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Infinite-step nilsystems, independence and complexity

Published online by Cambridge University Press:  09 December 2011

PANDENG DONG ,
SEBASTIÁN DONOSO ,
ALEJANDRO MAASS ,
SONG SHAO  and
XIANGDONG YE
PANDENG DONG
Affiliation:
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, University of Science and Technology of China, Hefei, Anhui, 230026, PR China Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
SEBASTIÁN DONOSO
Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email: sdonoso@dim.uchile.cl, amaass@dim.uchile.cl)
ALEJANDRO MAASS
Affiliation:
Centro de Modelamiento Matemático and Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile (email: sdonoso@dim.uchile.cl, amaass@dim.uchile.cl)
SONG SHAO
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
XIANGDONG YE
Affiliation:
Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China (email: dopandn@mail.ustc.edu.cn, songshao@ustc.edu.cn, yexd@ustc.edu.cn)
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Abstract

An ∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an ∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial 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 non-trivial open cover.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 1 , February 2013 , pp. 118 - 143
Copyright
Copyright © Cambridge University Press 2013

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