krieger type of such systems when the invariant measure is not equivalent to a homogeneous Markov measure.
• To provide a criterion for equivalence between Markov measures on Markov subshifts.
• To establish the uniqueness and structural constraints of absolutely continuous invariant measures in this class of dynamical systems.

Proposed method

• Analysis of the Doeblin condition to ensure uniform positive lower bounds on transition probabilities, ensuring stochastic stability.
• Application of ergodic theory tools to study shift-invariant measures on Markov subshifts.
• Use of the Krieger spectral classification to determine the type of the system based on the structure of the invariant measure.
• Reduction of the problem to the equivalence class of Markov measures via Radon-Nikodym derivatives.
• Leveraging topological mixing to ensure strong recurrence and ergodicity properties.
• Derivation of a criterion for equivalence of Markov measures using transition probability ratios and invariant densities.

Experimental results

Research questions

• RQ1 Under what conditions is an absolutely continuous shift-invariant measure on a Markov subshift necessarily a Markov measure?
• RQ2 What Krieger type does a conservative non-singular shift on a topologically mixing Markov subshift have if its invariant measure is not equivalent to a homogeneous Markov measure?
• RQ3 What conditions ensure equivalence between two Markov measures on a Markov subshift?
• RQ4 How does the Doeblin condition constrain the structure of absolutely continuous invariant measures in this setting?
• RQ5 Can the Krieger type be fully determined by the equivalence class of the invariant measure?

Key findings

• Any absolutely continuous shift-invariant measure on a non-singular conservative shift over a topologically mixing Markov subshift with the Doeblin condition must be a Markov measure.
• If such a Markov measure is not equivalent to a homogeneous Markov measure, the system is of Krieger type III₁.
• A criterion for equivalence of Markov measures is derived based on the ratio of transition probabilities and invariant densities.
• The Doeblin condition ensures uniform lower bounds on transition probabilities, which is essential for the uniqueness and Markov nature of the invariant measure.
• Topological mixing ensures strong ergodicity, which supports the classification of the Krieger type.
• The results provide a complete classification of absolutely continuous invariant measures in terms of their equivalence to homogeneous Markov measures and the resulting Krieger type.