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On the non-equivalence of the Bernoulli and $ K$ properties in dimension four
2018, 13: 251-270. doi: 10.3934/jmd.2018020
On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms
Zemer Kosloff
Einstein Institute of Mathematics, Edmond J. Safra Campus (Givat Ram), The Hebrew University, Jerusalem 91904, Israel
Dedicated to the memory of Roy Adler, whose work has been and still is a great inspiration for me
Received January 22, 2017 Revised June 28, 2017 Published December 2018
Figure(1) / Table(1)
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.
Keywords: Ratio set, Markov shifts, Anosov diffeomorphisms, Maharam extension.
Mathematics Subject Classification: Primary: 37A40, 37C40; Secondary: 37A20.
Citation: Zemer Kosloff. On manifolds admitting stable type Ⅲ$_{\textbf1}$ Anosov diffeomorphisms. Journal of Modern Dynamics, 2018, 13: 251-270. doi: 10.3934/jmd.2018020
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Figure 4.1. The construction of the Markov partition
$c_{M_{t-1}-3}$ $w$ $d_{M_{t-1}}$
$1$ or $2$ $11$ $1$ or $3$
1 or 2 13 2
3 21 1 or 3
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