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

Dedicated to the memory of Roy Adler, whose work has been and still is a great inspiration for me

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

    Mathematics Subject Classification: Primary: 37A40, 37C40; Secondary: 37A20.


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  • Figure 4.1.  The construction of the Markov partition

    Table 1.   

    $c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$
    $1$ or $2$ $11$ $1$ or $3$
    1 or 2 13 2
    3 21 1 or 3
    3 23 2
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