Math 254B: Entropy and Ergodic Theory, continued (UCLA, Winter 2018)

Summary: A continuation of 254A, studying entropy and its many roles in different branches of mathematics, especially information theory, probability, dynamics and geometry.

Currently planned for 254B:

  1. Topological entropy of a topological dynamical systems and its relationship to Kolmogorov--Sinai entropy in ergodic theory.
  2. Hausdorff dimensions of various dynamically-generated fractals.
  3. Statistical physics of lattice gases and the thermodynamic formalism.
  4. Sofic entropy and related notions for actions of non-amenable free groups.

Other topics may be added, depending on the time available and student demand.

I will present on the first topic above and give some of the lectures on the others. The rest of the lectures will be given by students, arranged into teams by subject.

Material will be taken from a wide variety of sources, including both books and papers. I will make these available to students as necessary.

Lectures: Mondays, Wednesdays and Fridays, 2--2.50, in Math Sciences 7608.

Office Hours: Mondays 10.30--11.30 and Wednesdays 3--4.

Communication: Email is preferred. I will try to answer emails within 24 hours, excluding weekends. I prefer the pronouns he/him/his.

Grading: Enrolled students will be assigned letter grades based on participation (including giving some of the lectures).

Students who did not attend 254A are welcome. Please refer to the 254A website for the material from that course. Only the more basic parts are needed for 254B. Please ask me directly for more details.

Lectures, with links to resources:

  1. Recap of some basic results from 254A
  2. The pointwise ergodic and Shannon--McMillan--Breiman theorems
After some overrun from Lectures 1 and 2, we'll cover some topological dynamics, largely drawing on this book by Viana and Oliveira (available within UCLA network). Below I cite this work as [VO].
  1. Definition and some examples of topological dynamical systems. Mostly from [VO] Section 1.3.
  2. Conjugacies and semi-conjugacies (= topological factors) [a bit hidden in VO, use index]; the existence of invariant measures [VO, Chap 2]; ergodicity and extreme points [VO, Sec 4.3 & 254A, Lec 20].
  3. K--S entropy in topological dynamics [VO, Secs 9.1 and 9.2]; affinity of K--S entropy [VO, Sec 9.6].
  4. Intro to topological entropy [VO, Sec 10.1].

Tim Austin,
tim AT math DOT ucla DOT edu