SUMS OF CONTINUOUS AND DIFFERENTIABLE
FUNCTIONS IN DYNAMICAL SYSTEMS
Pierre Liardet and Dalibor Volný
Abstract. Let T be a homeomorphism of a metrizable compact X, the sequence ck /k tend to 0 and ck tend
i
to infinity. We study the limit behaviour of the sums (1/ck )Σk−1
i=0 F ◦ T where F is from a space of continuous
functions - the central limit problem and the speed of convergence in the ergodic theorem.
The main attention is given to the case where X is the unit circle and T is an irrational rotation; in this case we
consider the spaces of absolutely continuous, Lipschitz, and k-times differentiable functions F .
1. Introduction.
Let (T, Ω, A, λ) be a dynamical system given by a measure preserving transformation T : Ω → Ω of the
Pn−1
probability space (Ω, A, λ). The limit behavior of the sums Sn (f ) = i=0 f ◦T i is a topic which is intensively
studied in ergodic theory and probability theory. The Birkhoff ergodic theorem says that if f is integrable,
(1/n)Sn (f ) converge pointwise a.e.. Burton and Denker proved that for each ergodic and aperiodic dynamical
system there exists f such that for the process (f ◦ T i ) the central limit theorem holds ([Burton-Denker], see
also [De La Rue-Ladouceur-Peškir-Weber]). Lacey proved a convergence to self-similar processes ([Lacey]).
On the other hand, Volný [Volný] proved that for a generic set of functions from Lp0 , 1 ≤ p < ∞ and any
sequence (cn )n such that limn cn = ∞ and limn cn /n = 0, the distributions of (1/cn )Sn (f ) converge (in
subsequences) to all probability laws (the case of p = ∞ was proved in [Keane-Volný]). This shows that in
the Birkhoff theorem, arbitrarily slow rate of convergence is not only possible but also generic (more results
on the speed of convergence in the ergodic theorem are given in the monography [Krengel]).
Here we shall first consider the situation when T is a homeomorphism of a topological space and f is
continuous. When considering the space of continuous functions C0 (X) on a metrizable compact space X,
with null integral for each T -invariant Borel probability measure, we get the same results as for the Lp0 spaces
on a general probability space with an aperodic measure preserving transformation (Theorem 1). Further
results concern the spaces of absolutely continuous, Lipschitz, and continuously differentiable functions for
an irrational rotation of the unit circle (our results thus treat one of the simplest differentiable dynamical
systems). The Koksma-Denjoy Theorem shows that for bounded variance functions, the rate of convergence
in the Ergodic Theorem cannot be arbitrarily slow. For example, if the rotation is given by the golden
number, the suprema of the partial sums of any zero mean absolutely continuous function grow with a
rate slower than the logarithmic one. As we shall see, the results depend on diophantine properties of the
rotation.
In the case of rotations x 7→ x + α mod 1 where the continued fraction expansion of α has unbounded
partial quotients, we shall show results similar to those which hold in Lp spaces or in the space of continuous
Typeset by AMS-TEX
1
2
PIERRE LIARDET AND DALIBOR VOLNÝ
functions; our problem can be considered as satisfactorily answered in this case. As a corollary we get a
(known) result giving conditions for the existence and genericity of k-times continuously differentiable zero
mean functions which are not coboundaries. We shall also prove the existence and genericity of ergodic
cocycles for a cylindric flow.
The results concerning the bounded partial quotients are much less complete. We shall prove that for a
generic set of absolutely continuous zero mean functions F , there exists a weak convergence to the standard
normal law in subsequences of (1/cn )Sn (F ), with limn cn = ∞. The partial sums Sn (F ) are thus not
stochastically bounded, hence F are not coboundaries (see [Moore, Schmidt]). We shall give bounds for the
suprema of Sn (F ).
[say that the paper has been elaborated during stay of the firt author in Praha and the second in Marseille]
2. Continuous functions on a compact metric space.
Let X be a compact metric space, T : X → X a homeomorphism of X onto itself and let λ be a T -invariant
Borel probability measure. We suppose that the measure λ is aperiodic (see [Denker-Grillenberger-Sigmund]).
C(X) denotes the space of all real valued continuous functions on X and C0 (X) is the space of all f ∈ C(X)
R
with f dλ̃ = 0 for any T -invariant probability measure λ̃. (For any given T -invariant probability measure,
C0 (X) contains all functions f with the conditional expectation E(f |I) = 0 where I is the σ-algebra of all
T -invariant measurable sets, cf. [Oxtoby]). By k.k∞ we denote the supremum norm in C(X).
Theorem 1. Let the measure λ be aperiodic. Let (cn )n be an unbounded increasing sequence of positive
integers with cn /n converging to 0. Then there exists a dense Gδ set of functions f ∈ C0 (X) such that for
each probability measure ν on R there exists a sequence of positive integers nk → ∞ for which the sequence
Pnk
f ◦ T j weakly converges to ν.
of distributions of (1/cnk ) j=1
As a corollary we get a result saying that for continuous functions, the convergence in the Birkhoff theorem
may be arbitrarily slow (cf. [Krengel]).
Corollary. Let (cn )n be as in Theorem 1. Then there exists a dense Gδ set of functions f ∈ C0 (X) for
Pnk
f ◦ T j diverges to infinity in the measure.
which there exist subsequences (cnk )k such that k 7→ (1/cnk ) j=1
Let U : C(X) → C(X) be the operator defined by U f = f ◦ T , and let I be the identity operator.
Lemma 1. (I − U )C(X) is a dense subset of C0 (X).
Proof. Let us suppose that (I − U )C(X) is not dense in C0 (X). Then there exists a measure λ̃ on X
R
R
and f ∈ C0 (X) such that f dλ̃ 6= 0 and g dλ̃ = 0 for all g ∈ (I − U )C(X). For h ∈ C(X) we thus
R
R
have (h − U h) dλ̃ = 0, hence λ̃ is a T -invariant measure. Therefore f dλ̃ = 0 which contradicts our
supposition. ¤
Proof of Theorem 1. The proof can be done in a similar way as the proof of Theorem 1 in [Volný]; Lemma 1
plays the same role as the Lemma in [Volný]. Notice that the density of (I − U )C(X) in C0 (X) is sufficient,
in fact we do not need the density of (I − U )C0 (X).
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
3
Let Γ be the set of all probability measures ν on R supported by finite sets of rational numbers, with
R
rational values, and x dν(x) = 0. One can easily see that Γ is a countable set, dense in the space of all
probability distributions on R (equipped with the topology of weak convergence).
The set of the weak limit points of the distributions of (1/cn )Sn (f ) is closed for each f ∈ C0 (X) hence for
the proof of Theorem 1 it suffices to prove that for an arbitrarily chosen ν ∈ Γ there exists a dense Gδ set of
f ∈ C0 (X) for which the distributions of (1/cnk )Snk (f ) weakly converge to ν for some sequence of positive
integers nk → ∞.
Let ν ∈ Γ and let n be a positive integer. There exists a positive integer m and rational numbers
x1 , . . . , xr , r ≤ m, such that ν({xj }) = m(j)/m, 1 ≤ j ≤ r, where m(j) are positive integers such that
Pr
Pr
j=1 m(j) = m and
j=1 xj m(j) = 0. The measure λ is regular, hence for any δ > 0 there exists a Rochlin
tower V, T −1 V, . . . , T −n·m+1 V where V is an open set, and a continuous function ϕ̃, 0 ≤ ϕ ≤ 1, which is
Pm·n−1
zero on X \ V such that λ( i=0 ϕ̃ ◦ T i = 1) > 1 − δ (i.e. the sets V, . . . , T −n·m+1 V exhaust most of X
and ϕ̃ is close to the indicator function of V ).
Define
m−1
[
W =
T −j·n V
j=0
and
X
X
m(1)−1
ϕ = x1
m(1)+...+m(r)−1
ϕ̃ ◦ T j·n + . . . + xr
j=0
ϕ̃ ◦ T j·n .
j=m(1)+...+m(r−1)
Hence, W, T −1 W, . . . , T −(n−1) W is a Rochlin tower, ϕ is zero on X \ W , and the distribution of
Pn−1
i=0
ϕ◦Ti
is close to ν. For any invariant measure λ̃ we have
Z
ϕ dλ̃ =
r
X
Z
xj m(j)
ϕ̃ dλ̃ = 0,
V
j=1
hence ϕ ∈ C0 (X).
Let ² > 0 be given. As limk (ck /k) = 0, we can choose k so big that (ck /k) max1≤j≤r |xj | < ². We can also
suppose that n is so big that k · λ(W ) < ². Define λW (.) = λ(. | W ), the conditional probability given W .
For δ sufficiently small we get
¯
¯
Z
Z
exp(itϕ(x)) dλW (x) −
Define
f=
¯
exp(itx) dν(x)¯ < ² for t ∈ (−k, k).
n−k
ck X
ϕ ◦ T j;
k j=0
we thus have kf k∞ < ². For s = k − 1, . . . , n − 1 we have
Z
T −s W
it
exp( Sk (f )) dλ =
ck
Z
Z
exp(it · ϕ ◦ T ) dλ = λ(W )
s
T −s W
exp(itϕ) dλW .
4
Therefore,
PIERRE LIARDET AND DALIBOR VOLNÝ
X Z
¯ n−1
¯
s=k−1
T −s W
hence
it
exp( Sk (f )) dλ − (n − 1 − k)λ(W )
ck
¯
¯E exp( it Sk (f )) −
ck
Z
Z
¯
exp(itx) dν(x)¯ ≤ ²
¯
exp(itx) dν(x)¯ ≤ 5².
There thus exists a sequence of positive real numbers (ηk )k , converging to 0, and functions fk ∈ C0 (X) such
that
Z
¯
¯E exp( it Sk (fk )) −
ck
¯
eitx dν(x)¯ < ηk for t ∈ [−k, k],
lim kfk k∞ = 0.
k→∞
For each k = 1, 2, . . . there exists an open neighborhood U(fk ) of fk such that for f ∈ U(fk )
¯
¯
¯E exp( it Sk (f )) − E exp( it Sk (fk ))¯ < 1 , t ∈ [−k, k],
ck
ck
k
i.e.
¯
¯E exp( it Sk (f )) −
ck
Z
¯ 1
eitx dν(x)¯ < + ηk , t ∈ [−k, k].
k
Let bk , k = 1, 2, . . . be positive real numbers such that limk bk = +∞ and limk (bk /ck ) = 0. For each k define
Uk = {g ∈ C0 (X) : kgk∞ ≤ bk }
and
Hk =
∞
[
[U(fn ) + (U − I)Un ], H =
n=k
The sets Hk are open and by Lemma 1,
∞
\
Hk .
k=1
∞
S
(U − I)Un is dense in C0 (X), so that H is a dense Gδ set.
n=k
Let f ∈ H. For any positive integer k we have f ∈ Hk , thus there exists n ≥ k and a decomposition
f = g 0 + g 00 − U g 00 with g 0 ∈ U(fn ), g 00 ∈ Un . Hence
¯
¢
¡
¯E exp it Sn (g 0 ) −
cn
Z
¯
1
eitx dν(x)¯ ≤ + ηn for t ∈ [−n, n]
n
and
k
1
2bn
1
Sn (g 00 − U g 00 )k∞ = kg 00 − U n g 00 k ≤
.
cn
cn
cn
Therefore there exists a strictly increasing sequence of integers nk such that the distributions of (1/cnk )Snk (f )
weakly converge to ν. ¤
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
5
3. Rotations on the circle.
3.1 Absolutely continuous and smooth functions on the circle.
Let T be the unit circle represented as the unit interval [0, 1) with the Borel σ-algebra and the Lebesgue
probability measure λ. For an irrational number α ∈ [0, 1), T = Tα denotes the rotation x 7→ x + α (mod 1)
on T. The functions on T will be often called cocycles. A cocycle F of the form F = G − G ◦ T is called a
coboundary, G is its transfer function.
We consider the following spaces:
– A0 is the space of all absolutely continuous cocycles F for which
R1
0
F (t) dt = 0,
– L0 is the space of all Lipschitz functions from A0 ,
– C0k is the space of all cocycles with continuous k−th derivative and with zero integral (1 ≤ k < ∞).
An absolutely continuous cocycle F has a derivative f = F 0 a.e.; for F ∈ A0 we have f ∈ L1 (T) and for
R1
F ∈ L0 ⊂ A0 we have f ∈ L∞ (T). By the definition of f, 0 f (x) dx = 0 and we have
Z 1Z u
Z t
f (x) dx −
f (x) dxdu , 0 ≤ t < 1 .
(2)
F (t) =
Therefore, F (0) = −
L0 and
R1Rt
0
0
0
0
0
f (x)dxdt. Formula (2) thus defines a bijection between A0 and L10 , and between
L∞
0 .
On the spaces A0 and L0 we introduce respectively the norms
Z 1
|F 0 (t)| dt
kF kA0 =
0
and
kF kL0 = ess sup|F 0 (t)|.
t∈T
When there is no danger of a confusion the norms will be used without subscripts. A0 and L0 with the
norms introduced above are Banach spaces and the relation (2) gives their isomorphisms onto L10 and L∞
0
respectively.
Let F ∈ C0k , 1 ≤ k < ∞. Then the k−th derivative F (k) is a continuous function on C0 (T) and (2) shows
that we can (by iterations) get F back from F (k) . The space C0k is thus isomorphic to C0 (T) and the metric
inherited from C(T) furnishes C0k with the standard topology of C0k as defined e.g. in [Volný 1]. On C0∞ we
have the coarsest topology with respect to which all the functionals from all C0k , 1 ≤ k < ∞, are continuous.
In the Proof of Theorem 1 we took advantage of the fact, that the coboundaries are dense in C0 . In the
next two lemmas we prove that for A0 , L0 and C0p , 1 ≤ p < ∞, this holds, too.
Lemma 2. Let E be any of the spaces A0 , L0 or C0k , 1 ≤ k ≤ ∞. If F ∈ E and F 0 is a coboundary, and
F 0 = G0 − G0 ◦ T where G ∈ E, then F is also a coboundary and F = G − G ◦ T .
Proof. Let F ∈ E. Following (2) we have
Z
Z t
0
F (x) dx −
F (t) =
0
Suppose that F 0 = f = g −g ◦T and G(t) =
Rt
0
0
1
Z
u
F 0 (x) dxdu.
0
g(x)dx. Then F (t) =
Rt
0
f (x) dx−G(α) = G(t)−G(t+α). ¤
6
PIERRE LIARDET AND DALIBOR VOLNÝ
Lemma 3. In the space A0 (resp. C0k , 1 ≤ k ≤ ∞), the set of the coboundaries
G − G ◦ T,
with G absolutely continuous (resp. G ∈ C0k ), is dense.
In L0 , the set of coboundaries with G measurable is dense.
Proof. The set of all coboundaries g − g ◦ T , g ∈ L10 , is dense in L10 . Their images in A0 in the isomorphism
between L10 and A0 are thus also dense and by Lemma 2 they are coboundaries with transfer functions from
A0 . In the same way (using Lemma 1) we prove that the coboundaries with continuously differentiable
transfer functions are dense in C01 . Now, Lemma 2 enables to extend the result recursively to all C0p , 1 ≤ p <
∞.
The set of coboundaries with integrable transfer function is not dense in L∞
0 (see [Katok] or [Keane-Volný])
and similarly, we can easily prove that the coboundaries with absolutely continuous transfer functions are
not dense in L0 . The proof of the Lemma for L0 is given in [Volný 1] (and is much more complicated than
in the previous cases).
¤
3.2 The Denjoy-Koksma’s theorem.
∗
(ξ) of ξ is by definition the quantity
Let ξ = (xn )n≥0 be a sequence in T. The discrepancy to the origin DN
¯
1
∗
(ξ) = max ¯t −
DN
0≥t<1
N
X
¯
1[0,t[ (xn )¯ .
0≤n<N
∗
(ξ) = o(1), i.e. ξ is uniformly distributed mod 1.
For xn = T n x = nα + x mod 1, x ∈ [0, 1) = T, we have DN
For all maps f : T → R of bounded variation V (f ) one has the so-called Denjoy-Koksma inequality
(3)
¯
¯N
Z
1
f (t) dt −
0
X
¯
∗
f (xn )¯ ≤ V (f )N DN
(ξ).
0≤n<N
∗
can be asymptotically bounded by a power or
(see e.g. [Kuipers-Niederreiter], Theorem 5.1, p.143). N DN
the logarithm of N , according to the number-theoretical properties of the rotation α. Therefore, the rate
of convergence in the Ergodic Theorem cannot be arbitrarily slow already for the functions with bounded
variation.
3.3 Diophantine approximation and discrepancy.
In the sequel we shall need results on the discrepancy of sequences (nα (mod 1))n . To this aim we recall
some basic facts on continued fraction expansion. By kxk, x ∈ [0, 1), we shall denote min{x, 1 − x}. For
x ∈ R let [x] denote the integer part of x and {x} = x − [x] the fractional part.
Let α = [0; a1 , a2 , . . . ] be the continued fraction expansion of α ∈ [0, 1). The convergents pn /qn of x are
given by the following recurrent formulas (see [Khintchine])
(4)
pn+1 = an+1 pn + pn−1
and qn+1 = an+1 qn + qn−1
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
7
for n ≥ 1, with p0 = 0, p1 = 1, q0 = 1, q1 = a1 . We recall the following basic formulas (see [Khinchine]):
1
(5)
2qn+1
≤ kqn αk ≤
(6)
1
,
kqn αk = |qn α − pn | =
min
kmαk
qn+1
1≤|m|<qn+1
kqn−2 αk = an kqn−1 αk + kqn αk,
qn+1 kqn αk + qn kqn+1 αk = 1 .
(7)
We thus have in particular
an kqn−1 αk ≥
(8)
1
4qn−1
and
qn−1 an kqn αk ≥ 1 −
(9)
2
.
an
Indeed, for an = 1 we have qn = qn−1 + qn−2 ≤ 2qn−1 , hence an kqn−1 αk ≥ 1/2qn ≥ 1/4qn−1 . For an ≥ 2
we have an kqn−1 αk ≥ an /2qn ≥ an /2(an qn−1 + qn−2 ) ≥ 1/4qn−1 . This proves (8). The inequality (9) is
also a consequence of (5) and (7). In fact qn−1 an kqn αk = qn kqn−1 αk − qn−2 kqn−1 αk = 1 − qn−1 kqn αk −
qn−2 kqn−1 αk ≥ 1 −
qn−1
qn+1
−
qn−2
qn
≥1−
2qn−1
qn
≥1−
2
an
as required.
We say that α is of type η if
η = inf{τ ∈ R : there exists c > 0 such that for all q ∈ N, q τ kqαk ≥ c}
(See [Kuipers-Niederreiter], Lemma 3.1, p. 121); if for every c > 0 and τ > 0 there exists q ∈ N with
q τ kqαk < c, we say that α is of type infinity. The next statement gives some useful characterizations of the
type.
Lemma 4. For any irrational number α ∈ [0, 1[, the followings are equivalent
(i) α is of type η,
(ii) η = inf{τ ∈ R : ∃ c > 0, ∀ n ≥ 0, an+1 ≤ qnτ −1 /c} .
(iii) η = inf{τ ∈ R : ∃ c > 0, ∀ n ≥ 0, qn+1 ≤ qnτ /c} .
Proof. Let τ > η and let c > 0 be such that q τ kqαk ≥ c for all natural numbers q. Then for all indices n one
has (cf. (4), (5))
c ≤ qnτ kqn αk ≤ qnτ
Therefore, an+1 ≤
qnτ −1 /c.
1
qn+1
≤
qnτ −1
.
an+1
Reciprocally, assume that for τ ≥ 1 there exists c > 0 such that an+1 ≤ c · qnτ −1
for all n ≥ 0. Then, for all q ∈ N and n such that qn ≤ q < qn+1 we get successively (cf. (5))
q τ kqαk ≥ qnτ kqn αk ≥ qnτ
1
2(an+1 qn + qn−1 )
≥
1 qnτ −1
1
≥ .
4 an+1
4c
Therefore (i) is equivalent to (ii). The proof of the equivalence between (ii) and (iii) is left to the reader. ¤
8
PIERRE LIARDET AND DALIBOR VOLNÝ
3.4 Rochlin towers for rotations.
Let us suppose that n is odd so that qn−1 α − pn−1 is positive. Then [0, 1) splits into two Rochlin towers:
[{jα}, {(qn−1 + j)α}) ,
j = 0, . . . , qn − 1
is the bigger one and
[{qn α}, 1), [{(j + qn )α}, {jα}) ,
j = 1, . . . , qn−1 − 1
is the smaller one.
For n even we get
[{qn−1 α}, 1) , [{(qn−1 + j)α}, {jα}) ,
j = 1, . . . , qn − 1
as the bigger tower and
j = 0, . . . , qn−1 − 1
[{jα}, {(j + qn )α})
as the smaller one.
4. Rotations with unbounded partial quotients.
In this section α will denote an irrational number in [0, 1) with unbounded partial quotients.
Theorem 2. Let E be one of the spaces A0 , L0 , C0p , 1 ≤ p ≤ ∞. Let nk , Bk , ck be positive integers and
limk nk = ∞. Let us assume (10) where
for E = A0 , L0 , we take p = 1,
for E = C0r , 1 ≤ r < ∞, we take p = r,
for E = C0∞ we take all positive integers p:
Bk /qnk → 0,
(10)
ck qnp k −1 /Bk → 0,
and ck → ∞ ,
ck qnp k −1 /Bk → 0,
ck = 1 for all k
or
Bk /qnk → 0,
and {Bk α} → 0
(in particular (10) implies limk ank = +∞). Then there exists a dense Gδ set of F ∈ E such that
(11)
the subsequences of (
1
SB (F )) converge in distribution to all
ck k
probability laws .
Remark. The sequences (nk )k , (Bk )k , (ck )k , for which the assumptions of Theorem 2 are fulfilled, can be
found:
for C = A0 , L0 if α has unbounded partial quotients,
for E = C0p if the type of α is greater than p,
for E = C0∞ if α is of type infinity.
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
9
Taking in (10) the sequence of ck constant has been motivated by the following application: For a real
cocycle F on the circle we define the skew product TF , i.e. the transformation
TF (x, y) = (T x, y + F (x))
on the cylinder T × R preserving the product measure. If the transformation TF is ergodic, we usually say
that the cocycle is ergodic.
If α is as in the Remark, the assumptions of Theorem 2 are fulfilled with ck = 1 for all k, hence for a
dense Gδ subset of F ∈ E the subsequences of SBk (F ) converge in probability to all constants. Hence, the
Essential Value Condition which is sufficient for ergodicity of F (see [Schmidt]) is fulfilled, namely:
For every set B of positive measure, a ∈ R and ² > 0, there exists n such that
λ(B ∩ T −n B ∩ {a − ² < Sn (F ) < a + ²}) > 0.
As a corollary to Theorem 2 we thus get
Theorem 3. Under the same assumptions as in Theorem 2, there exists a dense Gδ set of ergodic cocycles
F ∈ E.
The result has been known for smooth functions, for absolutely continuous ones it seems to be new.
Theorem 3 (as well as the corresponding version of Theorem 2 using ck ≡ 1) was found during the second
author’s stay in Toruń thanks to Professor Mariusz Lemanczyk.
Proof of Theorem 2. The C0p norm will be denoted by k.kC,p , 1 ≤ p < ∞. Similarly as in the proof of
Theorem 1, Γ denotes a countable and dense set of probability measures ν on R supported by finite sets of
R
(now, not necessarily rational) numbers and x dν(x) = 0.
We suppose that ν ∈ Γ is fixed; similarly as in the proof of Theorem 1 it suffices to show that (11) holds for
a dense Gδ set of F ∈ E. From ow on, functions on the circle will be understood as 1-periodic functions on
the real line.
Let H be a step function on [0, 1) with λ ◦ H −1 = ν; H is constant on the intervals (xi , xi+1 ), 0 = x0 <
x1 < · · · < xm = 1. Let 0 < δ <
min
0≤i≤m−1
(xi+1 − xi )/3, x0i = xi + δ, x00i = xi+1 − δ, 0 ≤ i ≤ m − 1.
We shall define a function H1 ∈ E which equals H on the intervals (x0i , x00i ), |H1 | ≤ |H| and H1 (0) = 0 =
(j)
(j)
(0)
H1 (1), 0 ≤ j ≤ p (H1
= H1 , p is from the definition of E). In fact, there exists a function h ∈ C ∞ which
is of constant sign on the intervals (xi , x0i ), and on (x00i , xi+1 ) (the sign of 0 is + by definition), equals zero
on (x0i , x00i ),
Z
x0i
Z
h dλ = −
xi
for i = 0, . . . , m − 1 and finally h
(k)
xi+1
x00
i
h dλ = H(
(0) = h(k) (1) for all k ≥ 0.
For any positive integer p, the function
Z
H1 (t) =
t
h(z) dz
0
xi + xi+1
),
2
10
PIERRE LIARDET AND DALIBOR VOLNÝ
fulfills our needs. Moreover, for a given p put h1 = h(p−1) for short, then we still have
Z tZ
xp−1
H1 (t) =
Z
···
0
0
x1
h1 (x) dxdx1 . . . dxp−1 .
0
Choosing δ sufficiently small, we can have H1 = H on [0, 1) up to a set of arbitrarily small (positive) measure.
The interval [0, 1) splits into two Rochlin towers as shown in Section 3.4. For simplicity (and without loss
of generality) we suppose that n = nk is odd and we define new Rochlin towers J0 , . . . , Jqn−1 −1 by
n −1
¡ a[
¢
T jqn−1 [0, kqn−1 αk) , Ji = T i J0 for i = 1, . . . , qn−1 − 1.
J0 = [0, an kqn−1 αk) =
j=0
The intervals Ji are mutually disjoint and by (9),
qn−1 −1
λ(
[
Ji ) = qn−1 an kqn−1 αk ≥ 1 −
i=0
2
.
an
Let us denote n = nk when this causes no confusion. Define
h2 (x) =
1
x
h1 (
), x ∈ J0 ,
(an kqn−1 αk)p
an kqn−1 αk
f (x) =



ck
−i
x)
Bk h2 (T
for x ∈ Ji , i = 0, . . . , qn−1 − 1


0
for x ∈ [01) \
Ji ,
i=0
Z tZ
xp−1
F (t) =
0
qn−1
S−1
Z
···
0
x1
f (x) dxdx1 . . . dxp−1 , t ∈ [0, 1) .
0
We thus have
F (p) = f,
kf k∞ =
kh1 k∞
ck
ck p p
≤
2 qn−1 kh1 k∞
p
Bk (an kqn−1 αk)
Bk
(as by (5), an kqn−1 αk ≥ 1/2qn−1 ). By the assumption limk ck qnp k −1 /Bk = 0 hence we can for every ² > 0
find k big enough so that kF kC,p = kf k∞ < ². A straightforward computation gives
x
Bk
) on J0 ,
F (x) = H1 (
ck
an kqn−1 αk
Bk
Bk
F (x) =
F (T −i x)
on Ji , i = 1, . . . , qn−1 − 1, and
ck
ck
qn
−1
[
Bk
F (x) = 0
on [0, 1) \
Ji .
ck
i=0
Denote bk = [Bk /qn−1 ]. By the definition, the functions f ◦ T ` 0 ≤ ` = −i − j · qn−1 < Bk , 0 ≤ i ≤
qn−1 − 1, 0 ≤ j ≤ bk , have the same values on the intervals T r (an kqn−1 αk(x0s , x00s )), 0 ≤ r ≤ qn−1 − 1,
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
11
0 ≤ s ≤ m − 1, except on an interval at both extremities of length at most bk kqn−1 αk = (bk /an )λ(J0 );
moreover limk (bk /ank ) = 0. Therefore, for any ² > 0 we can choose δ sufficiently small and k sufficiently
large such that Bk F = SBk (F ) with probability bigger than 1 − ².
For each of the intervals Jj , j = 0, . . . , qn−1 − 1,
Z
itBk ·F (x)/ck
e
Z
an kqn−1 αk
dx =
itBk ·F (x)/ck
e
Z
dx = an kqn−1 αk
0
Jj
1
eitH1 (x) dx;
0
from this and from 1 ≥ qn−1 an kqn−1 αk ≥ 1 − (2/an ) we get
¯
¯
Z
1
eitBk ·F (x)/ck dx −
0
Z
1
¯
eitH1 (x) dx¯ ≤
0
¯
¯
Z
qn−1 −1
[0,1)\∪j=0
Jj
eitBk ·F (x)/ck dx − (1 − qn−1 an kqn−1 αk)
Z
0
1
¯
4
eitH1 (x) dx¯ ≤
.
an
For any ² > 0, we can thus find F ∈ E and a positive integer k(²) such that kF kC,p < ² and
¯
¯
Z
Z
1
exp(itH1 ) dλ −
0
1
¯
exp(itSBk (F )/ck ) dλ¯ < ²
0
for all t ∈ R and k ≥ k(²). The function H1 can be found equal to H on [0, 1) up to a set of arbitrarily small
(positive) measure, hence there exist F and an new k(²) so that
¯
¯
Z
Z
1
exp(itH) dλ −
0
1
¯
| exp(itSBk (F )/ck ) dλ¯ < ² for t ∈ R and k ≥ k(²).
0
The trigonometric polynomials are dense in the space of continuous functions, therefore we can suppose that
h1 and hence also F are trigonometric polynomials.
For any probability distribution ν we can thus find a countable set K of positive integers k and trigonometric polynomials Fk , k ∈ K, such that
P
(i) the series F = k∈K Fk converges in C0p ,
(ii) the distributions of (1/ck )SBk (Fk ) converge weakly to ν as k → ∞.
The coboundaries are dense in C0p , hence we can suppose that Fk are coboundaries, i.e. Fk = Gk − Gk ◦ T
(see Lemma 3); we can thus guarantee
P
(iii) (1/ck ) j∈K,j<k SBk (Fj ) → 0 in the measure as k → ∞
(we can take advantage of {Bk α} → 0 or from ck → ∞) and choosing the norms of Fj , j ∈ K, j > k,
sufficiently small, we can get
P∞
(iv) (1/ck ) j>k,j∈K SBk (Fj ) → 0 in the C0p norm as k → ∞.
Using the same topological arguments as in the end of the proof of Theorem 1 we can prove the genericity
results. As kF kA0 ≤ kF kL0 ≤ kF kC,r , r =, 2, . . . we get the theorem in all spaces E = A0 , L0 , C01 , C02 , . . . .
P
We can choose the functions Fk so that k kFk kC,k < ∞. As kF kC,1 ≤ kF kC,2 ≤ . . . , we can get the
result in C0∞ , too. ¤
12
PIERRE LIARDET AND DALIBOR VOLNÝ
Theorem 4. Let F ∈ C0p . Then there exist positive numbers C, B1 , B2 , . . . such that
∞
X
Z
Bi2
≤
(F (p) )2 (x) dx
i=1
and for any positive integer q,
¡
q
qn−1
q1 ¢
|Sq (F )| ≤ C Bn p + Bn−1 p + · · · + B1 p
qn−1
qn−2
q0
(12)
and
Z
(13)
¡ ¡ q ¢2
¡ qn−1 ¢2
¡ q1 ¢2 ¢
2
+ Bn−1
+ · · · + B12 p
,
Sq2 (F )(x) dx ≤ C 2 Bn2 p
p
qn−1
qn−2
q0
where qn−1 < q ≤ qn .
As a corollary to Theorems 2 and 4 we get
Theorem 5. Let α be of type η, ² > 0. Then:
1. Let p ≥ 1 be an integer and F be an arbitrary function from C0p .
If η < p, then there exists a constant C such that
|Sn (F )| < C
for all n.
If η > p then there exists an integer q̄ such that for every q ≥ q̄,
p
|Sq (F )| ≤ q 1− η +² .
(14)
2. If η > p, then for every ² > 0 there exists a dense Gδ set of F ∈ C0p such that
p
λ(|Sq (F )| > q 1− η −² ) > 1 − ²
(15)
for infinitely many positive integers q.
Remark. If we take p = 1, Theorem 5 remains valid and the proof works for A0 and L0 as well as for
C01 .
The upper bound in this case can be also found in [Kuipers-Niederreiter] by combining Theorem 3.2,
p.123, which gives an estimate of the discrepancy of ξ = ({iα})i≥0 where α ∈ (0, 1) is an irrational number
∗
(ξ) = O(N (−1/η)+² ), and the theorem of Koksma-Denjoy which holds for all functions of
of type η, DN
bounded variance. Theorem 5 thus extends the result from [Kuipers-Niederreiter] to k-times continuously
differentiable functions and shows that the bound can be approached for a generic set of functions.
Proof of Theorem 5.
1. Let ², ²0 > 0,
p
η
−
p
η+²0
0
η+²
< ². By Lemma 4(iii) there exists c ≥ 1 such that qk ≤ c · qk−1
for every k ≥ 1.
We thus have
p
0
p
p
,
qkη+² ≤ c η+²0 qk−1
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
13
hence
qk
p
qk−1
p
1− η+²
0
p
p
1− η
+²
p
≤ c η+²0 qk
≤ c η qk
.
0
η+²
, therefore as above
Let n, q be positive integers, qn−1 ≤ q < qn . We have q ≤ c · qn−1
q
p
p
qn−1
p
≤ c η · q 1− η +² .
The numbers Bi are bounded, hence for some constant D (not depending on q),
|Sq (F )| ≤ D
¡ q
p¡
p
1− p +²
1+ p +² ¢
qn−1
q1 ¢
+ p + · · · + p ≤ D · c η q 1− η +² + qn−1η + · · · + q1 η
.
p
qn−1
qn−2
q0
From (4) we get qk ≤ fk , 2qk−1 ≤ qk+1 , k = 1, 2, . . . where (fk )k is the Fibonacci sequence (f0 = f1 and
fk+1 = fk + fk−1 ); if 1 − p/η + ² < 0, then
p
|Sq (F )| < 2Dc η
∞
X
p
1− η
+²
fn
< +∞.
k−1
If 1 − p/η + ² > 0, then for some constant E,
1− p +²
p
q 1− η +² + qn−1η
(16)
p
1− η
+²
+ . . . + q1
p
≤ E · q 1− η +² .
There thus exists a constant K,
p
|Sq (F )| ≤ K(1 + q 1− η +² ) for all q ∈ N.
The first statement of Theorem 5 easily follows.
2. By similar arguments as above we can derive that for any ² > 0 there exist infinitely many n for which
p
1− η
−²
(17)
qn
In fact, choose 0 < ²0 < η such that
p
1− η−²
0
qn
<
qn
p
qn−1
p
η−²0
−
p
η
<
qn
.
p
qn−1
η−²
< ². There are infinitely many n with qn−1
< qn so that
and (17) follows.
p
is infinity. Let (nk )k be an
We have supposed η > p, hence (see Lemma 4(iii)) the limes superior of qn /qn−1
increasing sequence of natural numbers such that limk qnk /qnp k −1 = +∞. Let 0 < δ < 1 and define
h
Bk =
h q
i
qnk i
n
, ck = ( p k )1−δ .
qnk
qnk −1
log qp
nk −1
where [x] denotes the integer part of x ∈ R. Then Bk /qnk → 0 and ck qnp k −1 /Bk → 0, hence (10) holds. By
Theorem 2, for every F from a dense Gδ subset of C0p there exists an increasing sequence of natural numbers
kj such that limj λ(|SBkj (F )| > ckj ) = 1. From this and from (17) it follows that for arbitrarily small ², δ > 0
with 1 −
p
η
−²>0
´
³
p
(1−δ)(1− η
−²)
>1−²
λ |Sq (F )| > qn
14
PIERRE LIARDET AND DALIBOR VOLNÝ
for infinitely many q, qn , q < qn (we take qn = qnk , q = Bk ). The second statement of theorem 5 follows. ¤
For η < p the partial sums Sq (F ) are thus bounded, hence F is a coboundary. On the other hand, if
η > p, there exists a dense Gδ set of F ∈ C0p for which the partial sums Sq (F ) are not stochastically bounded,
hence F are not coboundaries (according to [Moore-Schmidt]).
Using Theorem 2 and Theorem 4 more directly, we can get that
p
= ∞, there exists a dense Gδ set of F ∈ C0p which are not coboundaries;
– if lim supn→∞ qn /qn−1
p
< ∞, the L2 norms of the sums Sq (F ) are for every F ∈ C0p bounded, hence F
– if lim supn→∞ qn /qn−1
is a coboundary with a transfer function in L2 (see, e.g. [Parry and Tuncel]).
This reproves a well known result saying that if the limes inferior of q p kqαk is zero, all functions from C0p
are coboundaries, while in the other case there exists a Gδ set of F ∈ C0p which are not coboundaries (cf.,
e.g. [Baggett and Merrill]). Using methods from the proof of Theorem 7, a function F ∈ C0p can be found,
p
is positive and bounded. By Theorem 4, F is
for which |Sq (F )| are not bounded while lim supn→∞ qn /qn−1
a coboundary with an unbounded transfer function.
Proof of Theorem 4. Let
F (x) =
be the Fourier expansion of F . From
R1
0
X
bk e2πikx
k∈Z
F (x) dx = 0 follows b0 = 0. The function F is real, hence bk = b−k
for all k, b00 = 0. Let
b0k = |k|p bk ,
We thus have
F (x) =
as F (p) ∈ L20 ,
P
k∈Z
k ∈ Z.
X
k∈Z\{0}
b0k 2πikx
e
;
|k|p
|b0k |2 < ∞. Let
F1 (x) =
X
bk e2πikx and F2 (x) =
|k|≤qn−1 −1
For any integer q ≥ 2 we have
|Sq (F1 )| ≤
|k|≥qn−1
X
|bk | · |Sq (e2πikα )|.
1≤|k|≤q−1
Let qn−1 ≤ q < qn . Classically
Sq (e2πikα ) =
1 − e2πikqα
1 − e2πikα
and following [Kuipers-Niederreiter], pp.122-123, we have
1
1
≤
,
|e2πikα − 1|
2 · kkαk
hence
(18)
X
|Sq (e2πikα )| ≤
1
.
kkαk
bk e2πikx .
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
15
Therefore,
|Sq (F1 )| ≤
qj −1
n−1
X
X
j=1 |k|=qj−1
≤
n−1
X
|b0k |
· |Sq (e2πikα )|
|k|p
qj −1
X
1
qp
j=1 j−1 |k|=qj−1
|b0k |
kkαk
From the structure of the Rochlin towers for rotations (see 3.4) and (5) it follows that
qX
m −1
Let K 2 =
k=1
P∞
j=1
qX
qX
m −1
m −1
1
1
1
2
2
≤
·
≤
4q
,
m
kkαk2
kqm−1 αk2
k2
k2
k=1
1/j 2 then by the Schwartz inequality,
qj −1
X
|k|=qj−1
where Bj2 =
m = 1, 2, . . . .
k=1
Pqj −1
|k|=qj−1
|b0k |
≤ 2Kqj Bj
kkαk
|b0k |2 , j = 1, 2, . . . , hence
|Sq (F1 )| ≤ 2K
(19)
n−1
X
Bj ·
j=1
qj
.
p
qj−1
The functions x 7→ Sq (e2πikx ) are mutually orthogonal, hence for
qj −1
X
F1,j (x) =
|k|=qj−1
b0k
· Sq (e2πikx ),
|k|p
j = 1, . . . , n − 1, we have
(20)
E F12 = E
n−1
X
|F1,j |2 ≤ 4K 2
j=1
n−1
X
Bj2 ·
j=1
¡ qj ¢2
p
qj−1
Now, we shall give an estimate for F2 . From the the Rochlin tower described in 3.4, for any interval J of
length kqn−2 αk, all the intervals T j (J) j = 0, . . . , qn−1 −1 are disjoint and the same result is true if we replace
α by −α. Assume that n is even (the remaining case is analogous), n ≥ 2, and let Kn be the interval (mod 1)
[− 12 kqn−2 αk, 12 kqn−2 αk[. For each x ∈ Kn let `(x) be the smallest integer ` > 0 such that T ` (x) ∈ Kn . The
function `(·) takes only two values. In fact ` = qn−1 on the interval [− 12 kqn−2 αk + kqn−1 αk, + 12 kqn−2 αk[
and ` = qn−1 + qn−2 otherwise. Let (rj )j be the increasing sequence of all natural numbers r ≥ qn−1 such
that rα ∈ Kn . Hence, for each j = 1, 2, . . . we have rj+1 − rj ∈ {qn−1 , qn−1 + qn−2 } and for all x ∈ Kn we
still get from the Rochlin tower and (5)
X
`(x)−1
j=1
`(x)−1
X
1
1
1
≤
+
2
2
kx + jαk
{x
+
jα}
{−x
−
jα}2
j=1
≤
(21)
4
1
2
4 kqn−2 αk
≤ 16K 2
X 1
j2
j=1
`(x)−1
1
2
≤ 64K 2 qn−1
.
kqn−2 αk2
16
PIERRE LIARDET AND DALIBOR VOLNÝ
It follows from (18) and (21)
rj+1 −1
X
|k|=rj +1
|b0k |
|Sq (e2πikα )| ≤
|k|p
rj+1 −1
X
|k|=rj +1
|b0k |
≤ 16K · qn−1 · Bj00 /rjp
|k|p kkαk
Prj+1 −1
where Bj00 = ( |k|=r
|b0k |2 )1/2 , j = 1, 2, . . . . From rj ≥ j · qn−1 we get
j +1
∞
X
rj+1 −1
X
j=1 |k|=rj +1
∞
qn−1 X Bj00
|b0k |
qn−1
2πikα
|S
(e
)|
≤
16K
≤ 16B 00 K 2 p
q
p
p
|k|p
qn−1
j
qn−1
j=1
P∞
2
where B 00 = ( j=1 B 00 j )1/2 . Finally
∞
X
|b0rj |
j=1
q
|Sq (e2πirj α )| ≤
rjp
∞
X
|b0rj |
j=1
(j · qn−1 )p
0
≤ Bn−1
K
q
p
qn−1
P∞
0
where Bn−1
= ( j=1 |b0rj |2 )1/2 . Therefore,
0
K·
|Sq (F2 )| ≤ 2Bn−1
qn−1
q
+ 16K 2 B 00 p ;
p
qn−1
qn−1
from this and from (20) we get (12).
Similarly as in the case of F1 , we can show that
EF22 ≤ C12 ((
q 2
)
p
qn−1
+(
qn−1 2
) ),
p
qn−1
where C1 is a constant. From this and from (20) we get (13).
¤
5. Rotations with bounded partial quotients.
In the case of bounded partial quotients, the results will be much more meager than in the preceding
case. If α has bounded partial quotients, it is well-known that F ∈ A0 with a square integrable derivative
is a coboundary. Therefore, each function from L0 (and hence also from C0k , 1 ≤ k < ∞) is a coboundary.
R
The next result shows that for the functions with |F 0 |2 dx = ∞ this need not be the case. The proof of
this theorem works for any irrational number α but only the bounded partial quotients case is interesting
now. The other case follows from Theorem 2 and the fact that any measurable coboundary is stochastically
bounded.
Theorem 6. Let α have bounded partial quotients an . Then there exists a dense Gδ subset of A0 of
functions which are not coboundaries.
As we shall see in the proof, we can guarantee an existence of a dense Gδ set of functions F ∈ A0 such
that for some ck → ∞, nk → ∞ (depending on F ), the distributions of (1/ck )Snk (F ) weakly converge to the
standard normal law. At this moment, however, we are not able to prove a result as strong as Theorem 2.
It is even not clear whether a distribution which is not infinitely divisible can be a weak limit point. On the
other hand, the bounds for the partial sums can be found in a more satisfactory way.
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
17
Theorem 7. Let α be an irrational number with bounded partial quotients an . Then for each F ∈ A0 ,
Sn (F ) = o(log n).
Moreover, for any sequence of positive numbers (cn )n which converges to 0, there exists a dense Gδ set of
f ∈ A0 for which
kSn (f )k∞ ≥ cn log n
for infinitely many n.
The first part of Theorem 7 is well-known with several proofs; for completeness, we shall show one of
them. Before proving the theorems, let us state the followinf general lemma:
Lemma 5. For x, t ∈ [0, 1) and n = 1, 2, ... let
Fn (x, t) =
1
#{i : 1 ≤ i ≤ qn , {iα} < x, {qn {iα}} < t}.
qn
Then
lim |Fn (x, t) − x · Fn (1, t)| = 0.
n→∞
Proof of the lemma. We have {qn {iα}} = {iqn α} = {i{qn α}}. Without loss of generality we can suppose
that n is even (the “odd” case is similar); then {qn α} = kqn αk ≤ 1/qn+1 so that {qn {iα}} = i{qn α},
i = 1, . . . , qn . Let x, t ∈ [0, 1); if qn kqn αk < t, then Fn (1, t) = 1 and Fn (x, t) =
1
qn #{i
: 1 ≤ i ≤ qn , {iα} <
x, } so that the result follows from the uniform distribution mod 1 of the sequence (kα)k . Now, assume
t ≤ qn kqn αk and recall that the sequence (kα)k is well distributed mod 1 (see [Kuipers-Niederreiter]).
Therefore limM →+∞ r(M )/M = 0, where r(M ) = sup|x · M − #{i : j ≤ i ≤ j + M − 1, {iα} < x}|.
j≥1
Let M be given and put qn = k · M + p with 1 ≤ p ≤ M − 1. Given any ² > 0 we can choose
qn , M and k such that r(M )/M < ²/2, 1/k < ²/2. There exists one number j, 0 ≤ j ≤ k, such that
(j · M ){qn α} ≤ t < ((j + 1)M − 1){qn α}. Therefore,
|x · #{i : 1 ≤ i ≤ qn , {iqn α} < t} − #{i : 1 ≤ i ≤ qn , {iα} < x, {iqn α} < t}|
≤
j−1
X
|x · M − #{i : ` · M + 1 ≤ i ≤ (` + 1)M, {iα} < x}| + M
`=0
≤ k · r(M ) + M ≤ ²qn .
¤
For k = 1, 2, ... let dk be a positive number, I(k) = I = [0, δk ), νk be a positive integer and nk =
Pνk
i=1 q`i .
The concrete values of dk , δk , νk and `i will come out from the proof. We shall suppose that δk ¿ 1/2nk .
On [0, 1) we define
dk
χI (t) − dk ,
δk
Z t
Z
fk (x) dx, Fk (t) = F̃k (t) −
F̃k (t) =
fk (t) =
(22)
0
0
1
F̃k (x) dx.
18
PIERRE LIARDET AND DALIBOR VOLNÝ
We thus have Efk = 0, E|fk | = 2dk (1 − δk ). We define (for a more convenient use of the Rochlin towers we
replace the former definition here)
Sn (f ) = f + f ◦ T −1 + ... + f ◦ T −n+1 ;
thus,
Snk (fk ) = Sq`1 (fk ) + Sq`2 (fk ) ◦ T −q`1 + ... + Sq`ν (fk ) ◦ T
−q`1 −...−q`ν
k
k −1
.
Let us denote
Z
t
Fk,j (t) =
0
Z 1
Sq`j (fk ◦ T −q`1 −...−q`j−1 )(x) dx−
Z
0
Z
F̃k,j (t) =
0
u
0
t
Sq`j (fk ◦ T −q`1 −...−q`j−1 )(x) dxdu,
Sq`j (fk ◦ T −1 )(x) dx,
j = 1, ..., νk , t ∈ [0, 1).
R1
The functions Sq`j (Fk ) ◦ T −q`1 −...−q`j−1 , F̃k,j ◦ T 1−q`1 −...−q`j−1 − 0 F̃k,j (t) dt and Fk,j have the same derivaPνk
Fk,j .
tives and zero means, hence are equal. Similarly, Snk (Fk ) = j=1
In each of the intervals [i/q`j , (i + 1)/q`j ), 0 ≤ i ≤ q`j − 1, there is just one of the points T (0), ..., T q`j (0)
([Kesten]). The function F̃k,j is piecewise linear. In the sequel we consider the limit case δk = 0 (k is
0
=
considered as fixed), then F̃k,j has jumps of height dk at points T i (0), 1 ≤ i ≤ q`j , the derivative F̃k,j
−dk · q`j in all other points and F̃k,j (0) = 0. Therefore, F̃k,j (i/q`j ) = 0 for all 0 ≤ i ≤ q`j − 1 and we denote
by ξi the number {q`j {ui α}}(= q`j {ui α} where the integer ui is determined by {ui α} ∈ [i/q`j , (i + 1)/q`j )
and 1 ≤ ui ≤ q`j . Finally the function F̃k,j on the interval (ui , ui+1 ] is linear, decreasing with slope −dk · q`j
and its extremum are −dk ξi + dk and −dk ξi+1 .
Proof of Theorem 6. Using the auxiliary functions F̃k,j we shall show that Fk,j can be approximated by
independent random variables (which will be denoted F̄k,j ). From F̃k,j (i/q` ) = 0, 0 ≤ i ≤ q`j − 1, the
0
= −dk q`,j at all other points, we get
existence of jumps of height dk at points T i 0, 1 ≤ i ≤ q`j and F̃k,j
(23)
1 2
d ≤
12 k
Z
1
2
Fk,j
(x) dx ≤
0
1 2
d .
3 k
The probability that F̃k,j < t on [i/q`j , (i + 1)/q`j ) can be computed as
0
1
(ξi −
q `j
1
(ξi +
q `j
|t|
,
dk
|t|
for t < 0, ξi ≥
,
dk
for t < 0, ξi <
|t|
)
dk
t
) for 0 ≤ t, ξi < 1 −
dk
1
for 0 ≤ t, ξi ≥ 1 −
q `j
t
,
dk
t
.
dk
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
19
Therefore, for x ∈ [0, 1) of the type i/q`j , i ∈ {0, ..., q`j − 1}, we have
λ([0, x) ∩ {F̃k,j
P
 i/q` <x
j
< t}) = P

1
q`j
(ξi −
|t|
dk )χ[|t|/dk ,1) (ξi )
1
i/q`j <x q`j
(ξi +
t
dk )χ[0,1−t/dk ) (ξi )
for t < 0
+
P
1
i/q`j <x q`j
χ[1−t/dk ,1) (ξi )
for t ≥ 0.
Let us suppose that t is fixed, t < 0, and denote
g(i) = (ξi −
|t|
)χ[|t|/dk ,1) (ξi ),
dk
0 ≤ i ≤ q`j − 1.
We have 0 ≤ g ≤ 1. Considering g as a random variable on the probability space {0, ..., q`j − 1} with the
R
R1
Pn
measure µ(i) = 1/q`j and using the fact that g dµ = 0 µ(g > z) dz = limn→∞ (1/n) r=1 µ(g > r/n)
uniformly for all g with values in [0, 1), we get
1
q `j
X
g(i) = lim (1/n)
n→∞
0≤i<x·q`j
lim (1/n)
n→∞
n
X
1
q
r=1 `j
X
χ[r/n,1) (g(i)) =
0≤i<x·q`j
n
X
1
|t|
r
#{i : 0 ≤ i < x · q`j , ξi ≥
+ }.
q
d
n
`
k
j
r=1
Let the positive integer n and ² > 0 be fixed. From the definition of ξi and from Lemma 5, for `j sufficiently
big we get
n
¯1 X
|t|
r
1
¯
#{i : 0 ≤ i < x · q`j , ξi ≥
+ } −
¯
n r=1 q`j
dk
n
n
|t|
r ¯¯
xX 1
#{i : 0 ≤ i ≤ q`j − 1, ξi ≥
+ }¯ < ² .
n r=1 q`j
dk
n
For ² > 0 and q`j sufficiently big we thus have
(24)
|λ([0, x) ∩ {F̃k,j < t} − x · λ(F̃k,j < t)| < 2² .
The same result we get for t ≥ 0.
For any η > 0 we can find finite valued functions F̂k,j on [0, 1) such that kFk,j − F̂k,j k2 < η, and for each
R1
t ∈ R there exists an interval Jt ⊂ R, t ∈ Jt for Jt 6= ∅, λ({F̂k,j = t}) = λ({Fk,j ∈ Jt }), 0 F̂k,j (x) dx = 0,
j = 1, ..., νk . From (24) and the definition of F̂k,j follows that if the numbers `j+1 − `j are sufficiently big,
Xn ¡
¢
|λ F̂k,j = tj |F̂k,j−1 = tj−1 , . . . , F̂k,1 = t1 ) − λ(F̂k,j = tj :
λ(F̂k,1 = t1 , ..., F̂k,j−1 = tj−1 ) > 0} < η .
Therefore, for any ² > 0 and `j+1 − `j , 1 ≤ j ≤ νk − 1 sufficiently big, there exists a sequence F̄k,1 , ..., F̄k,νk
of independent random variables (functions on [0, 1)) with
Z
(25)
1
F̄k,h (x) dx = 0
0
kF̄k,j − Fk,j k2 < ²,
j = 1, ..., νk .
20
PIERRE LIARDET AND DALIBOR VOLNÝ
R1 2
By (23) we have (1/12)d2k ≤ 0 Fk,j
(x) dx ≤ (1/3)d2k . For ² sufficiently small we can suppose (1/20)d2k ≤
R1
R1 2
Pνk
F̄k,j (x) dx ≤ (1/2)d2k , so that for Σk = j=1
F̄k,j , σk2 = 0 Σ2k (x) dx, we have
0
νk 2
νk 2
d ≤ σk2 ≤
d .
20 k
2 k
Taking νk so big that d2k νk → ∞ as k → ∞ we thus get |F̄k,j |/σk → 0 as k → ∞ independently of j and by
the CLT (see, e.g., [Petrov, Thm.13]),
νk
1 X
F̄k,j → N (0, 1)
σk j=1
in distribution as k → ∞.
Letting `j increase sufficiently rapidly we can make the ² in (25) sufficiently small so that
νk
1
1 X
Fk,j =
Sn (Fk )
σk j=1
σk k
converge to N (0, 1) as well.
By (22), the A0 norm of Fk is not greater than 2dk . Hence, choosing dk so that
P∞
k=1
P∞
k=1
dk < ∞ we get
Fk = F ∈ A0 .
By Lemma 3, each of the functions Fk can be in A0 arbitrarily closely approximated by a coboundary with
a transition function in A0 ; without loose of generality, we thus can replace the Fk s by the coboundaries,
Pk−1
hence each of the partial sums i=1 Fi becomes a coboundary. We can thus choose the numbers νk so big
Pk−1
with respect to i=1 νi that
X ¢
¡ k−1
1
kSnk
Fi k2 = 0 .
lim
k→+∞ σk
i=1
When F1 , ..., Fk are given, we choose dk+1 , dk+2 , ... so small that
∞
X
1
kSnk (
Fi )k2 = 0.
k→+∞ σk
lim
i=k+1
This way, (1/σk )Snk (F ) → N (0, 1) in distribution as k → ∞. As σk → ∞, the partial sums Sn (F ) cannot
be stochastically bounded. By [Moore-Schmidt], F thus cannot be a coboundary.
The set of coboundaries G − G ◦ T , G ∈ A0 , is dense in A0 . Hence, for each N, M ∈ N the set
HN (M ) = {F ∈ A0 : (∃n ≥ N )(λ(|Sn (F )| > M ) > 1/2} contains a dense and open subset of A0 . The set
∞
∞ T
T
HN (M ) = H thus contains a dense Gδ subset of A0 and for each F ∈ H, the sequence Sn (F ) is not
N =1M =1
stochastically bounded, hence is not a coboundary. ¤
Proof of Theorem 7.
First, we shall prove the second part of the Theorem. Let K be a positive integer satisfying an ≤ K − 1
for all partial quotients an . Let m be a positive integer (the value will be specified later) and let k be fixed.
We put `i = i · m, i = 1, . . . , νk .
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
21
Let n = `j = m · j be a given even number, ξi = {qn {ui α}} where {ui α} ∈ [i/qn , (i + 1)/qn ), 0 ≤ ui ≤ qn .
Using the same argument as in the proof of Lemma 5, we have ξi = {ui {qn α}}, hence 0 ≤ ξi ≤ qn {αqn }
(≤ 1). Notice that by (4),
qu+2 = au+1 qu+1 + qu ≤ K · qu+1
for all integers u ≥ 0. By (7), (4), (5) we thus get
1 − qn kqn αk ≥ 1 − qn+1 kqn αk = qn kqn+1 αk ≥ qn /(2qn+2 ) ≥ qn /(2K 2 qn ) = 1/(2K 2 ),
hence
ξi ∈ (0, 1 − 1/(2K 2 )], 0 ≤ i ≤ qn − 1
so that the maximun dk ξi + dk of F˜k,j on the interval ({ui α}, {ui+1 α}) is greater than dk /(2K 2 ). From the
definition of the function F̃k,j we get that
F̃k,j (x) ≥
dk
1
for x ∈ ({ui α}, {ui α} +
]
4K2
4qn K 2
i = 0, . . . , qn − 1. A similar situation happens when n is odd.
For m big enough we have qu+m ≥ 12K 2 qu for all u = 1, 2, . . . and q`1 + · · · + q`j−1 < q`j . Hence, there
exist x ∈ [0, 1) such that
F̃k,j (T 1−q`1 −···−q`j−1 x) ≥
dk
for all j = 1, 2, . . . , νk , `j = j · m,
4K 2
hence
Snk (Fk (x)) ≥ νk
From qn+1 ≤ Kqn we get
nk =
νk
X
qm·j ≤
j=1
νk
X
dk
.
4K 2
K m·j ≤ K m(νk +1) ,
j=1
hence
1
logK nk ≤ νk + 1.
m
Therefore,
Snk (Fk (x)) ≥
dk
(logK nk − m).
4K 2 · m
We can choose νk as big as we need, hence we can get
Snk (Fk (x)) > 3cnk log2 nk
with limk cnk log2 nk = +∞. The functions Fk can be in A0 arbitrarily closely approximated by coboundaries
with transfer functions in A0 (see Lemma 3), we can thus replace them by the coboundaries. Having the
numbers n1 , . . . nk−1 fixed, we can thus choose nk so big that
k−1
X
¯
¯
¯ sup Sn (Fj )¯/(cn log2 nk ) ≤ 1 ;
k
k
2
j=1
22
PIERRE LIARDET AND DALIBOR VOLNÝ
choosing the number dk sufficiently small we get
∞
X
¯
¯
¯ sup Sn (Fj )¯/(cn log2 nk ) < 1 ,
k
k
2
j=k+1
and
F =
∞
X
Fk
k=1
converging in A0 . We have
sup Snk (F )/cnk log2 nk > 2
for infinitely many integers k.
For any G ∈ A0 we have
Snk (F + G − G ◦ T )/cnk log2 nk > 1
for infinitely many integers k. The set of coboundaries G−G◦T , G ∈ A0 , is dense in A0 (see Lemma 3). Using
the same approach as in the proof of Theorem 3 we can see that the set Hn = {F : ∃N ≥ n, sup SN (F ) >
∞
T
Hn is a dense Gδ subset of A0 and
cN log2 N } is dense and open in A0 and for each n = 1, 2, . . . , H =
n=1
for each F ∈ H, Sn (F ) > cn log2 n infinitely many times. This proves the second part of Theorem 7.
Let us suppose that for some c > 0 and F ∈ A0 we have
(26)
lim sup sup Sn (F )/ · log2 n > c.
n→∞
For any bounded function G we then have lim supn→∞ sup Sn (F + G − G ◦ T )/ log2 n > c. By Lemma 3 the
set of coboundaries G − G ◦ T with G ∈ A0 is dense in A0 , hence for every k > 0, in every open nonempty
subset of A0 there exists a function G − G ◦ T + k · F , G ∈ A0 . For any N ∈ N there exists n ≥ N with
sup Sn (k · F + G − G ◦ T )/ log2 n > k · c.
This property remains valid for a sufficiently small open neighborhood of G − G ◦ T + k · F , hence the set
∞
∞ T
T
HN,k is a
HN,k = {F ∈ A0 : ∃ n ≥ N, sup Sn (F )/ log2 n > k · c} is open and dense in A0 . Then H =
N =1k=1
dense Gδ subset of A0 , hence nonempty. For each F ∈ H we have
lim sup sup Sn (F )/ log2 n = ∞.
n→∞
∗
of the sequence {nα} is bounded by d · log2 N for some constant d (see [KuipersThe discrepancy N DN
Niederreiter], Theorem 3.4, p.125) and by the Denjoy-Koksma’s theorem, |Sn (F )| ≤ b · log2 n for all n =
1, 2, . . . and b = d · V (F ) where V (F ) denotes the total variation of F . Therefore, lim supn→∞ sup |Sn (F )|
must be finite. This contradiction shows that (26) cannot hold for any F ∈ A0 , hence
lim sup sup |Sn (F )|/ log2 n = 0 .
n→∞
¤
SUMS OF CONTINUOUS AND DIFFERENTIABLE FUNCTIONS IN DYNAMICAL SYSTEMS
23
Remark. The proof of the existence of Fk such that
Snk (F (x)) ≥
d k νk
4K 2
can be easily extended to a function F which has a jump of height d at 1/2 and is linear otherwise (with
constant slope and F (0) = 0). F is a zero mean bounded variation function and using the same approach
as in the preceding proof we can show that there exists a discontinuum of points x for which
Snk (F (x)) ≥
for all integers νk , nk =
Pνk
i=1 qi·m .
d · νk
4K 2
Therefore,
sup Sn (F ) > c · log2 n
for some c > 0 and infinitely many n. The proof of the first part of Theorem 7 thus cannot work for general
bounded variation functions, i.e. that the coboundaries G − G ◦ T with G bounded are not dense in that
space (with respect to the variation topology). In fact, even the coboundaries with measurable transfer
functions are not dense in that space (see [Lemanczyk, Mauduit]; it can be also shown using the methods
from [Volný]).
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