Examples of type ${\rm III}_{1}$ inhomogenous Markov shifts supported on topological Markov shifts

TL;DR

This paper constructs specific inhomogeneous Markov measures supported on topological Markov shifts, demonstrating the existence of type ${ m III}_{1}$ systems on hyperbolic toral automorphisms, with applications to smooth dynamical systems.

Contribution

It introduces new inhomogeneous Markov measures of type ${ m III}_{1}$ supported on topological Markov shifts related to hyperbolic automorphisms.

Findings

Constructed inhomogeneous Markov measures of type ${ m III}_{1}$

Supported measures on hyperbolic toral automorphisms

Applications to $C^{1}$ Anosov diffeomorphisms without ACIM

Abstract

We construct inhomogenous Markov measures for which the shift is of Kreiger type ${\rm III}_{1}$. These measures are fully supported on a toplogical markov shift space of the hyperbolic toral automorphism $f(x,y)=\left(\{x+y\},x\right):\mathbb{T}^{2}\to\mathbb{T}^{2}$ and are used in a subsequent paper in the construction of $C^{1}$ Anosov diffeomorphisms of $\mathbb{T}^{2}$ without an absolutely continuous invariant measure.