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New development of entropy theory: III: Continuity of exponents for SL(2,R) cocycles.
From our new understanding on the partial entropy and invariant principle, we can show the regularity of exponents for some SL(2,R) cocycles over (partially) hyperbolic base.
With Chao and Karina, we may show that among C² partially hyperbolic sympletic diffeomorphisms with 2-d center with the accessibility property, having center exponents different from zero is an open property.
With Marcelo Viana, we show that the exponents of SL(2,R) cocycles over expanding base in general vary continuously in the C⁰topology. This result is suprising since similar result is false for the cycles over diffeomorphisms, by the results of Bochi-Ma\~n\'e.
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New development of entropy theory: IV: Larger entropy and hyperbolicity for 3-dimensional derived from Anosov diffeomorphisms.
Let A be a 3-dimensional linear Anosov diffeofmorphism, with exponents k₃<0<k₂<k₁$. A diffeomorphism in the same isotopy class of A is called derived from Anosov (DA) if it is partially hyperbolic. We will show how the constant k₁ will provide a classification of the DA diffeomorphisms.
With Marcelo Viana, we show that for every DA diffeomorphism, the ergodic measures with metric entropy larger than k₁ are hyperbolic, have the same structure as the Anosov diffeomorphism, and its disintegration along the center foliation is not atomic.
Moreover with Shi and Viana, we show that every volume preserving, C¹⁺ DA diffeomorphism with metric entropy is larger than k₁admits a C¹ neighborhood, such that every C¹⁺ diffeomorphism in this neighborhood is transitive, and its center-unstable foliation is absolutely continuous. The physical measures are also studied for these diffeomorphisms.