
ВИДЕОТЕКА 
Конференция «Hyperbolic Dynamics and Structural Stability», посвященная 85летию Д. В. Аносова



Attractors of Direct Products S. S. Minkov^{a}, I. S. Shilin^{b} ^{a} Institute of Electronic Control Machines, Moscow ^{b} State University – Higher School of Economics 

Аннотация: For Milnor, statistical, and minimal attractors, we construct examples of smooth flows We are interested in attractors definitions of which rely on a natural (in our case, Lebesgue) measure on the phase space, which allows these attractors to capture asymptotic behavior of most points while possibly neglecting what happens with a set of orbits of zero measure. One type of such attractors was introduced by J. Milnor in [M] under the name “the likely limit set”. We refer to it as the Milnor attractor. The Milnor attractor Another way to define an attractor is via an “attracting” invariant measure: the attractor is its support. Most suitable are the notions of physical and natural measures, which may be viewed as analogues of SRBmeasures for general, nonhyperbolic dynamical systems (see, e.g., [BB]; we adapt the definitions from [BB] to the case of flows). Physical measures describe the distribution of For a flow $$\lim\limits_{T \to +\infty}\frac{1}{T}\int_0^T f(\varphi^t(x)) dt = \int_X f\, d\nu.$$ A measure $$\frac{1}{T}\int_0^T \varphi^t_*\tilde{\mu} \;dt \to \nu , T \to +\infty.$$ Statistical and minimal attractors are supports of these measures, respectively. When constructing examples of dynamical systems with required properties, it is not uncommon to utilize, at least as a piece of the construction, direct products of systems in lower dimensions. It is tempting to think that the attractor of the direct product of two systems always coincides with the direct product of their attractors. Although this holds, indeed, for socalled maximal attractors, this is not true for several other types of attractors, namely, for Milnor, statistical, and minimal attractors, and also for the supports of physical measures, when the latter exist. We present examples of smooth flows on [BB] Blank, M., Bunimovich L.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity, 16:1, 387–401 (2003) [M] Milnor, J.: On the concept of attractor. Comm. Math. Phys. 99, 177–195 (1985) Язык доклада: английский 