Dynamical Systems Seminar

The metric entropy of the SRB measure measures the level of complexity and is a smooth invariant. It defines a function defined in the space of smooth conjugacy classes. In this talk, I will mention a joint work with Huyi Hu and Miaohua Jiang about constructing two smooth paths in Axiom A (or Anosov) dynamical systems and area-preserving Anosov dynamical systems. The metric entropy takes any value from 0 to the topological entropy on these two paths. Our results lead to the recent Katok flexibility and rigidity program. I will discuss constructing a smooth path in expanding Blaschke products, with an almost expanding Blaschke product as a limit. It turns out that this almost expanding Blaschke product is smoothly conjugate to the famous Boole map on the real line. Thus, this gives a new explanation of Boole's formula, discovered more than one hundred years ago. By modifying this path, we construct another smooth path of area-preserving expanding Blaschke products with a totally degenerate map as a limit. These two paths represent the same smooth path in the space of all smooth conjugacy classes. We then obtain a global graph of the metric entropy, which looks like a bell. Finally, I will apply this global graph to the main cardioid of the Mandelbrot set.