# Category Archives: Ergodic Theory

## Entropy of measure preserving systems

A measure preserving system is a quadruple where is a set, is a -algebra, is a probability measure and is a measurable map satisfying for every . The notion of isomorphism in the category of measure preserving systems (defined, for … Continue reading

## Single and multiple recurrence along non-polynomial sequences

Vitaly Bergelson, Florian Richter and I have recently uploaded to the arXiv our new paper “Single and multiple recurrence along non-polynomial sequences”. In this paper we address the question of what combinatorial structure is present in the set of return … Continue reading

## Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension

This is the sixth and final post in a series about Szemerédi’s theorem. In this post I complete the proof of the Multiple Recurrence Theorem, which I showed in a previous post of this series to be equivalent to Szemerédi’s … Continue reading

## Gaussian systems

Examples of measure preserving systems with varied behaviours are vital in ergodic theory, to understand the general properties and to have counter examples to false statements. One classical method to craft examples with specific properties is the so-called Gaussian construction. … Continue reading

## The horocycle flow is mixing of all orders

— 1. Introduction — The main purpose of this post is to present a proof, due to Brian Marcus, that the horocycle flow is mixing of all orders. The precise definition of mixing of all orders for -actions is given … Continue reading

## Szemerédi’s Theorem Part V – Compact extensions

This is the fifth in a series of six posts I am writing about Szemerédi’s theorem. In the previous post I proved that the Sz property lifts through weak mixing extension and in this post I will prove that the … Continue reading

## Szemerédi’s Theorem Part IV – Weak mixing extensions

This is the fourth in a series of six posts I am writing about Szemerédi’s theorem. In the first three posts, besides setting up the notation and definitions necessary, I reduced Szemerédi’s theorem to three facts. Those three facts are … Continue reading