Examples of type ${\mathrm{III}}_1$ inhomogenous Markov shifts supported on topological Markov shifts

Abstract

We construct inhomogenous Markov measures for which the shift is of Kreiger type ${\mathrm{III}}_1$. 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.

arXiv commentThis paper was previously a section of arXiv:1410.7707 (which has now being separated into two parts)