Dynamical Systems and Ergodic Theory

At the end of the nineteenth century Poincaré becomes interested in the study of Celestial Mechanics seeking to understand more specifically the evolution of our Solar system.

While the approach until then had been directed at solving differential equations of movement analytically or numerically, Poincaré proposes the use of other tools from other areas, such as Topology, Geometry, and Algebra and Analysis, in order to obtain a qualitative and, whenever possible, quantitative description of the behavior of the system. This proposal, which goes back to his thesis, marks the birth of Dynamical Systems as a mathematical subject and is focused on developing a theory capable of foreseeing the evolution of natural and human phenomena observable in various fields of knowledge. This subject has received major contributions from some of the greatest mathematicians of the twentieth century, such as Lyapunov, Andronov, Birkhoff, and Kolmogorov.

Throughout, its scope has been greatly amplified, and now includes other models of evolution in time, aside from differential equations: iterations of transformations, equations of differences, partial differential equations of evolution, transformations and stochastic differential equations. Concomittantly, the application of results and methods of Dynamical Systems has intensified to explain complex phenomena in several sciences: Chemistry (chemical reactions, industrial processes), Physics (turbulent flow, phase transition, optics), Biology (competition among species, neurobiology), Economics (economic growth models, financial market) and many others.

Among the tools used by Poincaré was the study of invariant probability measures as regards the action of the system, which is the focus of the Ergodic Theory. Actually, this subject goes back to the work of Boltzmann, Maxwell and Gibbs, who established the Cynetic Theory of Gases in the latter part of the nineteenth century. The ergodic theorems proved by Birkhoff and Von Neumann in the early decades of the twentieth century were the basis for this subject, which would become notably successful within the scope of differentiable dynamical systems. The main reason was the verification, based on Lorenzs work on convection and time forecasts, that stochasticity is not an appanage of complex systems: even deterministic phenomenon with simple evolutionary laws can behave in apparently unpredictable ways, since their trajectory depends on the sensitive initial state (chaotic behavior). The Ergodic Theory can then lead to a very detailed description of this behavior in statistical or probabilistic terms.

The Dynamical Systems research group at IMPA studies the main areas of current interest in dissipative Dynamics (which focuses on general systems without making hypotheses about invariant measures) as well as on important directions in Conservative Dynamics (in which one supposes there is a special invariant measure, translating some conservation law).

The current lines of research at IMPA are:

• Strange Attractors, physical measures, stochastic stability;
• Homoclinical Bifurcation and Fractional Dimensions;
• Sympletic Dynamics;
• Uni-dimensional dynamics;
• Lyapunov Exponents and non uniform hyberbolicity;
• Partial hyperbolicity, dominated decomposition, dynamic robustness.