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].
  5. More on topological entropy [VO, Sec 10.1] and examples [Sec 10.2]. Intro to topological pressure [Sec 10.3].
  6. More on topological pressure, including the variational principle [VO, Secs 10.3--4].
  7. Completion of the variational principle [VO Sec 10.4]; equilibrium states [VO Sec 10.5]. The Brin--Katok local entropy formula [notes]. In this class I will use Shearer's inequality: see the references for 254A, Lecture 9.
In Lectures 10--13, we apply some of the theory we have learned to the study of fractal dimension for certain dynamically-generated sets and measures.
  1. Quick recap of fractal dimensions for sets and measures, and then theorems of Furstenberg [this paper, Prop III.1] and Billingsley [excerpt via CCLE] on sets and measures invariant under multiplication map of the circle.
  2. [Presented by Alex Mennen] Bowen's formula for Hausdorff dimension in the setting of cookie-cutter Cantor sets [following Bedford's survey, uploaded at CCLE, and along similar lines to VO Sec 12.4].
  3. [Presented by Angela Wu] The dynamical formula for Hausdorff dimension of certain fractals defined by iterated function systems, following theorems or Bowen and Ruelle [based on Section 6.5 of Keller's book ``Equilibrium States in Ergodic Theory'', uploaded at CCLE].
  4. [Presented by Redmond McNamara] Young's formula for the Hausdorff dimension of ergodic invariant measures for diffeomorphisms of surfaces [following the original paper; see also this survey for a more elementary sketch]. This lecture will assume a number of results from dynamical systems without proof.
In Lectures 14--21, we study the mathematical statistical mechanics of lattice models, starting with the general formalism and then looking closer at some well-known particular models. This builds on the introduction to statistical mechanics in 254A, lectures 12--14. The main reference up to lecture 18 is this survey by Lanford, cited as [Lan].
  1. A sketch of how some major results from earlier in the course (e.g. pointwise ergodic theorem, existence and properties of entropy rate) extend to the setting of measure-preserving actions of Z^d. Some of these are in [Lan, Section B.2]. Then: some classical background from convex analysis [notes].
  2. A recap of statistical mechanics for non-interacting particles, and then setting up the class of interacting models on lattices that we will be studying. For the former, see 254A, Lecs 12--14. For the latter I will follow [Lan], with a few modifications that are explained in these supplementary notes.
  3. Continued introduction to statistical mechanics of lattice models, following [Lan] and the supplementary notes above.
  4. [Presented by Fan Yang] The variational principle, equilibrium states and tangent functionals to the pressure. Following [Lan, Chapter B].

[No lectures on Feb 21 and 23. Lectures resume Feb 26.]

  1. [Presented by Andrew Krieger] Equilibrium states and Gibbs states; Dobrushin's uniqueness condition. Following [Lan, Chapter C].
  2. [Presented by Younghak Kwon] Peierl's argument for the existence of multiple phases in the 2D Ising model. [Based on Section 6.2 of Georgii's book "Gibbs Measures and Phase Transitions", uploaded at CCLE. These notes by Duminil-Copin are a good secondary reference.]
  3. [Presented by Tianqi Wu] The absence of phase transitions in one-dimensional lattice models. [Probably following Chapter 1, Sections A--D of this monograph by R. Bowen.]
  4. [Presented by Chris Shriver] A first look at dimer models of random surfaces. [Essentially following these notes by Chris, based on this survey by Kenyon.]
Lectures 22--27 are about entropy theory for measure-preserving actions for `big' non-amenable groups, particularly free groups of rank at least two. The central quantity here is Lewis Bowen's sofic entropy, with its many variants and consequences. Our guiding reference is Bowen's survey, cited as [Bow]; see also his shorter future ICM address for a quicker overview.
  1. Introduction to ergodic theory and entropy for non-amenable groups, I. I will draw material from [Bow] and also my own supplementary notes (which use my preferred notation and give some more careful proofs).
  2. Introduction to ergodic theory and entropy for non-amenable groups, II. Refs as above.
  3. [Presented by Assaf Shani] Sofic entropy and factor maps for Bernoulli shifts. Material taken from [Bow].
  4. [Presented by Josh Keneda] Introduction to the f-invariant. Material taken from [Bow] and also these original papers by Bowen. (Note: the ideas may be presented in non-historical order.)
  5. [Presented by Riley Thornton] Factors of free-group Bernoulli systems and IID processes on the tree. Largely following thus survey by Lyons, with some more results from [Bow].
  6. [Presented by David Jekel] Free entropy in random matrices and free probability. (Not part of sofic entropy, but intuitively related). Essentially following David's notes.
  7. [Presented by Puneet Sethi] Introduction to topological sofic entropy and the variational principle. Following [Bow] and the refs given there.


Tim Austin,
tim AT math DOT ucla DOT edu