Coboundaries in <span class="mathjax-formula">${L}_{0}^{\infty }$</span>

(1)

www.elsevier.com/locate/anihpb

Coboundaries in L ^∞ ₀

Dalibor Volný

^a,^∗

, Benjamin Weiss

^b

aLaboratoire de mathématiques Raphaël Salem, UMR CNRS 6085, université de Rouen, 76821 Mont-Saint-Aignan Cedex, France bDepartment of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

Received 6 February 2003; received in revised form 18 February 2004; accepted 24 March 2004 Available online 25 September 2004

Abstract

LetT be an ergodic automorphism of a probability space,f a bounded measurable function,S_n(f )=_n₋₁

k=0f◦T^k. It is shown that the property that the probabilitiesµ(|Sn(f )|> n)are of ordern⁻^proughly corresponds to the existence of an approximation inL^∞off by functions (coboundaries) g−g◦T,g∈L^p. Similarly, the probabilities µ(|Sn(f )|> n)are exponentially small ifff can be approximated by coboundariesg−g◦T whereghave finite exponential moments.

Résumé

SoitT un automorphisme ergodique d’un espace probabilisé,f une fonction bornée mesurable etSn(f )=_n₋₁ k=0f ◦T^k. Une correspondance est établie entre l’existence de l’estimation des probabilitésµ(|S_n(f )|> n)d’ordren⁻^pet l’existence de l’approximation dansL^∞de la fonctionf par des cobordsg−g◦T oùgest “presque” dansL^p. De manière similaire, les probabilitésµ(|S_n(f )|> n)sont d’ordre e⁻^cn, pour un certainc >0,n=1,2. . ., si et seulement sifadmet une approximation dansL^∞par des cobordsg−g◦T avecgayant des moments exponentiels.

MSC: 28D05; 60F10

Keywords: Coboundary; Probabilities of large deviations

* Corresponding author.

E-mail address: dalibor.volny@univ-rouen.fr (D. Volný).

doi:10.1016/j.anihpb.2004.01.004

(2)

1. Introduction and results

Let(Ω,A, µ)be a probability space andT:Ω→Ωa bijective, bimeasurable and measure preserving mapping.

Throughout the paper we shall suppose thatT is ergodic and aperiodic. For a measurable functionf onΩ we denote

Sn(f )=

n−1

i=0

f ◦Tⁱ.

L^p₀ denotes the space of allf ∈L^pwith zero mean, 1p∞. Iff =g−g◦T withga measurable function, then we say thatf is a coboundary. The functiongis then called the cobounding function. We shall study the approximation of functions fromL^∞₀ by coboundaries. The main results are presented in Theorems 1, 2, and 3;

they show a relationship between the moments of the cobounding functiongof the approximating coboundary and probabilities of large deviations of the stochastic process(f◦Tⁱ).

It is well known that for 1p <∞, the coboundaries with a cobounding function inL^p are a dense subset of L^p₀ (becauseL^∞is a dense subset ofL^p, the coboundariesg−g◦T withgbounded thus form a dense subset of allL^p₀, 1p <∞). As an immediate consequence of the density of the sets of coboundaries inL^pspaces we get the von Neumann’s ergodic theorem in these spaces:

1

nS_n(f )−Ef p

→0

for anyf ∈L^p, 1p <∞(cf. e.g. [10, p. 21]).

Forp= ∞the things are more complicated:

Theorem A. Letf∈L¹₀andε >0. Then there exists a measurable functiongsuch that f−(g−g◦T )

∞< ε.

The theorem follows from [7, Corollary 3] ([8] in English). For completeness, we shall show a proof the idea of which is due to Michael Keane.

There exist, however, bounded functions which cannot be (inL^∞) approximated by any coboundary with an integrable cobounding functiong:

Theorem B. Letϕ be a positive real function, limt→−∞ϕ(t)=limt→∞ϕ(t)= ∞. Then there exists a function f∈L^∞₀ withf∞=1 such that for each measurable functiongwith

ϕ◦g dµ <∞, f−(g−g◦T )

∞1/2.

In particular, for anyp >0 there existsf withf∞=1 s.t.f −(g−g◦T )∞1/2 for eachg∈L^p. This result is not completely new either; a version of it can be found in the work of A. Katok [6].

The main aim of this paper is to show that the set off ∈L^∞₀ which can be approximated by coboundaries whose cobounding functions have finite moments can be characterized by the probabilities of large deviations. The first two theorems show that the integrability of|f|^p is “almost equivalent” to the property that the probabilities µ(|S_k(f )|> xk)are of order 1/k^p. The third proposition extends the result to functions with exponential moments.

Theorem 1. Letf ∈L^∞₀ ,p1. If for everyδ >0 there exists ag∈L^pwithf −(g−g◦T )∞< δthen

(3)

(i) for eachε >0 ∞ k=1

k^p⁻¹µS_k(f )> εk

<∞, (ii) for eachε >0 there exists ac_ε>0 such that

µS_k(f )> εk

< cε· 1

k^p for allk=1,2, . . . .

Theorem 2. Letf ∈L^∞₀ ,p1. If for everyx >0 there exists a 0< cx<∞such that for allk, µS_k(f )> xk

< cxk⁻^p

then for allε >0 and every measurable functionv:R⁺→R⁺such that ∞

j=1

1

j v(aj )<∞ for alla >0 and v(x), x^p

v(x) are increasing there exists a measurable functiongsuch that

E |g|^p

v(|g|) <∞ and f −(g−g◦T )

∞< ε.

In particular, for anyδ >0 we can findg∈L^p⁻^δ. Theorem 3. Letf ∈L^∞₀ . It is equivalent

(i) For everyε >0 there exists ac_ε>0 andn_ε∈Nsuch thatµ(|S_n(f )|> εn) <e⁻^c^εⁿfor allnn_ε. (ii) For everyε >0 there exists a measurable functiongandc >0 such thatEe^c^|^g^|<∞and

f −(g−g◦T )

∞< ε.

For sequences of independent and weakly dependent random variablesX_i, the probabilities of{_n

i=1X_i> xn} have been analyzed in detail before. For example, by Azuma’s inequality [4] an exponential bound like in (i) of Theorem 3 exists whenever(X_i)is a uniformly bounded sequence of martingale differences. Therefore, iff ∈L^∞₀ and(f◦Tⁱ)is a martingale difference sequence (or even a sequence of mutually independent random variables) thenf can be approximated by coboundaries whose transfer functions have finite exponential moments.

Generic properties of sets of functionsf for which the probabilitiesµ(S_n(f ) > xn)have a particular asymptotic behaviour are studied in the paper [9].

2. Proofs

Proof of Theorem A. By the Birkhoff (almost sure) Ergodic Theorem for anyη >0 there exists A∈A with µ(A) >0 andn₀∈Nsuch that for eachω∈Awe have

(1/n)S_n(f )(ω)< η for allnn₀.

Let A^k be the set of points whose return time to A is k: A^k = {ω∈A: T^kω∈A, Tⁱω /∈Afor 0< i < k}, k=1,2, . . ., letA=_∞

k=1A^k. Because T is ergodic, the set_∞

k=1

k−1

i=0TⁱA^k has measure 1; without loss of generality we can suppose that it equalsΩ.

(4)

It is known that we can also suppose thatA^k= ∅for 1kn₀−1. This is the case ifA, T A, . . . , Tⁿ⁰⁻¹Ais a Rokhlin tower. If not, we can recursively find an adequate subset ofAof strictly positive measure because for anyAof positive measure andk0,B⊂A, 0< µ(B) < µ(A), we by ergodicity haveµ(B\T^k⁺¹B) >0, hence B, . . . , T^k⁺¹Bare disjoint.

For eachω∈Ωthere thus exist uniquekn0and 0ik−1 such thatω∈TⁱA^k. Define

ξ(ω)=T⁻ⁱω, g(ω)=f (ω)−1

k

k−1

j=0

f

T^jξ(ω) ,

h(ω)= i j=0

g(T^j⁻ⁱω)= i j=0

g

T^jξ(ω) .

Forω∈T^k⁻¹A^k,h(ω)=0. We thus have g=h−h◦T⁻¹.

From the definition of the setAit follows that f −g∞< η. 2

In the proof of Theorem B we shall use the result by A. Alpern (cf. [1–3,5]).

Theorem C (A. Alpern). Let 1n_k,k=1,2, . . ., be positive integers whose least common divisor is 1, p_k positive reals,_∞

k=1p_kn_k=1. Then there exist measurable setsF_k, such thatµ(F_k)=p_k,TⁱF_k, 0in_k−1, k=1,2, . . ., are pairwise disjoint andµ(_∞

k=1

n_k−1

i=0 TⁱF_k)=1.

Proof of Theorem B. Without loss of generality we can suppose thatϕis an even function. Fork=1,2, . . .letrk

be a positive integer such that

[(k−1)/2] j=1

ϕ(j/2)kr_k;

by the assumptions,rk→ ∞. There thus exist (strictly) positive numberspksuch that ∞

k=1

kp_k=1, ∞

k=1

krkpk= ∞.

By Theorem C there exist measurable setsA_k,k=1,2, . . ., such thatµ(A_k)=p_kand{A_k, T⁻¹A_k, . . . , T⁻^k⁺¹A_k}, k=1,2, . . .are mutually disjoint Rokhlin towers. LetA_k=B_k∪C_kwhereB_k∩C_k= ∅andµ(B_k)=µ(A_k)/2= µ(C_k),k=1,2, . . .. We define

f (ω)=







1 forω∈_∞

k=1

_k₋₁

i=0T⁻ⁱBk

−1 forω∈_∞

k=1

_k₋₁

i=0T⁻ⁱC_k 0 forω /∈_∞

k=1

_k₋₁

i=0T⁻ⁱA_k.

(5)

Suppose thatg, hare measurable functions,h∞<1/2 and f=g◦T −g−h.

Then

g◦T =f +g+h,

g◦T²=f◦T +g◦T +h◦T =f +f ◦T +h+h◦T+g, . . .

g◦Tⁿ=

n−1

i=0

f ◦Tⁱ+

n−1

i=0

h◦Tⁱ+g.

We have Eϕ◦g

∞ k=1

Ak

k−1

i=0

ϕ(g◦Tⁱ) dµ.

Let us denoteψ_j=j−1

i=0(f◦Tⁱ+h◦Tⁱ),j=1,2, . . .. Forω∈A_kwe get_k₋₁

j=0ϕ(g(T^jω))=_k₋₁

j=0ϕ(ψ_j+g).

We distinguish two possibilities:

1. The numbersψj(ω)+g(ω),j=0, . . . , k−1, are all of the same sign. Then_k₋₁

j=1ϕ(ψj+g)_k₋₁

j=1ϕ(j/2).

2. The numbersψ_j(ω)+g(ω),j =1, . . . , k−1, are not all of the same sign. Becausef (T^jω)are all 1 or all−1 while|h|1/2, the sequenceψ₁(ω), . . . , ψ_k₋₁(ω)is monotone. Hence, there exists 1nk−1 such that_k₋₁

j=1ϕ(ψ_j+g)=_n

j=1ϕ(ψ_j+g)+_k₋₁

j=n+1ϕ(ψ_j+g)_[(k−1)/2]

j=1 ϕ(j/2)where[x]denotes the integer value ofx.

We thus getEϕ(g)_∞

k=1p_k_[(k−1)/2]

j=1 ϕ(j/2)_∞

k=1kr_kp_k= ∞. This finishes the proof. 2

Proof of Theorem 1. Letε >0 be fixed. We put 0< δ < ε/2,gis a function fromL^pwithf−(g−g◦T )_∞< δ.

ThenSn

f −(g−g◦T )< δn < nε/2 hence

µS_k(f )> εk

µS_k(g−g◦T )> kε/2

2µ

|g|> kε/4 . Becauseg∈L^p,

∞ l=1

(l+1)^p−1 j=l^p

µ

|g|> l+1

^∞

l=1

(l+1)^p−1 j=l^p

µ

|g|> j

<∞ hence

∞ l=1

l^p⁻¹µ

|g|> l

<∞. The statement (ii) follows from µS_n(g−g◦T )> εn

2µ

|g|>ε 2n 2

|g|^p

(^ε₂n)^pdµ= c_ε n^p wherecε=2^p⁺¹

|g/ε|^pdµ. 2

For the proof of Theorem 2 we shall need the following statement:

(6)

Proposition. Letf ∈L^∞₀ ,ε >0. Then under the assumptions of Theorem 2 there exists an integern₀1 and a setF of positive measure with

F_k= {ω∈F|T^kω∈F,∀1ik−1, Tⁱω /∈F}, k=1,2, . . . , F_∞= {ω∈F| ∀1i, Tⁱω /∈F},

such that

(a) forω∈Fk, 1k <∞,

Sk(f )(ω)kε and Sj(f )(ω)> j ε for all 1jk−n0

and

(b) there exists a 0< c <∞such that for alln, ∞

k=n

µ(Fk) < cn⁻^p⁻¹.

Proof. As in the proof of Theorem A we can show that there exists an integer n₀1 and a setA of positive measure such that for allω∈A

(i) ifnn0then|_n₋₁

i=0f (T⁻ⁱω)|εn, (ii) if 1kn₀−1 thenT⁻^kω /∈A.

Forω∈Ωwe define

ψ(ω)=min{n2| |S_n(f )(ω)|εn} if∃n2,|S_n(f )(ω)|εn,

∞ otherwise.

Forω∈Awe recursively defineτ_k(ω),k=0,1, . . .by τ₀(ω)=0, τ_k₊₁(ω)=τ_k(ω)+ψ(T^τ^k^(ω)ω) and put

ϕ(ω)=

min{k1|T^kω∈A} if∃n1, Tⁿω∈A,

∞ otherwise,

t (ω)=

sup{0k|τk(ω)ϕ(ω)} ifϕ(ω) <∞,

∞ otherwise,

Observation. Letω∈A. Ifϕ(ω), t (ω) <∞then τ_{t (ω)}(ω)ϕ(ω),

ϕ(ω)−τ_{t (ω)}(ω)n₀−1, t (ω)1.

The first inequality follows immediately from the definition oft, the second follows from (i), the third from the preceding ones and (ii).

Letω∈A. Define

F=

ω∈A t (ω)−1

k=0

T^τ^k^(ω)ω.

By the construction, the setF is measurable and satisfies (a).

(7)

Let us prove (b). Letδ >0. If 0iδn,kn₀+n(1+δ),ω∈Fk, then by (a)

n+i−1 j=0

f (T^jω)

> (n+i)εnε and

i−1

j=0

f (T^jω)

δnf_∞. Therefore,

n−1

j=0

f (T^jTⁱω)

n+i−1 j=0

f (T^jω) −

i−1

j=0

f (T^jω)

nε−nδf∞=n

ε−δf∞ . Forn₀defined at the beginning of the proof, for eachδ >0,ω∈

kn₀+n(1+δ)F_k and 0iδn, we then have

|S_n(f )(Tⁱω)|n(ε−δf_∞). There thus exists a constantcdepending only onε−δf_∞, [δn]

kn₀+n(1+δ)

µ(F_k) < cn⁻^p

and the inequality (b) follows. 2

Proof of Theorem 2. Let 1k <∞,ω∈Fk, 0ik−1. We define h(Tⁱω)=f (Tⁱω)−1

k

k−1

j=0

f (T^jω),

g(Tⁱω)=

i−1

j=0

h(T^jω).

On the rest ofΩ we defineg=0. We then have h=g◦T−g

and f−(g◦T −g)

∞ε.

Let us denotea= h_∞. Using (b) we calculate E

|g|^p v(|g|)

∞ k=1

_k

j=1

(aj )^p v(aj )

µ(Fk) ∞ j=1

(aj )^p v(aj )

_∞

k=j

µ(Fk)

c

∞ j=1

(aj )^p v(aj )j⁻^p⁻¹

=ca^p ∞ j=1

1

j v(aj )<∞. 2

The proof of Theorem 3 is left as an exercise for the reader. For (i)⇒(ii) we can use the same construction as in the proof of Theorem 2, (ii)⇒(i) follows fromµ(|Sn(f )|> ε)µ(|Sn(f−(g−g◦T ))|> ε/2)+µ(|Sn(g− g◦T )|> ε/2).

References

[1] A. Alpern, Generic properties of measure preserving homeomorphisms, in: Ergodic Theory, in: Lecture Notes in Mathematics, vol. 729, 1979, pp. 16–27.

(8)

[2] A. Alpern, Return times and conjugates of an antiperiodic transformation, Ergodic Theory Dynam. Syst. 1 (1981) 135–143.

[3] A. Alpern, V.S. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms, Cambridge Univ. Press, Cambridge, 2000.

[4] K. Azuma, Weighted sums of certain dependent random variables, Tôhoku Math. J. 19 (1967) 357–367.

[5] S.J. Eigen, V.S. Prasad, Multiple Rokhlin Tower Theorem: A simple proof, New York J. Math. 3A (1997) 11–14.

[6] A. Katok, Constructions in ergodic theory, in preparation.

[7] A.V. Koˇcergin, On the homology of functions over dynamical systems, Soviet Math. Dokl. 17 (1976) 1637–1641 (in Russian).

[8] A.V. Koˇcergin, On the homology of functions over dynamical systems, Dokl. Akad. Nauk SSSR 231 (1976) (in English).

[9] E. Lesigne, D. Volný, Large deviations for generic stationary processes, Coll. Math. 84/85 (2000) 75–82.

[10] W. Parry, Topics in Ergodic Theory, Cambridge Tracts in Mathematics, vol. 75, Cambridge Univ. Press, Cambridge, 1981.

Further reading

I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, in: Ergodic Theory, in: Die Grundlehren der mathematischen Wissenschaften, vol. 245, Springer, Berlin, 1982.

Coboundaries in <span class="mathjax-formula">${L}_{0}^{\infty }$</span>

Coboundaries in L ∞ 0

Dalibor Volný

, Benjamin Weiss

Coboundaries in L ^∞ ₀