June 2nd, 2020
First lecture Thi Ndang
Title : Margulis-Ruelle inequality (2)
Abstract: Let M be a compact smooth manifold, f a smooth transformation, m a probability measure. This lecture centers on measurable entropy. I'll define this notion starting from the information function of a partition of M wrt m. Then I'll sketch Ruelle's proof of the inequality. Now giving ourselves a partition ensures an encoding by a subshift of f. Markov partitions allow a one to one correspondance between points of M and a biinfinite sequence of symbols. I'll sketch how to construct them for hyperbolic dynamical systems. Finally, basing myself on a survey of Young, I'll define SRB measures. In particular, they realize the equality in the Margulis Ruelle inequality.
Second lecture Alessandro Carderi:
Title: An introduction to percolation theory
Abstract: In this lecture we will see some basic facts about the theory of percolation on Cayley graphs. Percolations are random subgraphs of the Cayley graphs and we will interested in the case of the graphs obtained by randomly, independently and uniformly keeping or deleting each single edge with a fixed probability. We will study the connected component of the obtained graphs, try to understand when they are finite or infinite and consider the phase transitions between these phenomenons. Finally we will see how one can understand these probabilistic objects as graphings, or Cayley graphs of equivalence relations, introduced in the previous lecture.