Dmitrij V. Anosov Reihenfolge der Bücher

This volume focuses on the "hyperbolic theory" of dynamical systems (DS), specifically smooth DS's exhibiting hyperbolic behavior in their trajectories. Hyperbolicity refers to the property where a small displacement of a point on a trajectory leads to significant changes in the relative positions of the original and displaced points over time, akin to motion near a saddle. When there are "sufficiently many" such trajectories in a compact phase space, they tend to diverge yet remain constrained, resulting in complex behavior often associated with "chaos" in physical literature. This behavior contrasts with the more straightforward stability and regularity typically observed in other dynamical systems. While the ergodic properties of hyperbolic DS's have been explored in Volume 2 of this series, this volume primarily addresses topological properties. For further details, see section 2.

The Riemann-Hilbert problem (Hilbert's 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem in its original form.