We will talk about a paper by A. Moreira and J.C. Yoccoz, where they proved a conjecture by Palis according to which the arithmetic sums of generic pairs of regular Cantor sets on the line either has zero Lebesgue measure or contains an interval.

In this series of three lectures I will consider SRB measures (after Sinai, Ruelle and Bowen) which arguably form one of the most important classes of invariant measures with “chaotic” behavior in dynamics. This ensures a crucial role they play in applications of dynamical systems to science (this is why they are often called “physical measures”).

In the first lecture I introduce these measures and describe their ergodic properties. I also outline a construction of SRB measures for Anosov systems.

In the second lecture I consider the case of partially hyperbolic dynamical systems and outline a construction of SRB measures in this case. I also discuss an important role SRB measures play in the Pugh-Shub Stable Ergodicity problem (and I will also discuss this problem in a general setting).

The third lecture deals with the most general situation of the so-called “chaotic” attractors (or attractors with non-zero Lyapunov exponents), which appear in many models in physics, biology, etc. I will present a general rigorous definition of chaotic attractors and outline a construction to build SRB measures for this attractors.

While the first lecture will serve as an introduction to the subject and will be accessible for students, the other two lectures are more advanced.

In this series of three lectures I will consider SRB measures (after Sinai, Ruelle and Bowen) which arguably form one of the most important classes of invariant measures with “chaotic” behavior in dynamics. This ensures a crucial role they play in applications of dynamical systems to science (this is why they are often called “physical measures”).

In the first lecture I introduce these measures and describe their ergodic properties. I also outline a construction of SRB measures for Anosov systems.

In the second lecture I consider the case of partially hyperbolic dynamical systems and outline a construction of SRB measures in this case. I also discuss an important role SRB measures play in the Pugh-Shub Stable Ergodicity problem (and I will also discuss this problem in a general setting).

The third lecture deals with the most general situation of the so-called “chaotic” attractors (or attractors with non-zero Lyapunov exponents), which appear in many models in physics, biology, etc. I will present a general rigorous definition of chaotic attractors and outline a construction to build SRB measures for this attractors.

While the first lecture will serve as an introduction to the subject and will be accessible for students, the other two lectures are more advanced.

Inspired by a recent work of Marcin Sabok, we define a criterionfor a homogeneous metric structure X that implies that its automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space, regarded as metric structures, thus proving that their automorphism groups share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group with a left-invariant, complete metric, is trivial, and we verify it for the Urysohn space, and the Hilbert space.