Theorems and tools in one-dimensional dynamics

The first lecture will be an accessible introduction (peut-être en français) to smooth one-dimensional dynamics. After some historical examples and results, we shall consider the dynamics of some maps $f_a : x \mapsto ax(1-x)$ from the quadratic family. In what follows, we shall investigate typical behaviour from the probabilistic viewpoint. In particular, we shall show, under certain conditions, the existence of probability measures which describe the statistical behaviour of Lebesgue almost every orbit. Time permitting, the Markov extension, together with its utility in proving results concerning continuity of statistical properties, will be presented.