• Corpus ID: 119168266

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

@article{Kosloff2014ConservativeAD,
  title={Conservative Anosov diffeomorphisms of the two torus without an absolutely continuous invariant measure},
  author={Zemer Kosloff},
  journal={arXiv: Dynamical Systems},
  year={2014}
}
  • Z. Kosloff
  • Published 28 October 2014
  • Mathematics
  • arXiv: Dynamical Systems
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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References

SHOWING 1-10 OF 27 REFERENCES
Examples of expandingC1 maps having no σ-finite invariant measure equivalent to Lebesgue
In this paper we construct aC1 expanding circle map with the property that it has no σ-finite invariant measure equivalent to Lebesgue measure. We extend the construction to interval maps and maps on
Generic expanding maps without absolutely continuous invariant $\sigma$-finite measure
We show that a $C^1$-generic expanding map of the circle has no absolutely continuous invariant $\sigma$-finite measure.
Transformations without finite invariant measure have finite strong generators
The following theorem is proved: If T is a nonsingular invertible transformation in a separable probability space (Ω, F, μ) and there exists no T-invariant probability measure μo << μ, then the
SIMILARITY OF AUTOMORPHISMS OF THE TORUS
Abstract : The automorphisms of the Torus are rich mathematic objects. They possess interesting number theoretic, geometric, algebraic, topological, measure theoretic, probabilistic, and even
A generic C1 map has no absolutely continuous invariant probability measure
Let M be a smooth compact manifold of any dimension. We consider the set of C1 maps f:M → M which have no absolutely continuous (with respect to Lebesgue) invariant probability measure. We show that
On non-singular transformations of a measure space. I
We consider a Lebesgue measure space (M, ∇, m). By an automorphism of (M, ∇, m) we mean a bi-measurable transformation of (M, ∇, m) that together with its inverse is non-singular with respeot to m.
Exchangeable measures for subshifts
Abstract Let Ω be a Borel subset of S N where S is countable. A measure is called exchangeable on Ω , if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at
A Generic C1 Expanding Map¶has a Singular S–R–B Measure
Abstract: We show that for a generic C1 expanding map T of the unit circle, there is a unique equilibrium state for − log T′ that is an S–R–B measure for T, and whose statistical basin of attraction
ON THE QUESTION OF ABSOLUTE CONTINUITY AND SINGULARITY OF PROBABILITY MEASURES
The basic result in this paper (Theorem 1) generalizes the well-known criterion of Kakutani to measures corresponding to arbitrary random sequences. The proof is based on Theorem 6, which gives a
The C1 generic diffeomorphism has trivial centralizer
Answering a question of Smale, we prove that the space of C1 diffeomorphisms of a compact manifold contains a residual subset of diffeomorphisms whose centralizers are trivial.
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