Since approximately 1985 a rather complete theory has been developed for certain non-uniformly hyperbolic dynamical systems, which exhibit a complicated mixture of chaotic and periodic behaviour. In particular, now the dynamics of real quadratic polynomials on the real line is essentially fully understood. In particular for these H non maps we now know that the iterates of Lebesgue almost all initial points in the topological basin of attraction have the same statistics.
Many problems however remain open.
In the talk I will describe the development in this area and also some of the conjectures and open problems.