M.Viana, Dynamics: a probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin,  Home/Search   Document Details and Download   Summary   Related Articles   Check

....In the space of volume preserving di eomorphisms, the C 1 closure of the stably transitive di eomorphims coincides with the closure of the di eomorphisms admitting a dominated splitting. 3 For a discussion of the dominated splitting condition and some results related to Conjecture 0. 3 see [Vi]. Even though the results of this paper could be useful in attacking this conjecture some other ideas (possibly ones from the paper [BonDi] are necessary to solve this problem. Here we note only that in [BV] a volume preserving example is presented which is stably transitive yet not partially ....

Viana, M., Dynamics: a probabilistic and geometric perspective Proc. ICM{98, Documenta Math. Extra Vol. I (1998) 557-578. 35

....but maintaining strong the unstable space. She proves the bounded distortion of backward iterates of unstable volume elements, to conclude that there exists a SRB measure. In this case, the H older continuitiy of the unstable foliation is used to compare unstable volume elements of nearby points. In [E 1998] a heteroclinic intersection of an Anosov di eomorphism is perturbed to obtain a cubic heteroclinic tangency. Also using that the distortion of unstable volume elements in backward iterates remains bounded, and the techniques of Pesin and Sinai in [PS 1982] the author proves that the SRB measure ....

....to prove that there exists an ergodic attractor, that the di eomorphism is Bernoulli, and that Lebesgue almost all regular point is in the Pesin region. These facts and the existence of the SRB measure are the content of the second part of Theorem 1, which is proved in Sections 3 and 4 of this work. In [V 1998] Viana includes the conjecture which states that smooth maps with only non zero 2 Lyapounov exponents for Lebesgue almost all points, admit SRB measures. The almost hyperbolic di eomorphisms studied in this work, in particular, are not hyperbolic with Lebesgue almost all regular points in the ....

[Article contains additional citation context not shown here]

M. Viana. Dynamics: a probabilistic and geometric perspective. Proc. of the Int. Congress of Math. (Berlin) Doc. Math. Extra Vol. I. (1998), pp 395-416. 25

....goes to zero we show (Theorem B) that time averages of every continuous taken along almost all random orbits of M are close, for small noise levels and in the weak sense, to the closed convex hull of the space averages f R d i ; i = 1; 2; g. This property has been proposed, e.g. [24], as a way to de ne stochastic stability for several attractors with respect to a given map f . The proofs provide another statement (Theorem C) for systems f : M M with nitely many stochastically stable attractors ( 1 ; 1 ) N ; N ) such that every trapping region an open set U ....

....in addition to hypothesis A, the attractors are stochastically stable, then averages of every continuous along almost all random orbits of m almost every point x 2 M are near R d i for some i 2 f1; Ng when the noise level is close to zero. This behavior has been proposed, cf. e.g. [24], as a de nition of stochastic stability for a family of attractors with respect to the same map f . 8 Moreover the orbital averages of any xed point x de ne a probability measure (x) on the manifold that is close, in the weak sense, to the closure of the convex hull co f i g N i=1 ....

Viana, M., Dynamics: A Probabilistic and Geometric Perspective, Proceedings ICM 1998 { Documenta Mathematica.

....if and only if the zeta function has a simple pole at 1 as only singularity in the larger disc. See also [75] and references therein. 3.3. Partially hyperbolic di eomorphisms Partially hyperbolic di eomorphisms naturally generalise uniformly hyperbolic diffeomorphisms (see e.g. Sections 7 8 of [98] for an overview, including references to fundamental work by Brin Pesin and Pesin Sinai, as well as links with the theory of robust transitivity) The tangent bundle TM is assumed to split as either E c E u , with E u uniformly expanding and dominating the expansion in E c (the central ....

....such as good unimodal maps, results almost as strong, including L 1 stability of the density and some spectral stability, were obtained by Baladi Viana [17] Weaker stability had been previously proved by Katok Kifer, Collet, and Benedicks Young in similar settings. Benedicks and Viana [98] proved stochastic stability (weak convergence of the Markov chain invariant measure towards the SRB measure) of good H enon maps. Instead of considering the averaged behaviour given by a Markov chain, one may investigate almost sure random behaviour, i.e. consider the two sided Spectrum and ....

M. Viana, Dynamics: a probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians held in Berlin, August 18-27 1998, Doc. Math., Extra Vol. I (electronic), (Bielefeld), 1998, 557-578.

....0 2 M and any y 0 close to it, almost surely, the corresponding trajectores diverge from each other in the future, exponentially fast. More precisely, all the Lyapunov exponents are positive, almost everywhere. This result motivates the following conjecture that I formulated for the rst time in [Via98]. 4 Conjecture 3. Any smooth system whose Lyapunov exponents are non zero almost everywhere (with respect to volume) has physical measures (a nite number of them if the Lyapunov exponents are bounded from zero) ....

M. Viana. Dynamics: a probabilistic and geometric perspective. In Procs. International Congress of Mathematicians ICM98-Berlin, Documenta Mathematica, vol I, pages 557-578. DMV, 1998.

No context found.

M.Viana, Dynamics: a probabilistic and geometric perspective, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin,

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC