Impossibility results in classical dynamical systems.

In 1932, motivated by questions in statistical and celestial mechanics, von Neumann proposed classifying the statistical behavior of dynamical systems. In the 1960’s, motivated by work of Poincaré, Smale proposed classifying the qualitative behavior of dynamical systems. These questions laid the groundwork for enormous amounts of work, but the fundamental questions remain open. This talk shows that they are impossible to answer in a rigorous sense. The talk will discuss various kinds of impossibility results and describe how they apply to von Neumann’s program and Smale’s program.