Infinite-step nilsystems, independence and complexity

2011/05/18 by Pandeng Dong, Sebastián Donoso, Dong, P. D. +7 · 2 citations
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.1105.3584

openalex publication_date 2011/05/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no nontrivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without nontrivial pairs with arbitrarily long finite IP-independence sets is an almost one to one extension of its maximal ∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some ∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an ∞-step nilsystem is computed, showing that it is polynomial for each nontrivial open cover.

Citations

• Convergence of multiple ergodic averages

Cited by

• Almost Countable Spectrum and Logarithmic Sarnak Conjecture
• CF-Nil systems and convergence of two-dimensional ergodic averages

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