L.-S. Young, Ergodic theory of chaotic dynamical systems, XIIth International Congress of Mathematical Physics (ICMP'97) (Brisbane), 131-143, Internat. Press, Cambridge, MA, 1999. 33  Home/Search   Document Details and Download   Summary   Related Articles   Check

....by L. S. Young [Y1] The success in these studies had signi cant impact on modern statistical mechanics. The methods and ideas originally developed for the planar Lorentz process were applied to many other classes of physical models see recent reviews by Cohen, Gallavotti, Ruelle and Young [GC, Ru, Y2]. On the other hand, the progress in the study of the multidimensional Lorentz process (where d 2) has been much slower and somewhat controversial. Relatively few papers were published covering speci cally the case d 2, especially in contrast to the big number of works on the 2 D case. ....

L.-S. Young, Ergodic theory of chaotic dynamical systems, XIIth International Congress of Mathematical Physics (ICMP'97) (Brisbane), 131-143, Internat. Press, Cambridge, MA, 1999. 33

....say, extended dynamical systems are usually regarded as topological dynamical systems. In contrast, most of the very deep results in finite dimensional dynamical systems use measure theoretic ideas, namely ergodic theory (as advocated for instance in the review by L. S. Young at the ICMP in 1997, [Y2]) One of the favourite models of infinite dimensional dynamical systems studied recently is the Ginzburg Landau equation. It appears as a generic normal form describing the amplitude of periodic bifurcated solutions (see [C2] and it is also believed to be a good example of spatiotemporally ....

Young L.-S., Ergodic Theory of Chaotic Dynamical Systems, in XIIIth International Congress of Mathematical Physics (ICMP'97), Brisbane, Internat. Press, Cambridge, MA, 1999.

....Sinai Ruelle Bowen (SRB) measures. Those measures are characterized by smooth conditional distributions on unstable manifolds. The SRB measures are the only physically observable measures, they are called nonequilibrium steady states in the language of statistical mechanics. We refer the reader to [GC, Ru, Y2] for more discussion on SRB measures and their role in hyperbolic dynamics and physics. There is a remarkable example of the system (1.1) 1.2) well studied in the literature. Let F be a small constant electric force, possibly combined with a small magnetic force, with a Gaussian thermostat added, ....

L.-S. Young, Ergodic theory of chaotic dynamical systems, XIIth International Congress of Mathematical Physics (ICMP'97) (Brisbane), 131--143, Internat. Press, Cambridge, MA, 1999. 40

....is a finite Lebesgue equivalent f invariant Sigma invariant ergodic measure. Lebesgue equivalent means that the sets of measure zero are the same for the ergodic measure and the Lebesgue measure induced on the one dimensional sets) Moreover, the measure is a SinaiBowen Ruelle measure. See [16] for information about SBR measures. 5 There are several related issues to those discussed in this paper that we have passed over. For example, we could ask what additional restrictions are imposed on the symmetry groups of (connected) attractors for differentiable maps, homeomorphisms, ....

L-S. Young. Ergodic theory of chaotic dynamical systems, preprint, 1990. 28

....material was presented and discussed in his 1 We take some of the conclusions of the theorems of Sinai, Ruelle, and Bowen as a de nition and warn the reader that the use of SRB measure or attractor is not uniform in the literature. For a survey of SRB measures (using a di erent de nition) see [You2]. 7 seminar during March 1998. The question of whether perturbing a skew product di eomorphism over an Anosov could produce nonzero exponents was raised by Lai Sang Young during a conversation about these questions. We thank her for reminding us that examples such as the ones we construct might ....

Young, L.-S., Ergodic Theory of Chaotic Dynamical Systems. International Congress of Mathematical Physics, 1997. 17

....point spaces relative to which A Gamma still remains stable after the bifurcation. For this we recall a specific version of a result from [1] For an introduction of the notions SBR measure or nonuniformly hyperbolic invariant sets occurring in the following theorem the reader is refered to [31]. Theorem 3.8 Let V be a flow invariant subspace of R n of dimension m n and let A be a nonuniformly hyperbolic invariant set inside V . Suppose that there exists a corresponding SBR measure for which the n Gamma m normal Lyapunov exponents are negative. Then there is a set of positive ....

L.-S. Young. Ergodic theory of chaotic dynamical systems, (1992), preprint.

....was generic in Diff 2 (M ) 1 We take some of the conclusions of the theorems of Sinai, Ruelle, and Bowen as a definition and warn the reader that the use of SRB measure or attractor is not uniform in the literature. For a survey of SRB measures (using a different definition) see [You2]. Pesin s question is answered in the negative by Cheng and Sun [CS] and Herman. In [He] see also [Yoc] Herman showed that on any manifold M of dimension at least 2, and for sufficiently large r, there are open sets of volume preserving C r diffeomorphisms of M all of which posess positive ....

Young, L.-S., Ergodic Theory of Chaotic Dynamical Systems. International Congress of Mathematical Physics, 1997.

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