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Vladikavkaz Mathematical Journal 2024, Volume 26, Issue 2, P. 95-102
УДК 517.987.5+519.216.8 DOI 10.46698/w0408-5668-5674-e
ON THE RATE OF CONVERGENCE OF ERGODIC AVERAGES FOR FUNCTIONS OF GORDIN SPACE#
I. V. Podvigin1
1 Sobolev Institute of Mathematics of the Siberian Branch of the RAS, 4 Ac. Koptug Ave., Novosibirsk 630090, Russia E-mail: ipodvigin@math.nsc.ru
To Georgii Georgievich Magaril-Il'yaev in occasion of his 80th birthday
Abstract. For an automorphisms with non-zero Kolmogorov-Sinai entropy, a new class of L2-functions called the Gordin space is considered. This space is the linear span of Gordin classes constructed by some automorphism-invariant filtration of a-algebras Fn. A function from the Gordin class is an orthogonal projection with respect to the operator I — E(-|3n) of some Fm-measurable function. After Gordin's work on the use of the martingale method to prove the central limit theorem, this construction was developed in the works of Volny. In this review article we consider this construction in ergodic theory. It is shown that the rate of convergence of ergodic averages in the L2 norm for functions from the Gordin space is simply calculated and is It is also shown that the Gordin space is a dense set of the first Baire
category in L2(Q, F,m) © L2(Q, n(T, F),m), where n(T, F) is the Pinsker a-algebra. Keywords: rates of convergence in ergodic theorems, filtration, martingale method. AMS Subject Classification: 37A30, 60G42.
For citation: Podvigin, I. V. On the Rate of Convergence of Ergodic Averages for Functions of Gordin Space, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 95-102. DOI: 10.46698/w0408-5668-5674-e.
1. Introduction
In the famous work of Gordin [1] the martingale approximation method was first used to prove the central limit theorem for stationary sequences. Subsequently, this approach was developed both for the central limit theorem (see, for example, [2-4]), and in other problems (see, for example, [5] on the convergence of series and [6-8] on large deviations); see also review [9]. The key idea is to consider the filtration of a-algebras associated with a measure-preserving transformation. We will use this construction in the theory of convergence rates in ergodic theorems.
Let (Q, F, be a standard probability space, and let T : Q ^ Q be a measurable invertible measure-preserving transformation (automorphism). Let Fo be a a-subalgebra of a-algebra F, such that T-1Fo C Fo- Thus, a filtration of a-algebras Fn := TnFo, n € Z arises, i. e.,
Fn C Fn+1, n € Z.
#The work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics, project FWNF-2022-0004.
© 2024 Podvigin, I. V.
Let us denote by En the conditional expectation operator with respect to the a-algebra Fn, acting in L2(Q, F,»), i. e., Enf = E(f |Fn). The operator En orthogonally projects L2(Q, F,») onto L2(Q, Fn,»). We will also use the symbol T to denote the Koopman operator acting in L2(Q, F, ») according to the rule Tf = f o T. Let us also denote ergodic averages as Anf, i. e.,
1 n— 1
n~ n
k=0
Von Neumann's ergodic theorem states that Anf norm-converges to E(f |J) for n ^ to, where J is the a-algebra of T-invariant sets, i. e., such A € F that T—1A = A.
It is well known that functions f from the class of coboundaries (or cohomologous to zero), i. e., / € (/— T)L2(Q, F, ¿¿), are characterized by the rate of convergence ^ in von Neumann's ergodic theorem. Namely for such and only such functions the asymptotic relation ||^ra/||2 = for rw oo holds [10]. It is worth noting that for non-unitary operators T
the asymptotics = 0(^7=) as n —> 00 with some additional condition will also imply
zero cohomology [11].
The class of coboundaries is believed to be the only simple construction for the abstract transformation T, where an estimate of the rate of convergence in von Neumann's ergodic theorem is easily obtained. There are also more complex classes of functions for which it is possible to obtain estimates of the rates of convergence of ergodic averagess, for example, the class of fractional coboundaries [12].
Our goal in this somewhat survey article is to consider in some sense new class of functions for which it is quite simple to obtain an estimate of the rate of convergence in von Neumann's ergodic theorem. We also prove that when the transformation is a K-automorphism, this class is dense in the space of of L2 functions with zero integral.
We say that a function f belongs to the Gordin class G(T, F0) generated by the a-algebra Fo, if
f € (I - E«)L2(Q, Fm,»), i.e., f € L2(fi, Fm,») © L2(fi, Fn,») for some m,n € Z, m> n. Thus,
G(T, Fo) = U Hn,m, Hn,m = (I - E„)L2(Q, Fm, »).
m>n
The Gordin space G(T) is the linear span of all the Gordin classes G(T, F0), i. e.,
G(T) =span{©(T, Fo)}.
Let us mention several results where functions from the Gordin space are found.
Remark 1. The proof of Kolmogorov's theorem that K-automorphisms have infinite Le-besgue spectral type (see, for example, [13, Theorem 5.13]) involves functions from the Gordin class of the form xa — E(xa|F—1), A € Fo-
Remark 2. It is shown in [14, Theorem 6.1] that functions of the form g = xa—E(xa|F—n), A € Fm, m, n > 0, are Wiener-Wintner functions of power type 1/4 in L2(Q, F,»). This means that
M— 1
1
sup
£
1 N—1
-Y go Tk- e2mk£ k=0
Now let us present the main result.
Theorem 1. For any function f from the Gordin space ©(T) the following estimate is true:
1
— /y I —
\A%f = as N—¡>00.
2. The Proof of Theorem 1
Let us use the following properties of the conditional expectation. Let A be a 0-algebra, then
E(E(f |A)) = Ef, (1)
E(f ■ g|A) = g ■ E(f |A), g - A-measurable, (2)
E(f o T|A) = E(f |TA) o T, (3)
where Ef = J f d^.
< The set of functions / with ||^.jv/||2 = ^("T^f) dearly linear. To prove Theorem 1, it is enough to consider the function f from some Gordin class ©(T, F0). Then there are n, m € Z with m > n and g € Fm, such that f = g — Eng.
To estimate the norm of ergodic averages, we use the following well-known formula (see, for example, [15, § 1]):
\Kf\\l = j-2 £ ^-\k\)(foT\f)L2=MM + ^{N.k)Re{foT\f)L2.
When calculating scalar products, we will use the properties (1), (2) and (3). Taking in account that g o Tk will be Fm-k-measurable, and Fm-k C Fn for k ^ m — n, for such k we obtain
(/ o Tk, f) l2 = (g oTk — E„g o Tk,g — Kg)l2 = E(g o )k ■ g) — E(g o Tk ■ E„g) + E(E„g o Tk ■ E„g) — E(E„g o Tk ■ g) = E(go Tk ■ g) — E(E„ (g o Tk-g)) + E(E„-k (g o Tk) ■ E„g) — E(E„-k (g o Tk) ■ g) = E(g o Tk ■ g) — E(g o Tk ■ g) + E)E„(E„-k(g o Tk) ■ g)) — E(E„-k(g o Tk) ■ g) = 0 + E(E„-k (g o Tk) ■ g) — E(E„-k (g o Tk) ■ g) = 0.
Thus, for all N ^ m — n > 0
K/II2 = ^ + Ja " *) Mf o T\ f)L2 < J® + ■Ml "^f\n - k)
k=1 k=1
_ 11/111 | 2II/IH / m-nx 2||/||2(m-n)
iV iV1 2N J ^ N
The proof of Theorem 1 is complete. > Let us present several corollaries.
Corollary 1. For any function f from the Gordin class, the spectral measure 0/ satisfies the estimate 0/((—5, 5]) = O(5) as 5 ^ 0.
< This follows from the well-known Kachurovskii criterion (see, for example, [15, Theorem 3]), as well as from the fact that the spectral measure 0/ has a continuously differentiable
density (see, [15, Theorem 7]), since most of the correlation coefficients (f o Tk, f )l2, k € Z, vanish. >
Corollary 2. For uniform convergence on the space Hnm there is the estimate
IA
T
N I I Hn,m , F, m)
<
2(m — n)
N
for all N ^ m — n.
In connection with Corollary 2, we note a recent paper [16], in which subspaces with power-law uniform convergence in von Neumann's discrete-time ergodic theorem were studied.
Corollary 3. For any function f from Gordin space the following asymptotic relation holds a. e.:
AN f = o
In iV(ln In iV)" \
7n J
as N —> oo
for any ft > 1/2.
< This follows from Theorem 4.5 in [17]. >
Corollary 4. If f is in the Gordin space G(T), then f € (I — T)aL2(Q, F,^) for every 0 < a < 1/2. For a = 1/2 a similar statement is not true.
< The first statement follows from Theorem 1 and [12, Theorem 2.17]. For a = 1/2 we will use the following criterion [12, Theorem 2.11 and Corollary 2.12]:
f e (I — T)1/2l2(q,f,u)
sup
N >1
N
^ Vj
< oo.
For any nonzero f € Hn,m with m — n = 1, using the fact that (Tkf, f) = 0 for k = 0 we have
sup
N >1
N ^
^ VI
.7=1 v,y
= sup> > -=-= / sup> - = oo. >
2 ^léièi V*J ^ifc
k
2
3. Additional Properties of Gordin Space
Let us now discuss the question for which automorphisms the Gordin class exists, i. e., it does not degenerate into a zero function.
Proposition 1. The Gordin space G(T) = {0} if and only if T has zero Kolmogorov-Sinai entropy h(T).
< It is clear that Gordin classes consist only of zero function if and only if all a-algebras Fn in the filtration coincide. This is equivalent to Fo = T-1Fo, i. e., the a-algebra Fo is invariant under T, or is a factor. Thus, we need to find a condition on the automorphism T, equivalent to the statement: for any a-algebra Fo Ç F
T-1Fo Ç Fo Fo = T-1Fo.
Such a condition was found in the works of Adler [18] and Sinai [19], namely: h(T) =0. >
Proposition 1 shows that L2 coboundaries need not be included in the Gordin space. Theorem 1, Corollary 3, and Theorem 2 below apply to K-automorphisms, which have positive entropy [13, Theorem 18.9], in particular to Bernoulli shifts [13, Proposition 3.51]. Ergodic
automorphisms of n-dimensional tori (n ^ 2) are isomorphic to Bernoulli shifts [20], so have positive entropy, but irrational rotations of the circle have zero entropy [21, p. 252].
In addition, recall that the condition h(T) = 0 can also be expressed by the following equality of a-algebras (see, for example, [13, p. 320]):
n(T, F) = F,
where n(T, F) is the Pinsker a algebra, i. e., n(T, F) = {A € F : h({A, Q\ A} = 0)}. Thus, for automorphisms with positive entropy, the Pinsker a-algebra n(T, F) is a proper a-subalgebra of the a-algebra F- If n(T, F) = {0, Q}, then the automorphism T is called a K-automorphism.
For F0 satisfying T-1F0 ^ F0, let's put
= f^j Fra, = \f Fn.
n<0 ra^0
Proposition 2. Let h(T) > 0. The Gordin class ©(T, F0) is a linear space invariant under the Koopman operator; and it is also dense subset of the first Baire category in
L2(Q,e L2(Q,F-^,^).
< For the Gordin class to be linear, it is sufficient to check that the sum of two functions from the Gordin class is a function from the Gordin class generated by the same a-algebra. Let f € and g € HP;q, i. e.,
f = p — E„p, g = ^ — Eq ^
for some functions p € L2(Q,Fm,^) and ^ € L2(Q,Fp,^). Then, assuming for definiteness that q ^ n, we obtain
f + g = P + ^ — Era^ — Eq ^ = £ — Eq where £ = p + ^ — Enp. It is clear that
Eq £ = Eq (p + ^ — E„p) = Eq p + Eq ^ — Eq E„p = Eq Thus, it showed that
Hm,n f Hp,q C Hmax{m,p}, min(ra,q}.
For the Gordin class to be invariant with respect to the Koopman operator, it is sufficient to show that THn>m = Hn-1;m-1. Let g € L2(Q, Fm,^), then g o T € L2(Q, Fm-1,^) and
(g — E„g) o T = g o T — E(g|Fn) o T = g o T — E(g o T|T-1Fn) = g o T — E„-1(g o T).
Let now f € L2(Q, e L2(Q, By definition (Emf)m>0 is a martingale
and (Enf)n<0 is a reverse martingale. Then the two-parameter family of functions fm,n = Emf — EnEmf = Emf — Enf will approximate the function f as m ^ and n ^ —to. This follows from Doob's theorem on the convergence of direct and reversed martingales (see, for example, [13, Theorem 14.26]).
The first Baire category for the Gordin class follows from the fact that it is a countable union of closed spaces HTO;ra. >
Theorem 2. Let h(T) > 0. The Gordin space ©(T) is invariant under the Koopman operator; and it is also dense subset of the first Baire category in L2(Q, F, e L2(Q, n(T, F),J").
< Invariance follows from Proposition 2. By the Rokhlin-Sinai theorem (see, for example, [13, Theorem 18.9]) on the characterization of K-automorphisms, there is an extremal a-sub-algebra Fo C F satisfying T-1Fo C Fo, such that
F-rc = n(T, F), F+rc = F.
From Proposition 2 it follows that the Gordin class ©(T, F0) is an everywhere dense subset of L2(Q, F, ¿) © L2(^, n(T, F),u). On the other hand, for any a-subalgebra F0 C F satisfying T-1F0 C Fo, from the work of Volny [22, Theorem 2] it follows that
L2(H,F+rc,u) © L2(Q,F-rc,u) C L2(Q,F,u) © ¿2(H,n(T,F),u).
Indeed, let f = g — E(g|F-rc) for g € L2(Q, F+rc,u). Then for any h € L2(Q, F, ¿) we have
(f, E(h|n(T, F)))L2 = (E(f |n(T, F)),h)La =0
since
E(f |n(T, F)) = E(g|n(T, F)) — E(E(g|F-rc)|n(T, F)) = E(g|n(T, F)) — E(E(g|F-rc V n(T, F))|n(T, F)) = E(g|n(T, F)) — E(g|n(T, F)) = 0.
In terminology of Volny the closure cl©(T) is the set of all difference decomposable functions (see [22, p. 116]).
The first Baire category for ©(T) follows from [23, Theorem 2] and Corollary 3. > Corollary 5. If T is a K-authomorphism, then the Gordin space ©(T) is dense in
L0(H,F,¿) := {f € L2(Q,F,u) : J f du = 0}.
It shows that for a K-automorphism on a standard probability space, the Gordin space is not closed. Otherwise we have, by Theorem 1, a uniform rate of convergence in the mean ergodic theorem, contradicting the well-known Krengel's result on arbitrary slow convergence in the mean ergodic theorem.
In conclusion, we note that it would be interesting to find a natural dense class of functions for which the rate of convergence of ergodic averagess for an abstract automorphism with zero entropy can be calculated.
The author thanks the referee for remarks and suggestions for improving the paper.
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Received December 21, 2023 IVAN V. PODVIGIN
Sobolev Institute of Mathematics of the Siberian Branch of the RAS,
4 Ac. Koptug Ave., Novosibirsk 630090, Russia,
Senior Researcher
E-mail: ipodvigin@math.nsc.ru
Владикавказский математический журнал 2024, Том 26, Выпуск 2, С. 95-102
О СКОРОСТИ СХОДИМОСТИ ЭРГОДИЧЕСКИХ СРЕДНИХ ДЛЯ ФУНКЦИЙ ИЗ ПРОСТРАНСТВА ГОРДИНА
Подвигин И. В.1
1 Институт математики им. С. Л. Соболева Сибирского отделения РАН, Россия, 630090, Новосибирск, проспект ак. Коптюга, 4 E-mail: ipodvigin@math.nsc.ru
Аннотация. Для автоморфизмов с ненулевой энтропией рассмотрен естественный класс функций, названный пространством Гордина. Это пространство есть линейная оболочка классов Гордина, построенных по некоторой инвариантной относительно автоморфизма фильтрации а-алгебр Fn. Функция из класса Гордина представляет собой ортогональную проекцию относительно оператора I — E(f |Fn) некоторой Зт-измеримой функции. После работы Гордина о применении мартингального метода для доказательства центральной предельной теоремы, эта конструкция получила свое развитие в работах Далибора Волны. В этой обзорной статье мы рассматриваем эту конструкцию в эргодической теории. Показано, что скорость сходимости эргодических средних в L2 норме для функций из пространства Гордина просто вычисляется и равна Также показано, что пространства Гордина есть плотное
множество первой катеогрии по Бэру в L2(0, З, м) © L2(0, П(Т, F), м), где П(Т, З) — а-алгебра Пинскера.
Ключевые слова: скорости сходимости в эргодических теоремах, фильтрация, мартингальный метод.
AMS Subject Classification: 37A30, 60G42.
Образец цитирования: Podvigin I. V. On the Rate of Convergence of Ergodic Averages for Functions of Gordin Space // Владикавк. мат. журн.—2024.—Т. 26, № 2.—C. 95-102 (in English). DOI: 10.46698/w0408-5668-5674-e.
