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arXiv · 1410.7707

Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure

Zemer Kosloff
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Abstract

We construct examples of $C^{1}$ Anosov diffeomorphisms on $\mathbb{T}^{2}$ which are of Krieger type ${\rm III}_{1}$ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative $C^{1+\alpha}$ Anosov diffeomorphisms have a smooth invariant measure does not hold true in the $C^{1}$ setting.

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Zemer Kosloff. 2014-10-28. Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure. https://arxiv.org/abs/1410.7707

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