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

Zemer Kosloff

doi:10.3934/jmd.2018020

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2018, Volume 13: 251-270. Doi: 10.3934/jmd.2018020

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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 21, 2017

Revised: June 27, 2017

Published: December 2018

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Abstract

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.

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

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Figure 4.1. The construction of the Markov partition

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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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