A class of dynamical systems with a compact phase manifold was studied in detail. For systems of this class the behavior of all trajectories is maximally unstable (technically, these systems are completely and uniformly hyperbolic). These systems are now called "Anosov systems"; their theory is the prototype of a number of subsequence works on the systems with the hyperbolic behavior of trajectories, for which the compete and uniform hyperbolicity condition is weakened or modified in some way. A smooth version of the method of approximating by periodic transformations was suggested (jointly with A. B. Katok). This lead to construction of dynamical systems with unexpected ergodic properties. Constructions in the theory of regular linear systems of ordinary differential equations in the complex domain were simplified. This fascilitated the work in this area. Last time I studied geometric questions related to the behavior of trajectories of flows on surfaces when lifted to the covering plane.
Main publications:
D. V. Anosov, Geodezicheskie potoki na zamknutykh rimanovykh mnogoobraziyakh otritsatelnoi krivizny, Trudy MIAN, XC, Nauka, M., 1967
D. V. Anosov, A. B. Katok, “Novye primery v gladkoi ergodicheskoi teorii. Ergodicheskie diffeomorfizmy”, Trudy Moskovskogo matematicheskogo obschestva, 23, 1970, 3–36
D. V. Anosov, A. A. Bolibruch, The Riemann–Hilbert problem, A publication from the Steklov Institute of Mathematics, Aspects Math., E22, Friedr. Vieweg & Sohn, Braunschweg, 1994
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