Abstract

By Birkhoff's ergodic theorem, for any dynamics preserving a finite invariant measure, the average of an integrable function is well defined for almost every orbit. However, for large classes of dynamics the set of points for which the averages do not exist may be very large from the points of view of topological entropy and Hausdorff dimension. Using examples from symbolic dynamics and hyperbolic dynamics I want to illustrate this and other peculiar phenomena beyond the apparent simplicity of Birkhoff's ergodic theorem. In particular, local quantities such as local entropy, pointwise dimension and Lyapunov exponents may possess a very complicated local structure. The necessary material from dimension theory and multifractal analysis will be recalled along the way.