Dynamical Systems & Ergodic Theory
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Description
Dynamical systems and ergodic theory are two relatively new areas in Mathematics that study how the information of a system changes over time. They are related and, in many contexts, inseparable areas, and began at the end of the 19th century with the celebrated work on celestial mechanics by Henri Poincaré entitled Les méthodes nouvelles de la mécanique céleste, and with the work of Ludwig Boltzmann on statistical mechanics. Since then, both areas have witnessed tremendous advances. One of the great discoveries was the change from a quantitative perspective (unanimous until the 19th century) to a qualitative perspective, where the change of information over time is described by means of a probabilistic language (instead of analytical). This made it possible to better understand systems that are sensitive to initial conditions (in these systems it is impossible to predict, with accuracy, the behavior in a pre-determined period of time). In the qualitative approach, it is possible to measure various stochastic properties, such as mixing times, central limit theorem, and convergence in distribution.
Dynamical systems and ergodic theory make use of a broad language, therefore they interact with many other areas of Mathematics such as combinatorics, geometry, probability, number theory, topology, and also with other domains of knowledge such as astronomy, biology, economics, physics. By means of these two areas, nowadays we have a better understanding of the dynamics of gases, geodesic flows, atmosphere convection, biological populations, economic models, Schrodinger equation, and also, surprisingly, some properties on diophantine approximations of real numbers by rationals.
The research group of Dynamical Systems and Ergodic Theory at UFC studies, with mathematical rigor, dynamical phenomena related to the concept of chaos. The dynamical phenomena include the classification of systems and their invariant measures, the counting of periodic points, among others. The word chaos refers to the presence of hyperbolicity and its variants, such as partial hyperbolicity and non-uniform hyperbolicity. The tools that are used to understand the dynamical phenomena of such chaotic systems include linear cocycles, fractal dynamics, symbolic dynamics, invariance and rigidity, among others.
