Ergodic Theory

Reading Group for Summer 2025

What

This is the webpage for the summer 2025 reading group on Ergodic Theory, organized by McGill students.
We are following notes by Professor A. Tserunyan, available on her website.
We meet twice a week, Monday and Wednesday from 4 to 5 in BURN 1B23.

Here are notes summing up what we did.

For more information, email ludovic.rivet [at] mail.mcgill.ca

Next Talks

Frédéric Kai, 09/07: Moore's Ergodicity Theorem
Description: Ergodicity of actions of lattices in Lie groups.

Past Talks

Samy Lahlou, 06/05: Crash Course on Measure Theory (Part 1) [Exercises]
Description: \(\sigma\)-algebras; measures; Lebesgue and Bernoulli measures; Borel \(\sigma\)-algebra.

Samy Lahlou, 08/05: Crash Course on Measure Theory (Part 2) [Exercises]
Description: Measurable functions; Isomorphism Theorems; Lebesgue integral; Convergence Theorems; Radon-Nikodym Derivative.

Bo Peng, 12/05: An introduction to Ergodic Theory
Description: Overview of dynamical systems theory, with emphasis to topological and measure systems; open questions and applications.

Zhaoshen Zhai, 14/05: Examples of Ergodic Transformations
Description: Ergodic equivalence relations; PMP and ergodic transformations; 99% lemma and ergodicity of irrational rotations, with an application to graph colouring; mixing and ergodicity of Bernoulli shifts; a functional characterization of ergodicity.
Exercise Session, 21/05: [Exercises].

Zhaoshen Zhai, 26/05: Birkhoff's Pointwise Ergodic Theorem
Description: Poincaré Recurrence; density characterization of ergodicity; Proof of Birkhoff's Pointwise Ergodic Theorem; applications.
Exercise Session, 28/05: [Exercises].

Ludovic Rivet, 04/06: An overview of Szemeredi's theorem
Description: Erdos and Turan conjectured that "dense" subsets of the naturals always contain arbitrarily long arithmetic progressions. Szemeredi proved it using a long combinatorial argument, and Furstenberg gave another proof using ergodic theory. We will present Szemeredi's theorem and neighboring results, and give an overview of Furstenberg's proof.
Exercise Session, 09/06: [Exercises].

Ludovic Rivet, 11/06: The actors in the Furstenberg-Zimmer theorem
Description: The Furstenberg-Zimmer theorem gives a classification of measure-preserving systems. We define the relevant objects.

Zhaoshen Zhai, 16/06: Szemerédi's Theorem for compact and weak-mixing systems
Description: We prove the Furstenberg Recurrence Theorem for compact systems and weak-mixing systems. Using the splitting of \(L^2\) into almost-periodic and weakly-mixing functions, we deduce Roth's Theorem (corresponding to the case \(k=3\)).
Exercise Session, 18/06 + 23/06: [Exercises].

Zhaoshen Zhai, 25/06: The Furstenberg-Zimmer Structure Theorem
Description: We prove the Furstenberg-Zimmer Structure Theorem for measure preserving dynamical systems. This is done by studying the dichotomy between conditionally almost periodic and weak mixing functions, which we formulate in the language of Hilbert modules. Added: We also proved that the Szemerédi property lifts up compact and weak mixing extensions, hence finishing the proof of Furstenberg's Multiple Recurrence.
Exercise Session, 30/06: [Exercises].