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

2014/10/28 by Zemer Kosloff, Kosloff, Zemer · 1 citation
Mathematics · Physics and Astronomy · #37A40 #37C40 #37D20 #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1410.7707

openalex publication_date 2014/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct examples of C¹ Anosov diffeomorphisms on \mathbbT² which are of Krieger type \rm III₁ with respect to Lebesgue measure. This shows that the Gurevic Oseledec phenomena that conservative C^1+α Anosov diffeomorphisms have a smooth invariant measure does not hold true in the C¹ setting.

Cited by

• Examples of type \rm III₁ inhomogenous Markov shifts supported on topological Markov shifts

Related

• Rigidity of U-Gibbs measures near conservative Anosov diffeomorphisms on \mathbbT³
• Two remarks on C^∞ Anosov diffeomorphisms
• Realization of Anosov Diffeomorphisms on the Torus
• Smooth rigidity for very non-algebraic Anosov diffeomorphisms of codimension one
• Diffeomorphisms Holder conjugate to Anosov diffeomorphisms