TY - JOUR

T1 - Norm convergence of continuous-time polynomial multiple ergodic averages

AU - Austin, Tim

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/4

Y1 - 2012/4

N2 - For a jointly measurable probability-preserving action τ:R D(X,μ) and a tuple of polynomial maps pi:R→R D, i=1,2,..,k, the multiple ergodic averages 1/T0T (f1τ(t) p1}) (f2 τp2(t).. (f kτ pk(t))dt converge in L 2(μ) as T→∞ for any f1,f2,..,fk∈L ∞(μ). This confirms the continuous-time analog of the conjectured norm convergence of discrete polynomial multiple ergodic averages, which in its original formulation remains open in most cases. A proof of convergence can be given based on the idea of passing up to a sated extension of (X,μ,τ) in order to find a simple partially characteristic factor, similarly to the recent development of this idea for the study of related discrete-time averages, together with a new inductive scheme on tuples of polynomials. The new induction scheme becomes available upon changing the time variable in the above integral by some fractional power, and provides an alternative to Bergelsons polynomial ergodic theorem induction, which has been the mainstay of positive results in this area in the past.

AB - For a jointly measurable probability-preserving action τ:R D(X,μ) and a tuple of polynomial maps pi:R→R D, i=1,2,..,k, the multiple ergodic averages 1/T0T (f1τ(t) p1}) (f2 τp2(t).. (f kτ pk(t))dt converge in L 2(μ) as T→∞ for any f1,f2,..,fk∈L ∞(μ). This confirms the continuous-time analog of the conjectured norm convergence of discrete polynomial multiple ergodic averages, which in its original formulation remains open in most cases. A proof of convergence can be given based on the idea of passing up to a sated extension of (X,μ,τ) in order to find a simple partially characteristic factor, similarly to the recent development of this idea for the study of related discrete-time averages, together with a new inductive scheme on tuples of polynomials. The new induction scheme becomes available upon changing the time variable in the above integral by some fractional power, and provides an alternative to Bergelsons polynomial ergodic theorem induction, which has been the mainstay of positive results in this area in the past.

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U2 - 10.1017/S0143385711000563

DO - 10.1017/S0143385711000563

M3 - Article

AN - SCOPUS:84858690804

VL - 32

SP - 361

EP - 382

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 2

ER -