2015/05/18 by Terence Tao, Tao, Terence
Mathematics · #37A30 #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #math.DS #msc:37A30
paper · pdf · doi:10.48550/arxiv.1505.04725
16 pages, 3 figures
arxiv created 2015/05/18 · openalex publication_date 2015/05/18 · arxiv updated 2015/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F2 denote the free group on two generators a,b. For any measure-preserving system (X, \mathcal X, μ, (Tg)g ∈ F2) on a finite measure space X = (X,\mathcal X,μ), any f ∈ L1(X), and any n ≥ 1, define the averaging operators \mathcal An f(x) := \frac14 × 3n-1 ∑g ∈ F2: |g| = n f( Tg-1 x ), where |g| denotes the word length of g. We give an example of a measure-preserving system X and an f ∈ L1(X) such that the sequence \mathcal An f(x) is unbounded in n for almost every x, thus showing that the pointwise and maximal ergodic theorems do not hold in L1 for actions of F2. This is despite the results of Nevo-Stein and Bufetov, who establish pointwise and maximal ergodic theorems in Lp for p>1 and for L log L respectively, as well as an estimate of Naor and the author establishing a weak-type (1,1) maximal inequality for the action on ℓ1(F2). Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator.