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Nonuniform hyperbolicity for C 1 generic diffeomorphisms
, 2008
"... We study the ergodic theory of nonconservative C 1generic 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 C 1ge ..."
Abstract

Cited by 4 (1 self)
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We study the ergodic theory of nonconservative C 1generic 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 C 1generic diffeomorphisms are nonuniformly 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 C 1generic 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 C 1.
Dominated splitting and zero volume for incompressible threeflows
 Nonlinearity
"... ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able ..."
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Cited by 3 (1 self)
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ABSTRACT. We prove that there exists an open and dense subset of the incompressible 3flows of class C 2 such that, if a flow in this set has a positive volume regular invariant subset with dominated splitting for the linear Poincaré flow, then it must be an Anosov flow. With this result we are able to extend the dichotomies of BochiMañé (see [26, 13, 9]) and of Newhouse (see [30, 10]) for flows with singularities. That is we obtain for a residual subset of the C 1 incompressible flows on 3manifolds that: (i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii) either the flow is Anosov or else the elliptic periodic points are dense in the manifold.