Category Archives: Ergodic Theory

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

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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

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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

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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

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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

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Szemerédi’s Theorem Part III – Precise definitions

This is the third in a series of six posts on Szemerédi’s theorem. In the previous post I outlined the ideas of the ergodic theoretical proof by Furstenberg. In this post I will set up the machinery and give the … Continue reading

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Szemerédi’s Theorem Part II – Overview of the proof

This is the second in a series of posts about Szemerédi’s theorem. In the first post I presented the first step in the proof of Szemerédi’ theorem, namely applying the correspondence principle of Furstenberg to transform the problem into one … Continue reading

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