J.F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....setting, one cannot expect to obtain a description from the topological dynamic point of view. In the former, there is no general theory regarding its topological dynamic consequences (although there are many important results from the ergodic point of view, see for instance [BP] PuSh] [ABV], CY] There is also another category which includes the partial hyperbolic systems: dominated splitting. An f invariant set is said to have dominated splitting if we can decompose its tangent bundle in two invariant subbundles T M = E F; for all x 2 ; n 0: with C 0 and 0 1: ....

J. Alves, C. Bonatti, M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140 (2000), 351-398.

....Theorem 10. Let f be a partially hyperbolic C diffeomorphism on a manifold M . We have 1. 18] If the central direction is mostly contracting, then the Gibbs u states of f are SRB measures, there are finitely many of them, and their basins cover a full Lebesgue measure subset of M . 2. [3]) If the central direction is mostly expanding, then Lebesgue almost every point is in the basin of some SRB measure. If the central Lyapunov exponents are bounded away from zero then there are finitely many SRB measures. Pushing part 1 of the theorem further on, Castro [24] has just proved ....

....measure and, indeed, the basins of such measures cover almost all of M . There is a version of this last result for piecewise smooth maps, assuming that most points do not visit the singular set (where the map fails to be smooth, or the derivative fails to be surjective) too close too often; see [3]. Such results suggest that non uniform hyperbolicity may suffice for a system to have good statistical properties. In this spirit, I state the following Conjecture: If a smooth map has only non zero Lyapunov exponents at Lebesgue almost every point, then it admits some SRB measure. ....

J. Alves, C. Bonatti, and M. Viana. SRB measures for partially hyperbolic systems whose central direction is mostly expanding. In preparation.

....(jj jj 2 ) 2 d 1 d 2 d 3 : See [40, 38] for other proofs, all proofs proceed by using Taylor series for sine and cosine etc. Applying Proposition 1 we obtain Corollary 3. If L( 6= 0 then for almost all x lim N 1 1 N ln jjdF N n jE c jj(x) 6= 0: Combining this corollary with [1, 4] we obtain Corollary 4. If L( 6= 0 then for large n F n has unique u Gibbs states and its basin of attraction has total Lebesgue measure in M: Lemma 6. If f; preserve a smooth measure m and L( 6= 0 then W c (F n ) is not absolutely continuous for large n: Proof. Without the loss of ....

....manifolds (see [22] In particular the results of Sections 1 and 2 are not needed. cf. 3, 38] where non ergodic examples with singular center are given. However to understand the dynamics of these examples theory given above is helpful. For more detailed description of this dynamics see [1, 4, 15, 39, 40]. 20 DMITRY DOLGOPYAT 2.4. Fractional parts of linear forms. Here we will describe an application of u Gibbs states to number theory. This example is taken from [33] It will use translation on SL d (R) SL d (Z) which is has nite volume but is not compact. However Theorem 2 and Corollary 2 ....

Alves J. F., Bonatti C. & Viana M. SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000) 351-398.

....that there are no robustly transitive di eomorphisms on the sphere S . DPU, Theorem H] veri es this conjecture in a more restrictive context. State the ergodic properties of these systems, in particular, the existence and niteness of SRB measures (for results on this subject see [BV, ABV, Do] In this paper we will begin the topological description of the strong stable or unstable foliation, motivated by the two following ideas: For describing the global dynamics of strongly partially hyperbolic di eomorphisms one often attempts to propagate topological or measurable local ....

J. Alves, Ch. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Inventiones Math., 140, 351-398, (2000).

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J.F. Alves, C. Bonatti and M. Viana, SRB measures for partially hyperbolic systems whose central direction is mostly expanding, Invent. Math. 140 (2000), 351-398.

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