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Category Archives: Ergodic Theory
Single and multiple recurrence along nonpolynomial sequences
Vitaly Bergelson, Florian Richter and I have recently uploaded to the arXiv our new paper “Single and multiple recurrence along nonpolynomial sequences”. In this paper we address the question of what combinatorial structure is present in the set of return … Continue reading
Posted in Ergodic Theory, paper
Tagged Bergelson, Haland, Nonpolynomial sequences, Richter, Syndetic sets, Thick sets
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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
Posted in Ergodic Theory, Ramsey Theory
Tagged Compact extensions, joinings, Szemeredi's theorem, weak mixing
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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 socalled Gaussian construction. … Continue reading
Posted in Classic results, Ergodic Theory
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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
Posted in Analysis, Classic results, Ergodic Theory
Tagged flows, horocycle flow, Marcus, mixing of all orders, van der Corput
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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
Posted in Ergodic Theory
Tagged almost periodic, Compact extensions, Furstenberg, Szemeredi's theorem
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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
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
Posted in Combinatorics, Ergodic Theory, Ramsey Theory
Tagged almost periodic, skew product, Szemeredi's theorem, weak mixing
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