Nonuniform hyperbolicity for C1-generic diffeomorphisms
By F. Abdenur, C. Bonatti, S. Crovisier. 2008
146 citations recorded by the source corpus.
We study the ergodic theory of non-conservative C1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C1-generic diffeomorphisms are non-uniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set Λ of any C1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.In addition, confirming a claim made by R. Mañé in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin’s Stable Manifold Theorem, even if the diffeomorphism is only C1.
Subfields: Cell Biology, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/6fd08717c2799d3bf58b38e9b61891c4fecc0822
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Citations in the corpus
- Essential hyperbolicity and homoclinic bifurcations: a dichotomy phenomenon/mechanism for diffeomorphisms (2010, background): “Proof Let us choose a smooth map φ : Rs+1 → [0, 1] supported on U1 which takes the value 1 on U2 and a smooth map ψ : Ru → [0, 1] supported on the”