Failure of the L1 Pointwise and Maximal Ergodic Theorems for the Free Group

Let F2 denote the free group on two generators a and b. For any measure-preserving system (X,X , μ, (Tg)g∈F2 ) on a finite measure space X = (X,X , μ), any f ∈ L1(X), and any n > 1, define the averaging operators An f (x) := 1 4× 3n−1 ∑ g∈F2 :|g|=n f (T g 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 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 L p 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 `(F2). Our construction is a variant of a counterexample of Ornstein concerning iterates of a Markov operator. 2010 Mathematics Subject Classification: 37A30 (primary)