Universal Characteristic Factors and Furstenberg Averages

Abstract

Let X=(X^0,μ,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X is characteristic for averaging schemes of length k (or k-characteristic) if for any non zero distinct integers a_1,...,a_k, the limiting L^2(μ) behavior of the averages in (*) is unaltered if we first project the functions f_j onto the factor. A factor of X is a k-universal characteristic factor (k-u.c.f)} if it is a k-characteristic factor, and a factor of any k-characteristic factor. We show that there exists a unique k-u.c.f, and it has a structure of a (k-1)-step nilsystem, more specifically an inverse limit of (k-1)-step nilflows. Using this we show that the averages in (*) converge in L^2(μ). This provides an alternative proof to the one given by Host and Kra in 2002.

47 pages

References in corpus (2)

• A Non Conventional Ergodic Theorem for a Nil-System
• Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

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• Norm convergence of multiple ergodic averages on amenable groups
• Convergence of Diagonal Ergodic Averages
• Strictly ergodic distal models and a new approach to the Host-Kra factors
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• A spectral refinement of the Bergelson-Host-Kra decomposition and new multiple ergodic theorems
• Seminorms for multiple averages along polynomials and applications to joint ergodicity
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• Topological Wiener-Wintner ergodic theorem with polynomial weights
• Multiple recurrence and convergence for certain averages along shifted primes
• On nilspace systems and their morphisms
• Khintchine-type recurrence for 3-point configurations
• Nilsystems and ergodic averages along primes
• A good universal weight for nonconventional ergodic averages in norm
• Host-Kra theory for ${⨁}_{p\in P}{\mathbb{F}}_p$- Systems and multiple recurrence
• Generic properties of extensions
• Disjointness for measurably distal group actions and applications
• Value patterns of multiplicative functions and related sequences
• Host-Kra factors for ${⨁}_{p\in P}\mathbb{Z}/p\mathbb{Z}$ actions and finite dimensional nilpotent systems
• Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups
• Weakly mixing sets and polynomial equations
• Multiple recurrence and popular differences for polynomial patterns in rings of integers
• Van der Corput's difference theorem for amenable groups and the left regular representation