On manifolds admitting stable type III$_{\textbf1}$ Anosov diffeomorphisms

Abstract

We prove that for every \begin{document}$d≠3$\end{document} there is an Anosov diffeomorphism of \begin{document}$\mathbb{T}^{d}$\end{document} which is of stable Krieger type \begin{document}${\rm III}_1$\end{document} (its Maharam extension is weakly mixing). This is done by a construction of stable type \begin{document}${\rm III}_1$\end{document} Markov measures on the golden mean shift which can be smoothly realized as a \begin{document}$C^{1}$\end{document} Anosov diffeomorphism of \begin{document}$\mathbb{T}^2$\end{document} via the construction in our earlier paper.