aa r X i v : . [ m a t h . D S ] M a y COBOUNDARIES OF NONCONVENTIONAL ERGODIC AVERAGES
I. ASSANIA bstract . Let ( X , A , µ ) be a probability measure space and let T i , 1 ≤ i ≤ H , be invertible bi measurablemeasure preserving transformations on this measure space. We give a sufficient condition for the product of H bounded functions f , f , ..., f H to be a coboundary. This condition turns out to be also necessary when oneseeks bounded coboundaries.
1. I ntroduction
The purpose of this short article is to answer a question brought to our attention by S. Donoso ( ) duringthe 2017 ETDS workshop held at Chapel Hill.To this end we refine the setting in [2]. Definition 1.1.
A probability measure preserving system ( X , F , µ , T , T , . . . , T H ) is a combination of a probabilitymeasure space ( X , F , µ ) and T i , 1 ≤ i ≤ H bi-measurable invertible measure preserving maps acting on thisprobability space.
Given a probability measure preserving system ( X , F , µ , T , T , . . . , T H ) , µ ∆ is the diagonal measure on X H , Φ = T × T × · · · × T H , and ν is the diagonal-orbit measure of Φ , i.e. ν ( A ) = ∑ n ∈ Z | n | µ ∆ ( Φ − n A ) .We note that ν is nonsingular, since ν ( A ) ≤ ν ( Φ − A ) ≤ ν ( A ) .. Definition 1.2.
The diagonal orbit system of the probability measure preserving system ( X , F , µ , T , T , . . . , T H ) isthe system ( X H , F H , ν , Φ ) . Remarks (1) The maps T i do not necessarily commute.(2) The nonsingularity of Φ with respect to ν implies the following simple but key lemma (Thislemma does not seem to hold when one replaces ν with the diagonal measure µ ∆ on (cid:0) X H , F H ) , defined by the equation R F ( x , x , . . . , x H ) d µ ∆ = R F ( x , x , . . . , x ) d µ (cid:1) .Lemma 1.3. Let F n be a sequence of measurable functions defined on X H . If F n converges ν a.e. then the sequenceG n = F n ◦ Φ converges ν a.e. as well. Department of Mathematics, UNC Chapel Hill, NC 27599, [email protected]. He indicated that this question was mentioned to him by J.P. Conze and Y. Kifer
Proof.
Let A = { z ∈ X H ; F n ( z ) converges } and B = { z ∈ X H ; G n ( z ) converges } . We have B = Φ − ( A ) .Therefore if ν ( A c ) = ν ( Φ − ( A c )) = ν . (cid:3) We wish to prove the following proposition.
Proposition 1.4.
Let ( X , F , µ , T , . . . , T H ) be a measure preserving system, and f , f , . . . f H ∈ L ∞ ( µ ) and ≤ p < ∞ .(1) Let us assume that the supremum of the nonconventional ergodic sums is L p -bounded, i.e. sup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T ni (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( ν ) < ∞ .(2) Then the product of the functions is a coboundary in L p ( X H , ν ) , i.e. if Φ = T × T × · · · × T H , thereexists V ∈ L p ( X H , ν ) such that H O i = f i = V − V ◦ Φ , ν -a.e.Therefore, for µ ∆ -a.e. ( x , x , ..., x H ) ∈ X H , we havef ( x ) f ( x ) · · · f H ( x H ) = V ( x , x , . . . , x H ) − V ( T x , T x , . . . , T H x H ) .We give only the proof for the case p =
1. We use the following a.e.-convergence result obtained byKomlós in 1967. When 1 < p < ∞ the reflexivity of L p ( ν ) allows to bypass this lemma. For p = ∞ the assumptions (1) and (2) in the statement of Theorem 1.4 are equivalent . We state it separately as acorollary. Lemma 1.5 ( [3]) . Let ( X , F , µ ) be a probability measure space, and ( g n ) be a sequence in L ( µ ) . Assume that lim inf n k g n k L ( µ ) < ∞ . Then there exists a subsequence ( g n k ) k and a function g ∈ L ( µ ) such that for µ -a.e.x ∈ X, lim K → ∞ K K ∑ k = g n k ( x ) = g ( x ) . Proof of Proposition . We show that the techniques used for an invariant measure can be applied to ourcurrent nonsingular setting. The assumption made tells us that, if F = f ⊗ f ⊗ · · · ⊗ f d , we havelim N → ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N N ∑ n = F ◦ Φ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( ν ) = N k of natural numbers such thatlim N k → ∞ N k N k ∑ n = F ◦ Φ n ( z ) = OBOUNDARIES OF NONCONVENTIONAL ERGODIC AVERAGES 3 for ν -a.e. z ∈ X H . Because Φ is nonsingular with respect to ν , we also know thatlim N k → ∞ N k N k ∑ n = F ◦ Φ n + ( z ) = ν -a.e. z ∈ X H for the same subsequence ( N k ) . Set D N k = N k N k ∑ n = n − ∑ j = F ◦ φ j .Since sup n (cid:13)(cid:13)(cid:13) ∑ n − j = F ◦ φ j (cid:13)(cid:13)(cid:13) L ( ν ) < ∞ , lim inf k (cid:13)(cid:13) D N k (cid:13)(cid:13) L ( ν ) < ∞ . Thus, we may apply Lemma 1.5 to show thatthere exists a subsequence of ( D N k ) (which remain denoted as ( D N k ) ) such that the averages K ∑ Kk = D N k converge ν -a.e to a function V ∈ L ( ν ) . Similarly, by Lemma 1.3 we have V ◦ Φ ( z ) = lim K K K ∑ k = D N k ◦ Φ ( z ) for ν -a.e. z ∈ X H . Therefore, V − V ◦ Φ = lim K K K ∑ k = F − K K ∑ k = N k N k ∑ n = F ◦ Φ n !! = F .Since V − V ◦ Φ = F for ν -a.e., the equality certainly holds for µ ∆ -a.e. (the construction of ν guaranteesthat V ∈ L ( µ ∆ ) ). (cid:3) Corollary 1.6.
Let ( X , F , µ , T , . . . , T H ) be a measure preserving system, and f , f , . . . f H ∈ L ∞ ( µ ) The followingstatements are equivalent. (1)
The supremum of the nonconventional ergodic sums is L ∞ -bounded, i.e. sup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T ni (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ν ) < ∞ .(2) The product of the functions is a coboundary in L ∞ ( X H , ν ) , i.e. if Φ = T × T × · · · × T H , there existsV ∈ L ∞ ( X H , ν ) such that H O i = f i = V − V ◦ Φ , ν -a.e.Therefore, for µ -a.e. x ∈ X, we havef ( x ) f ( x ) · · · f H ( x ) = V ( x , x , . . . , x ) − V ( T x , T x , . . . , T H x ) . Proof.
The implication 1) implies 2) can be obtained by following the same path as in the proof of theprevious proposition. The only thing to check is that V ∈ L ∞ ( ν ) . But this follows from the fact that ifsup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T ni (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( ν ) < C < ∞ I. ASSANI then k D N k k L ∞ ≤ C . From this observation one can conclude that the limit function V is also in L ∞ ( ν ) .For the reverse implication, if F = f × f ... × f H is a coboundary in L ∞ ( ν ) (i.e. F = V − V ◦ Φ where V ∈ L ∞ ( ν ) ) then N ∑ n = F ◦ Φ n = V − V ◦ Φ N + and k N ∑ n = F ◦ Φ n k L ∞ ( ν ) = k V − V ◦ Φ N + k L ∞ ( ν ) ≤ k V k L ∞ ( ν ) (cid:3) Remark
The assumption sup N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T ni (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( ν ) < ∞ .is satisfied when sup N , m ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T n + mi (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) < ∞ .This last condition is equivalent to sup N ∈ Z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N ∑ n = H ∏ i = f i ◦ T ni (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( µ ) < ∞ .which may be easier to check in the applications.R eferences [1] I.Assani :“A note on the equation y = ( I − T ) x in L " Illinois J. of Math. vol 43, 3, (1999) p.540-541.[2] I. Assani :“Pointwise recurrence for commuting measure preserving transformations" Preprint , arXiv:1312.5270v2, 2015[3]