FAILURE OF THE L1 POINTWISE AND MAXIMAL ERGODIC THEOREMS FOR THE FREE GROUP

Abstract

Let F2 denote the free group on two generators a and b. For any measure-preserving system (X, X, μ (Tg)g2F2) on a finite measure space X = (X, X, μ), any f ∈ L1 (X), and any n ≥ 1, define the averaging operators (Equation Presented) 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 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.