Central limit theorems for the shrinking target
problem.
Nicolai Haydn
∗
Matthew Nicol †, ‡Sandro Vaienti§, Licheng Zhang
¶
October 24, 2013
Abstract
Suppose Bi := B(p, ri ) are nested balls of radius ri about a point p in a
dynamical system (T, X, µ). The question of whether T i x ∈ Bi infinitely often (i.o.) for µ a.e. x is often called the shrinking target problem. In many
P
dynamical settings it has been shown that if En := ni=1 µ(Bi ) diverges then
P
there is a quantitative rate of entry and limn→∞ E1n nj=1 1Bi (T i x) → 1 for
µ a.e.
x ∈ X.
This is a self-norming type of strong law of large num-
bers. We establish self-norming central limit theorems (CLT) of the form
P
limn→∞ a1n ni=1 [1Bi (T i x) − µ(Bi )] → N (0, 1) (in distribution) for a variety
of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalP
ization constants are a2n ∼ E[ ni=1 1Bi (T i x) − µ(Bi )]2 . Dynamical systems to
∗
Mathematics Department, University of Southern California, Los Angeles, 90089-1113, email:
nhaydn@math.usc.edu
†
Department of Mathematics, University of Houston, Houston, TX 77204-3008, email:
nicol@math.uh.edu
‡
MN was partially supported by NSF grant DMS 1101315.
§
Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France and Université de
Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France, Email: vaienti@cpt.univ-mrs.fr.
¶
Department of Mathematics, University of Houston, Houston, TX 77204-3008, email:
xiyao.fei@gmail.com
1
which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions,
piecewise expanding maps. For such central limit theorems the main difficulty
is to prove that the non-stationary variance has a limit in probability.
Contents
1 Introduction
2
2 The setup.
3
2.1
Some lemmas on variance . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Property (SP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Short returns and Assumption (C) . . . . . . . . . . . . . . . . . . .
9
2.4
Bounds on wj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
3 Decay in Lipschitz versus L 1
13
4 Applications to dynamical systems.
20
5 Decay in BV (X) versus L 1
20
6 Applications of Proposition ??.
21
6.1
Piecewise expanding maps in higher dimensions . . . . . . . . . . . .
22
7 Discussion.
27
8 Appendices
28
8.1
Gal-Koksma Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . .
28
8.2
Assumption (C) for expanding systems . . . . . . . . . . . . . . . . .
28
2
1
Introduction
Suppose (T, X, µ) is an ergodic dynamical system and Bi (p) is a nested sequence
of balls about a point p ∈ X. Recently there have been many papers concerning
P
the behavior of the almost sure limit of the normalized sum E1n ni=1 1Bi (p) (x) where
Pn
En :=
i=1 µ(Bi (p)) diverges [4, 26, 11, 13, 14, 19, 24]. If the limit is known to
exist almost surely then {Bi (p)} is said to satisfy the Strong Borel Cantelli property.
Many of the references we mentioned consider more general sequences of sets than
nested balls. The study of hitting time statistics to a sequence of nested balls is
sometimes called the shrinking target problem. In this paper we study self-norming
central limit theorems for the shrinking target problem, namely the distribution limit
P
of a1n ni=1 [1Bi (p) ◦ T i − µ(Bi )] where an is a sequence of norming constants. One
important case for applications is the case where µ(Bi (p)) = 1i , and hence En = log n.
P
Our results are stated for balls satisfying i µ(Bi (p)) = ∞ and µ(Bi (p)) ≤ Ciγ2 where
C2 is a positive constants and 0 < γ ≤ 1. The main difficulty is to establish that
the non-stationary variance has a limit in probability. Our results are limited to nonuniformly expanding systems i.e. those without a contracting direction and are based
upon the Gordin [12] martingale approximation approach (see also [27]).
More generally, this paper is also an attempt to study the statistics of nonstationary stochastic processes arising as observations (which perhaps change over
time) on an underlying dynamical system (which may change over time). Conze
and Raugi [6] studied similar problems for sequential expanding dynamical systems.
Somewhat related results were obtained by Nándori, Szász and Varjú [30] who obtained central limit theorems in the setting in which a fixed observation φ : X → R
was considered on a space on which a sequence of different transformations acted
Ti : X → X, preserving a common invariant measure µ. The main difficulty in [30]
was also controlling the variance, but the setting in which the underlying maps change
but the observation is fixed is simpler in some respects and more difficult in others.
We obtain fairly complete results in the case in which the transfer operator with
respect to the invariant measure is quasicompact in the bounded variation norm.
These results are contained in Proposition 5.1 and Theorem 6.4. For systems in which
3
the transfer operator is quasicompact in a Hölder or Lipschitz space we show that
under the assumption we call (SP) (derived from a Gal-Koksma lemma as formulated
by Sprindzuk [38]) or a form of short returns assumption called Assumption (C) we
have a central limit theorem (Theorem 3.1). Assumption (C) and the SP property
have been shown to hold for generic points in a variety of non-uniformly expanding
systems [5, 15, 21].
In Section 2 we discuss the set-up, describe the martingale approach we use, prove
some general results on variance and discuss the SP property and Assumption (C).
Section 3 gives our results under the assumption of quasi-compactness in Hölder
norms and also some applications. In Section 4 we give our results when we have
quasi-compactness of the transfer operator in the bounded variation norm, and we
give applications to piecewise expanding maps in higher dimensions. The last section
is a concluding discussion, while the Appendices describe the Gal-Koksma lemma
we use and show that Assumption (C) is satisfied for generic points in many of our
applications.
2
The setup.
We suppose that (T, X, µ) is an ergodic dynamical system. Let the transfer operator
R
R
P be defined by φψ ◦ T dµ = P φψ dµ for all φ, ψ ∈ L 2 (µ) so that P is the
adjoint of the Koopman operator U φ := φ ◦ T with respect to the invariant measure
µ. Suppose Bα is a Banach space of functions and kφk1 ≤ Ckφkα where k.kα is the
Banach space norm and k.k1 is the L 1 norm with respect to µ. We assume P restricts
to an operator P : Bα → Bα such that kP n φkα ≤ C1 θn kφkα for all φ ∈ Bα such that
R
φ dµ = 0. This implies exponential decay of correlations of the form, that for some
0 < θ < 1,
Z
n
φψ ◦ T dµ −
Z
Z
φ dµ
ψ dµ ≤ Cθn kφkα kψk1
for all φ ∈ Bα , ψ ∈ L 1 (µ). In our applications we will have the pairs (BV (X), L 1 (µ))
or (Hγ (X), L 1 (X)) where BV (X) is the space of function of bounded variation and
Hγ (X) is the space of Hölder functions of exponent γ. For example if T is a smooth
4
uniformly expanding map of the unit interval X then Bα could be taken as the Banach
space of functions of bounded variation BV (X). In this paper we will consider Lipschitz rather than Hölder functions, as our results and proofs immediately generalize
to the Hölder setting with the obvious changes.
Remark 2.1 The weaker assumption of exponential decay of correlations
Z
Z
Z
n
φψ ◦ T dµ − φ dµ ψ dµ ≤ Cθn kφkα kψk∞
implies that kP n φk1 ≤ Cθn kφkα (by taking ψ to be sign(P n φ)) and hence P contracts
exponentially in the L 1 norm. This assumption is sufficient for all our results on
variance in Section 2, with the exception of the proof of the boundedness of the
terms wj , given in Lemma 2.10 which seems to require our stronger assumption that
kP n φkα ≤ C1 θn kφkα . These estimates on the growth of wj are used in the proof of
Theorem 3.1. If Bα is the space of functions of bounded variation then the wj terms
are easily seen to be uniformly bounded under the assumption kP n φkBV ≤ Cθn kφkBV .
Let p ∈ X and let Bn (p) be a sequence of nested balls about p such that µ(Bn (p)) ≤
C2
nγ
for constants C2 > 0 and 0 < γ ≤ 1. Let 1Bn (p) be the characteristic function of
R
R
Bn (p). We will sometimes write E[φ] or φ for the integral φ dµ when the context
P
is understood. Our standing assumption is that nj=1 µ(Bj (p)) → ∞ as n → ∞.
1Bn (p) may not lie in Bα but we assume we may take an approximation to it, φ̃αn
such that:
(i) |1Bn (p) − φ̃αn |1 ≤
1
n3
and;
(ii) kφ̃αn kα ≤ Cnk where C, k are independent of n;
(iii) φ̃αn ≥ 0, φ̃αn ≥ φ̃αn+1
Remark 2.2 If we are taking a Hölder approximation then condition (ii) is satisfied
for the balls Bi = B(p, ri ) if there exists δ(p) > 0 and C > 0 such that µ{x : r <
d(x, p) < r + } < Cδ(p) . This condition is satisfied if the invariant measure µ has
a density h with respect to Lebesgue measure m such that h ∈ L 1+η (m) for some
η > 0.
5
We define φαn = φ̃αn −
R
φ̃αn so that
R
φαn = 0. For ease of notation we will subse-
quently drop the superscript α on φαn and φ̃αn .
Define φ0 = 0 and for n ≥ 1
wn = P φn−1 + P 2 φn−2 + . . . + P n φ0 =
n
X
P j φn−j
j=1
so that w1 = P φ0 , w2 = P φ1 + P 2 φ0 , w3 = P φ2 + P 2 φ1 + P 3 φ0 etc... For n ≥ 1 define
ψn = φn − wn+1 ◦ T + wn
Recall our assumptions kφ̃n kα ≤ Cnk (so kφn kα ≤ C̃nk ) and kP n φkα ≤ C1 θn kφkα
R
for all φ ∈ Bα such that φdµ = 0; moreover we have the monotonicity property
||φn−j ||α ≤ ||φn ||α , for j < n. These facts immediately imply that kwn kα ≤ C2 kφn kα
(where the constant C2 takes care of the sum of the geometric series of in θ), kwn ◦
T kα ≤ C3 kφn kα (since kU P φkα ≤ Ckφkα for all φ ∈ Bα ) and hence kψn kα ≤ C4 kφn kα .
Using the fact that P (wn+1 ◦ T ) = wn+1 P 1 = wn+1 one may show that P ψn = 0.
Since U P (·) = E[·|T −1 B], P ψj = 0 implies that E[ψj |T −1 B] = 0 and in turn
E[ψj ◦ T j |T −1−j B] = 0 (since T preserves µ). Furthermore ψj ◦ T j is T −j B measurable
for all j ≥ 0.
Following the approach of Gordin we will express
Pn
j=1
φj ◦ T j as the sum of a
(non-stationary) martingale difference array and a controllable error term and then
use the following Theorem 3.2 from Hall and Heyde [16]:
Theorem 2.3 (Theorem 3.2 [16]) Let {Sn,i , Fn,i , 1 ≤ i ≤ kn , n ≥ 1} be a zeromean square-integrable martingale difference array with differences Xn,i and let η 2 be
an almost sure finite random variable. Suppose that:
(a) maxi |Xn,i | → 0 in probability;
P 2
(b) i Xn,i
→ η 2 in probability;
2
(c) E(maxi Xn,i
) is bounded in n;
(d) the σ-fields are nested: Fn,i ⊂ Fn+1,i for 1 ≤ i ≤ kn , n > 1.
Then Sn,kn → Z (in distribution) where the random variable Z has the characteristic function E(exp(− 21 η 2 t2 )).
6
As is common in the application of martingale theory to non-invertible dynamical
systems we will have to consider the natural extension so that we have a martingale
in backwards time. We outline our scheme of proof.
Let (σ, Ω, m) be the natural extension of (T, X, µ). Each ψj lifts to to a function
ψj∗ on Ω in a natural way, ψj∗ (. . . ω−2 ω−1 .ω0 ω1 . . .) := ψj (ω0 ). To simplify notation we
write simply ψj instead of ψj∗ .
We define scaling constants by a2n = E(
Pn
j=1
φj ◦ T j )2 . This sequence of constants
play the role of non-stationary variance. Giving estimates on the growth and nondegeneracy of an in this non-stationary setting is more difficult than in the usual
stationary case.
We define a triangular array Xn,i = a1n ψn−i ◦ σ −i , i = 1, . . . , n, n ∈ N, and
Pi
i
put Sn,i =
j=1 Xn,j for the partial sums (along rows). Then Xn,i is Fi := σ B0
measurable where B0 is the σ-algebra B lifted to Ω. Note that in Theorem 2.3
we take Fn,i := Fi for all n and kn = n. The Fi form an increasing sequence
of σ-algebras. We obtain E[Sn,i+1 |Fi ] = Sn,i + E[Xn,i+1 |Fi ] where by stationarity
E[Xn,i+1 |Fi ] = E[ψn−i−1 |σB0 ] = 0. Hence E[Sn,i+1 |Fi ] = Sn,i and for every n ∈ N
Xn,i is a martingale difference array with respect to Fi .
We will then verify conditions (a), (b), (c) and (d) of Theorem 2.3. The hard part
lies in establishing (b). This is in contrast with the stationary setting where condition
(b) is usually a straightforward consequence of the ergodic theorem. Condition (b) is
established in [30] by using [37, Lemma 3.3.], however in our setting the Lipschitz
norms of the observations φ̃i are unbounded and other techniques have to be used.
P
Once we have established (a), (b), (c) and (d) it follows that limn→∞ a1n n−1
j=0 ψj ◦
Pn
1
j
j
T → N (0, 1) in distribution. In the final step we show that an j=1 [wj ◦ T − wj ◦
Pn−1
T j+1 ] → 0 in L 1 which implies that limn→∞ a1n j=0
φj ◦T j → N (0, 1) in distribution.
2.1
Some lemmas on variance
In this section we establish some preliminary results on the growth of the variance
P
E[( nj=1 φj ◦ T j )2 ] that will be useful in determining the scaling constants an .
For further reference let us notice that kP n φkα ≤ C3 θn kφkα and kφk1 ≤ C3 kφkα
7
and that there exists a constant a such that
X
k
1
.
i3
P j φi k1 ≤
j>a log i
Lemma 2.4
where En =
(2.1)
n
X
1
E(
φi ◦ T i )2 ≥ 1
lim sup
n→∞ En
i=1
Pn
j=1
E(φ2j ).
Proof: By exponential decay of correlations and (2.1) we get for the long term
interactions:
Z
X
φi ◦ T i φj ◦ T j ≤
j>a log i+i
c1
,
i2
where we used exponential decay and our bound kφj k1 ≤ C3 kφj kα ≤ c1 j k , where c1 ,k
R
are independent of j. This bound is from assumption (ii). Recall φj = φ̃j − φ̃j and
kφ̃j k1 ≤
c2
jγ
(for some c2 ). Thus for the short term interactions we get
i+a log i Z
X
i
j
i+a log i Z
X
φi ◦ T φj ◦ T =
j=i+1
φ̃i ◦ T i φ̃j ◦ T j + O(a log iE(φ̃i )2 )
j=i+1
whence
n X
X
i
j
E[φi ◦ T φj ◦ T ] =
i=1 j>i
i
j
E[φ̃i ◦ T φ̃j ◦ T ] +
i=1 j=i+1
Since
n
X
O(a log iE(φ̃i )2 ).
i=1
n
n
n X
X
X
X
i 2
2
E(
φi ◦ T ) =
E(φi ) + 2
E[φi ◦ T i φj ◦ T j ]
i=1
and
log i
n i+a
X
X
Pn Pi+a log i
i=1
j=i+1
Lemma 2.5
i=1
i=1 j>i
E[φ̃i ◦ T i φ̃j ◦ T j ] +
n−1 X
n Z
X
Pn
i=1
i
a log iE(φ̃i )2 ≥ 0 the lemma is proved.
j
φi ◦ T φj ◦ T =
i=1 j=i+1
n Z
X
i=1
8
(φi wi ) ◦ T i
Proof: Recalling that φ0 = 0 this follows by a direct calculation and rearrangement
of terms as
n−1 X
n Z
X
i
φi ◦ T φj ◦ T
j
=
i=1 j=i+1
j−1 Z
n X
X
φi ◦ T i φj ◦ T j
j=2 i=1
=
j−1 Z
n X
X
P j−i φi φj
j=2 i=1
j−1
n Z X
X
P j−i φi )φj
=
(
=
j=2
n Z
X
i=1
wj φj .
j=2
The following lemma is the main result of this subsection:
Lemma 2.6
a2n
= E(
n
X
i=1
i 2
φi ◦ T ) =
n
X
E[ψi2 ]
Z
−
w12
Z
+
2
wn+1
i=1
Proof: Let us first observe that factoring out yields
ψj2 = φ2j + 2φj (wj − wj+1 ◦ T ) + (wj − wj+1 ◦ T )2
2
= φ2j + 2φj (wj − wj+1 ◦ T ) + wj2 + wj+1
◦ T − 2wj wj+1 ◦ T
which when integrated leads to
Z
Z
Z
Z
Z
Z
2
2
2
2
ψj =
φj + 2 φj (wj − wj+1 ◦ T ) + wj + wj+1 − 2 wj wj+1 ◦ T
Z
Z
Z
Z
Z
Z
2
2
2
=
φj + 2 φj wj − 2 P φj wj+1 + wj + wj+1 − 2 P wj wj+1
Z
Z
Z
Z
Z
Z
2
2
2
=
φj + 2 φj wj − 2 P φj wj+1 + wj + wj+1 − 2 (wj+1 − P φj )wj+1
Z
Z
Z
Z
2
2
2
=
φj + 2 φj wj + wj − wj+1
.
9
Since by Lemma 2.5
a2n
=
n
X
E(φ2i )
+2
i=1
n−1 X
n Z
X
Z
n X
2
i
φi ◦ T φj ◦ T =
E(φi ) + 2 (φi wi ) ◦ T
i
j
i=1 j=i+1
i=1
the statement follows by substituting
R
ψj2 −
R
wj2 +
R
2
wj+1
for the terms inside the
sum on the RHS and then telescoping out the expected values of wj2 .
2.2
Property (SP)
Several authors [25, 4] have used a property derived from the Gal-Koksma theorem
(see Appendix) to prove the SBC property for sequences of balls. Later we will show
that in certain settings the (SP) property also implies a CLT. Property SP (where SP
stands for Sprindzuk Property), states that E(S(m,n) )2 ) ≤ CE(S(m,n) ) where fi ≥ 0
P
and S(m,n) = ni=m fi , it is a condition that appears in one guise or another often in
proofs of the Borel Cantelli property (which is easily deduced from this).
Suppose Bi are balls and let fi = 1Bi ◦ T i . If
n X
n
X
E(fi fj ) − E(fi )E(fj ) ≤ C
i=m j=i+1
n
X
E(fi )
(SP )
i=m
for arbitrary integers n > m then the balls are said to have the (SP) property.
2.3
Short returns and Assumption (C)
In this section we discuss a condition on short return times first considered, to
our knowledge, by P. Collet [5].
We have called it Assumption (C), after Col-
let. This condition has been used to establish extreme value statistics [5, 21, 15]
and dynamical Borel-Cantelli lemmas [14, 19]. Assumption (C) is a strong control on measure of points making short returns. If Assumption (C) holds for a
point p then the measure of a set of points in nested balls Bi (p) about p returning to Bi (p) in a time interval smaller than an integer power of − log µ(Bi (p)) is
smaller than µ(Bi (p))η where η is greater than one. Note by Kac’s theorem the
expected return time to µ(Bi (p)) is of order µ(Bi (p))−1 , so an integer power of
10
− log µ(Bi (p)) is indeed a very short return. This condition fails for periodic points,
as a fixed fraction of the mass of a ball returns after the period. Heuristically if
the first return times to Bi (p) follow an exponential law (which one somehow expects for generic points) then limi→∞
and hence limi→∞
1
µ{x
µBi (p))
1
µ{x
µBi (p))
∈ Bi (p) : τ (x) ≤
t). Suppose now we could solve for
t
µ(Bi (p))
k
∈ Bi (p) : τ (x) >
t
}
µ(Bi (p))
t
}
µ(Bi (p))
→ e−t
→ 1 − e−t ∼ t (for small
= (− log(µ(Bi (p)))k , we would then have
µ{x ∈ Bi (p) : τ (x) ≤ (− log(µ(Bi (p))) } ∼ (− log(µ(Bi (p)))k µ(Bi (p))2 . Note that
our assumption µ(Bi ) ≤
C2
iγ
implies that (− log µ(Bi ))k ≥ C(log(i))k for large i.This
train of thought makes Assumption (C) seem reasonable for generic points.
Suppose therefore that p ∈ X and Bi (p) is a nested sequence of balls centered at
a point p, with limi µ(Bi (p)) = 0.
Assumption (C): We say (Bi (p)) satisfies Assumption (C) if there exists η(p) ∈
(0, 1) and κ(p) > 1 such that for all i sufficiently large
µ(Bi (p) ∩ T −r Bi (p)) ≤ µ(Bi (p))1+η
for all r = 1, . . . , (log i)κ .
If (Bi (p)) satisfies Assumption (C) then we can say more about the behavior of
the constants an .
Remark 2.7 Note that our assumption µ(Bi ) ≤
C2
iγ
implies that (− log µ(Bi ))k ≥
C(log(i))k for large i.
Lemma 2.8 Under Assumption (C) there exists a constant C4 and some large a so
that
Z
|φj wj | ≤ C4 µ(Bj−a log j )1+η log j.
Proof: By the contraction property of the transfer operator one has as in (2.1) for a
sufficiently large constant a
X
Z
φj P j−i φi ≤ µ(Bj−a log j )2 .
i<j−a log j
11
Let φj = φ̃j −
R
φ̃j where φ̃j is the Bα approximation to 1Bj (p) . Hence we obtain in
the L 1 -norm: (as φ̃j ≥ 0)
a log j
Z
|φj wj | ≤
X Z
Z
n
φ̃j P φ̃j−n +
Z
φ̃j−n
Z
Z
φ̃j +
φ̃j
n
Z
P φ̃j−n +
Z
φ̃j
φ̃j−n
n=1
+O (j − a log j)µ(Bj−a log j )2
a log j Z
X
n
φ̃j P φ̃j−n + 3µ(φ̃j−n )µ(φ̃j ) + O(j)µ(Bj−a log j )2
=
n=1
a log j
=
XZ
φ̃j P n φ̃j−n + O(j)µ(Bj−a log j )2 ,
n=1
Now by Assumption (C) we have
Z
Z
n
φ̃j P φ̃j−n ≤ φ̃j−n ◦ T n φ̃j−n ≤ µ(Bj−n ∩ T −n Bj−n ) ≤ c1 µ(Bj−n )1+η ,
for n ≤ a log j, and thus
a log j
XZ
φ̃j P n φ̃j−n ≤ c2 aµ(Bj−a log j )1+η log j,
n=1
proving the lemma.
Lemma 2.9 If (Bi (p)) satisfies Assumption (C) then
lim (E(
n→∞
n
X
n
X
φi ◦ T ) )/(
E[φ2i ]) = 1
i=1
i=1
i 2
Proof: Rearranging the sums yields by Lemma 2.5
n
n
n−1 X
n Z
X
X
X
i 2
2
E(
φi ◦ T ) =
E[φi ] + 2
φi ◦ T i φj ◦ T j
i=1
=
i=1
n
X
E[φ2i ] + 2
i=1
i=1 j=i+1
n Z
X
wj φj
j=2
and hence the result follows by Lemma 2.8 as η > 1.
12
2.4
Bounds on wj
We now assume that kφk∞ ≤ Ckφkα which under our assumption on the transfer
operator implies that for a mean-zero function φ ∈ B, kP n φk∞ ≤ Cθn kφkα for some
C, 0 < θ < 1 independently of φ. For example if k.kα were the Banach space of
Hölder functions of exponent α on the unit interval then kφk∞ ≤ Ckφkα . In the
BV or quasi-Hölder norm indicator functions are bounded, and the proof that wj is
uniformly bounded is straightforward in this case; we would like to stress that the
next result is obtained under the general assumption that µ(Bn (p)) ≤
C2
.
nγ
Lemma 2.10 Assume kP n φk∞ ≤ Cθn kφkα then there exists a constant C5 such that
kwj k∞ < C5 for all j.
Proof: For some a > 0 we can achieve
Pn
j=ba log nc
−2
|P j φn−j |∞ ≤ c1
Pn
j=ba log nc
θj (n −
j)k = O(n−2 ) and in particular |P j φn−j |∞ = O(n ) for all j ≥ ba log nc and all n. As
in the previous lemma let φ̃j be the Bα approximation for 1Bj and φj = φ̃j − µ(φ̃j ). In
P log nc j
P φn−j independently
view of the tail estimate it is only necessary to bound ba
j=1
of n.
(i) Bound from below: Since φj ≥ −µ(φ̃j ) ≥ −c2 µ(Bj ) ≥ − jcγ3 (c2 , c3 > 0) one obP log nc c3
P log nc j
n
P φn−j ≥ ba
tains ba
≥ −c4nlog
for some constant c4 independent
γ
j=1
j=1
(n−j)γ
of j and n. Hence wn ≥ −c5 for some c5 > 0 and all n.
(ii) Bound from above: Since 1Bj+1 ≤ 1Bj one has φ̃j+1 ≤ φ̃j and in particular
µ(φ̃j+1 ) ≤ µ(φ̃j ). Hence φj+1 − φj ≤ µ(φ̃j ) − µ(φ̃j+1 ) and (as φ0 = 0)
wm − wm−1 =
m−1
X
P j (φm−j − φm−1−j ) + P m φ0
j=1
≤
m−1
X
µ(φ̃m−1−j ) − µ(φ̃m−j )
j=1
≤
ba log mc X
µ(φ̃m−1−j ) − µ(φ̃m−j ) + O(m−2 ).
j=1
13
Consequently (w1 = P φ0 = 0)
wn =
n
X
(wm − wm−1 ) + w1
m=2
≤
n
X

ba log mc X

m=2
j=1
ba log nc
=
=
n
X
X
j=1
µ(φ̃m−1−j ) − µ(φ̃m−j ) + O(m−2 )
j
m=2∨de a e
ba log nc X

µ(φ̃m−1−j ) − µ(φ̃m−j ) + O(m−2 )
µ(φ̃
) − µ(φ̃n−j ) + O((2 ∨ e ) )
j
a
j
2∨de a e−j
−1
j=1
≤ c6
for a constant c6 independent of n because
ba log nc
X
µ(φ̃n−j ) ≤ c7
j=1
as n → ∞ and
ba log nc
ba log nc
X
a log n
→0
nγ
µ(φ̃
j
2∨de a e−j
) ≤ c8
X
j
(e a )−γ = O(1)
j=1
j=1
for constants c7 , c8 independent of n.
3
Decay in Lipschitz versus L 1
We take Bα to be the space of Lipschitz functions, the arguments we give hold for
Hölder norms with obvious modification. We assume that the transfer operator P ,
when restricted to Lip(X), contracts exponentially:
||P n φ||Lip ≤ Cθn ||φ||Lip
for all Lipschitz functions φ such that
R
(3.1)
φ dµ = 0 where θ ∈ (0, 1) and C are inde-
pendent of φ.
14
This implies
Z
φψ ◦ T n dµ − E[φ]E[ψ] ≤ Cθn kφkLip kψkL 1
(3.2)
for the same θ ∈ (0, 1) and C independent of φ, ψ.
P
P
For a sequence of (nested) balls Bi we put En = ni=1 µ(Bi ) and Sn = ni=1 1Bi ◦T i
for the ‘hit counter’ for an orbit segment of length n. The sequence of balls Bi satisfies
the strong Borel-Cantelli (SBC) property if
Sn (x)
=1
n→∞ En
(3.3)
lim
for almost every x ∈ X.
Theorem 3.1 Assume that the transfer operator, when restricted to Lip(X), contracts exponentially as in (3.1) for some θ ∈ (0, 1).
Suppose Bi (p) be nested balls about a point p such that
µ(Bi (p)) ≤
i
C2
iγ
2
for constants C2 > 0 and 0 < γ ≤ 1. Let
P
i
2
an
µ(Bi ) = ∞ and
P
= E( ni=1 (1Bi ◦
T − µ(Bi ))) .
(I) If the nested sequence of balls (Bi (p)) satisfies Assumption (C) and the SBC property (3.3) then
n
1 X
1Bi ◦ T i − µ(Bi ) → N (0, 1)
an j=1
in distribution.
(II) If (Bi (p)) has the SP property then
n
1 X
(1Bi ◦ T i − µ(Bi )) → N (0, 1).
an i=1
R
Proof: We will let φj = φ˜j − φ˜j ,where φ˜j be a Lipschitz approximation to 1Bj , such
that

1
˜


 kφj − 1Bj k1 < j 3
kφ˜j kLip ≤ Cj k .



φ̃j ≥ 0
15
We define wn = P φn−1 + P 2 φn−2 + . . . + P n φ0 and put ψn = φn − wn+1 ◦ T + wn .
R
P
Then P ψn = P φn − wn+1 + nj=2 P j φn−j+1 = 0 which corresponds to ψn χ ◦ T dµ =
R
χP ψn dµ = 0 for any integrable χ. Note that ||φj ||∞ ≤ ||φj ||Lip , ||φj ||1 ≤ ||φj ||Lip .
Lemma 3.2 There exist constants C6 , k so that
(I) kwn kLip ≤ C6 nk ,
(II) kwn k∞ ≤ C6 ,
n
(III) kwn k1 ≤ C6 log
.
nγ
Proof of Lemma 3.2. (I) By the contraction of the transfer operator for Lipschitz
continuous functions one obtains
kwn kLip ≤
n
X
j
kP φn−j kLip ≤
j=1
n
X
C1 θj kφn−j kLip ≤ c1 nk
j=1
(II) Is a consequence of Lemma 2.10.
(III) For sufficiently large a we get
a log n
||wn ||1 ≤
X
||P j φn−j ||1 +
j=1
≤
≤
||φn−j ||1 +
n
X
j=1
j=a log n+1
a log n
n
X
X
||P j φn−j ||1
j=a log n+1
a log n
X
n
X
||φn−j ||1 +
j=1
||P j φn−j ||Lip
C1 θj ||φn−j ||Lip
j=a log n+1
log2 n
≤ C(a log n)µ(Bn−a log n ) + c4 2
n
log n
≤ c5 γ
n
for some c4 , c5 independent of n.
Now put C6 = max(c1 , c5 ).
P
As before let (σ, Ω, m) be the natural extension of (T, X, µ) and put a2n = E( nj=1 φj ◦
T j )2 for the rescaling factors where the ψj lift to Ω in a natural way. Again we put
Xn,i =
1
ψ
an n−i
◦ σ −i , i = 1, . . . , n which are Fi = σ i B0 measurable where B0 is the
16
σ-algebra B lifted to Ω. The Fi form an increasing sequence of σ-algebras. We put
P
Sn,i = ij=1 Xn,j , i = 1, . . . , n (kn = n), and obtain E[Sn,i+1 |Fi ] = Sn,i + E[Xn,i+1 |Fi ]
but by stationarity E[Xn,i+1 |Fi ] = E[φn−i−1 |σB] = 0. Hence E[Sn,i+1 |Fi ] = Sn,i and
Xn,i is a martingale difference array with respect to Fi .
Condition (d) clearly holds; instead conditions (a) and (c) simply follow since
||φn ||∞ is bounded and an tends to infinity.
We now prove (I) and show that under Assumption (C),
Pn
i=1
2
→ 1 in probXn,i
ability and hence condition (b) holds.
Lemma 3.3
n
1 X 2
ψ ◦ Tj → 1
a2n j=1 j
in probability as n → ∞.
Proof. We follow an argument given by Peligrad [31]. As ψj = φj + wj − wj+1 ◦ T
we obtain
2
ψj2 = φ2j + 2φj wj + wj2 + wj+1
◦ T − 2wj+1 ◦ T (φj + wj )
2
= φ2j + 2φj wj + wj2 + wj+1
◦ T − 2wj+1 ◦ T (ψj + wj+1 ◦ T )
2
= φ2j + (wj2 − wj+1
◦ T ) − 2ψj wj+1 ◦ T + 2φj wj .
We want to sum over j = 1, . . . , n and normalize by log n and wish to estimate the
2
◦ T are
error terms which are the last four terms on the RHS. The terms wj2 − wj+1
bounded and telescope so may be neglected.
In order to estimate the third of the error terms, ψj wj+1 ◦ T we proceed like
Peligrad (page 9) using a truncation argument. Let wj = wj 1{|wj |≤an } , where for
simplicity of notation we have left out the dependence on n. Then
!2
Z X
n Z
n
n Z
X
X
j j+1 2
2 2
j j+1
=
ψj ◦ T wj+1 ◦ T
≤ an
ψj2
ψj ◦ T wj+1 ◦ T
j=1
j=1
j=1
17
since the cross terms vanish (for j > i), as
Z
Z
j
i
(ψj wj+1 ◦ T ) ◦ T (ψi wi+1 ◦ T ) ◦ T =
(ψj wj+1
◦ T ) ◦ T j−i (ψi wi+1
◦ T)
Z
=
(ψj wj+1
◦ T ) ◦ T j−i−1 P (ψi wi+1
◦ T)
Z
P ψi = 0
◦ T ) ◦ T j−i−1 wi+1
=
(ψj wj+1
P ψi .
◦ T ) = wi+1
as P (ψi wi+1
For any a > we obtain using Tchebycheff’s inequality (on the second term):
!
n
1 X
µ
ψj ◦ T j wj+1 ◦ T j+1 > a
a2n j=1
!
n
X
1
≤ µ max wj+1 ◦ T j+1 > an + µ
ψj ◦ T j wj+1
◦ T j+1 > a
1≤j≤n
a2n j=1
n Z
2 X
j+1
≤ µ( max |wj+1 ◦ T | > an ) + 2 2
ψj2
1≤j≤n
a an j=1
= µ( max |wj ◦ T j+1 | > an ) + c1
1≤j≤n
In the last line we used
Pn
j=1
2
.
a2
E[ψj2 ] ∼ an by Lemmata 2.6 and 2.8. By boundedness
of the wj (Lemma 2.10) one gets that P (max1≤j≤n |wj+1 ◦ T j+1 | > an ) → 0 for every
P
1
> 0 as n → ∞. Choosing a = 2 we conclude that a12 nj=1 ψj ◦ T j wj+1 ◦ T j+1
n
converges to zero in probability as n → ∞.
P
For the fourth error term a12 2 nj=1 (φj wj ) ◦ T j we obtain by Lemma 2.9:
n
n
X
(φj wj ) ◦ T j
j=1
Thus
2
a2n
≤
1
n
X
kφj wj k1 ≤ c2
j=1
∞
X
µ(Bj−a log j )1+η log j
j=1
Pn
Since
j
j=1 (φj wj ) ◦ T → 0 in probability.
P
the term a12 nj=1 φ2j ◦ T j converges to
n
1 almost surely by the SBC property
the proof is complete.
Lemma 3.3 completes the proof of part (I) of the theorem. In order to show (II)
we proceed as in the proof of (I) except for the verification of condition (b). We
18
will prove an SBC property for φ2j + 2wj φj . Decomposing φj = φ̃j − µ(φ̃j ) and
defining w̃j = P φ̃j−1 + ... + P [a log j] φ̃j−[a log j] we see that kφ2j − φ̃2j k1 ≤ Cµ(Bj )2
and kwj φj − w̃j φ̃j k1 ≤ Cµ(Bj−a log j )2 log j. Note that both w̃j and φ̃j are positive
functions and moreover that similarly to Lemma 3.2 there exists a constant C̃6 such
P
that kw̃j k∞ ≤ C̃6 and kw̃j kLip ≤ C̃6 j k . Let En := nj=1 E[φ̃2j + 2w̃j φ̃j ]. It suffices to
P
P
consider the sequence φ̃2j +2w̃j φ̃j . This is because nj=1 E[φ̃2j +2w̃j φ̃j ] = nj=1 E[φ2j +
P
2wj φj ] + nj=1 (Cµ(Bj )2 + Cµ(Bj−a log j )2 log j) and hence µ almost surely,
n
n
1 X 2
1 X 2
j
j
(φ̃ ◦ T + 2(w̃j φ̃j ) ◦ T ) =
(φ ◦ T j + 2(wj φj ) ◦ T j ).
En j=1 j
En j=1 j
We will use Proposition 8.1, a form of the Gal and Koksma theorem as stated by
P
Sprindzuk to show that E1n nj=1 φ̃2j ◦ T j + 2(w̃j φ̃j ) ◦ T j → 1 almost surely. For this we
R
want to use Proposition 8.1 with fj = φ̃2j + 2w̃j φ̃j , gj = fj and hj to be determined
2
R 2
R Pn
2
+
2
w̃
φ̃
)
+
2
w̃
φ̃
−
φ̃
.
(
φ̃
below. We need to estimate the terms in
j j
j j
j
j
i=m
In order to verify the condition of the proposition we look at the three individual
R
R
R
sums of terms φ̃j ◦ T j−i φ̃i , (φ̃j w̃j ) ◦ T j−i φ̃i w̃i and (φ̃j w̃j ) ◦ T j−i φ̃i as follows:
(i) The fact that condition (SP) holds for the functions φ̃j implies
n
n X
n Z
X
X
φ̃j ◦ T j−i (φ̃i ) − E[φ̃j ]E[φ̃i ] ≤ c1
E[φ̃j ].
(3.4)
i=m
i=m j=i+1
Since E(φ̃2j ) − E(φ̃j ) = O(j −k ) we obtain
n X
n Z
n
n
X
X
X
2
j−i 2
2
2
φ̃j ◦ T φ̃i − E[φ̃j ]E[φ̃i ] ≤ c1
E[φ̃j ] +
O(i−k+1 ).
i=m j=i+1
i=m
(ii) Here we estimate the sums of the terms
R
i=m
(φ̃j w̃j ) ◦ T j−i (φ̃i w̃i ) − E[φ̃j w̃j ]E[φ̃i w̃i ].
By exponential decay of correlations one has for some constant β > 0
i+β
log i Z
n Z
X
X
j−i
j−i
˜
φ̃j ◦ T φ̃i − E[φ̃j ]E[φ̃i ] =
φj ◦ T φ̃i − E[φ̃j ]E[φ̃i ] + O(i−k )
j=i+1
j=i+1
and therefore
i+β log i Z
i+β log i
n Z
X
X
X
j−i
−k
j−i
φ˜j ◦T φ̃i ≤
E[φ̃j ]E[φ̃i ]+O(i )+
φ̃j ◦ T (φ̃i ) − E[φ̃j ]E[φ̃i ]
j=i+1
j=i+1
j=i+1
19
Using the uniform bound on |w̃j |∞ this implies
i+β log i Z
X
(φ˜j w̃j ) ◦ T j−i (φ̃i w̃i )
j=i+1
i+β log i
X
≤ C̃62
E[φ̃j ]E[φ̃i ] + O(i−k ) +
j=i+1
n Z
X
!
φ̃j ◦ T j−i (φ̃i ) − E[φ̃j ]E[φ̃i ]
j=i+1
By decay of correlation and since kw̃j φ̃j kα ≤ C̃6 j k we get with a possibly larger β
that
n Z
X
(φ̃j w̃j ) ◦ T j−i (φ̃i w̃i ) − E[φ̃j w̃j ]E[φ̃i w̃i ]
j=i+1
i+β log i Z
=
X
j−i
˜
˜
(φj w̃j ) ◦ T (φ̃i w̃i ) − E[φj w̃j )]E[φ̃i w̃i ] + O(i−k )
j=i+1
i+β log i
≤
C̃62
X
E[φ̃j ]E[φ̃i ] +
j=i+1
≤ C̃62 E[φ̃i ]2 β log i +
n Z
X
C̃62
φ̃j ◦ T
j−i
(φ̃i ) − E[φ̃j ]E[φ̃i ] + O(i−k )
j=i+1
n Z
X
!
φ̃j ◦ T j−i (φ̃i ) − E[φ̃j ]E[φ̃i ]
+ O(i−k̃ )
j=i+1
for some k̃ > 1. Since E[φ̃i ]2 log i ≤ c2 E[φ̃i ] ∀i and some c2 and since for every m, n
inequality (3.4) holds we now obtain
n
n X
n Z
X
X
j−i
(φ̃j w̃j ) ◦ T (φ̃i w̃i ) − E[φ̃j w̃j ]E[φ̃i w̃i ] ≤ c3
E[φ̃i ] + O(i−k̃ )
i=m
i=m j=i+1
(iii) A similar argument shows that for the ‘mixed’ terms
n X
n Z
n
n X
X
X
j−i 2
2
(φ̃j w̃j ) ◦ T φ̃i − E[φ̃j w̃j ]E[φ̃i ] ≤ c4
E[φ̃j ] +
O(i−k̃ ) .
i=m j=i+1
i=m
i=m
Combining (i), (ii) and (iii) yields for all m < n and some constant c5 :
!2
Z X
Z
n n X
φ̃2j + 2w̃j φ̃j − φ̃2j + 2w̃j φ̃j
≤ c5
E[φ̃i ] + O(i−k̃ ) .
i=m
i=m
We choose
hi = E[φ̃i ] + O(i−k̃ ), and so Proposition 8.1 implies that
Pn
1
2
j
j
→ 1 almost surely, provided k ≥ 2.
j=1 φ̃j ◦ T + 2(w̃j φ̃j ) ◦ T
En
20
4
Applications to dynamical systems.
Theorem 3.1 applies to a variety of dynamical systems including Gibbs-Markov
maps [1] and rational maps [17]. For Gibbs-Markov maps it has been shown [14,
Theorem 1] that nested sequences of balls (Bi (p)) satisfy both the Strong Borel Cantelli property and Assumption (C), so that (I) applies. For rational maps [17, Theorem
10] shows that the transfer operator contracts exponentially in the L ∞ norm hence if
the (SP) property is also proved then (II) holds. More generally (II) shows that proving the (SP) property for systems whose associated transfer operator has exponential
decay suffices to prove the SBC property and the CLT for shrinking targets.
5
Decay in BV (X) versus L 1
It is known that summable decay of correlations in BV (X) versus L 1 implies the
SP property by work of Kim [26, Proof of Theorem 2.1] (see also Gupta et al [14,
Proposition 2.6]). Hence the statement in this setting is simpler.
R
R
Let the transfer operator P be defined by φψ ◦ T dµ = P φψ dµ for all φ, ψ ∈
L 2 (µ), that is P is the adjoint of the Koopman operator U φ := φ ◦ T .
We assume that the restriction of P to the space BV (X) is exponentially contracting, i.e. P : BV (X) → BV (X) satisfies
kP n φkBV ≤ Cθn kφkBV
for all φ ∈ BV (X) such that
R
(5.1)
φ dµ = 0.
This implies that (T, X, µ) has exponential decay of correlations in BV versus
L 1 , so that for some 0 < θ < 1,
Z
Z
Z
n
φψ ◦ T dµ − φ dµ ψ dµ ≤ Cθn kφkBV kψk1
(5.2)
for all φ ∈ BV (X), ψ ∈ L 1 (µ). In particular the measure µ is ergodic.
Proposition 5.1 Assume the transfer operator P contracts exponentially as given
by (5.1)
21
Let Bi := B(p, ri ) be nested balls of radius ri about a point p such that
∞ and µ(Bi (p)) ≤
C2
iγ
P
i µ(Bi ) =
P
for constants C2 > 0 and 0 < γ ≤ 1. Let a2n = E( nj=1 (1Bi ◦
T i − µ(Bi )))2 . Then:
(I)
n
1 X
(1Bi ◦ T i − µ(Bi )) → N (0, 1).
an j=1
(II) If the nested sequence of balls (Bi (p)) about p satisfies Assumption (C) then
n
1 X
(1Bi ◦ T i − µ(Bi )) → N (0, 1)
an j=1
in distribution.
Proof: The proof is the same as for Theorem 3.1 with the simplification that the SP
property holds automatically as we have summable decay of correlations in BV (X)
versus L 1 (see proof of [26, Theorem 2.1]). Furthermore Lemma 2.4 shows that the
variance is unbounded and Lemma 2.9 gives a precise rate of growth in the case that
Assumption (C) holds.
Remark 5.2 For one-dimensional maps of the interval, Proposition 5.1 is basically
a consequence of Conze and Raugi [6, Theorem 5.1]. Follow the proof of [6, Theorem
5.1] taking Tk = T for all k, m to be the invariant measure µ and choosing fn =
1Bn (p). The rates of growth are given by Lemma 2.4 which shows that the variance is
unbounded. Lemma 2.9 gives a precise rate of growth in the case that Assumption (C)
holds. In Proposition 6.2 we extend these results to piecewise expanding maps in
higher dimensions.
6
Applications of Proposition 5.1.
Proposition 5.1 applies to certain classes of one-dimensional maps such as piecewise
expanding maps of the interval T : X → X with
1
T0
of bounded variation and pos-
sessing an absolutely continuous invariant measure with density bounded away from
22
zero (those maps satisfying the assumptions of [26, Theorem 2.1], see also [14]). For
these systems, Assumption (C) has been shown to hold for nested balls about µ
a.e. p ∈ X [21, 15]. In the next subsection we generalize these results to piecewise
expanding maps in higher dimensions.
6.1
Piecewise expanding maps in higher dimensions
In this section we prove the Strong Borel Cantelli property and the CLT for shrinking
balls in a class of expanding maps in higher dimensions. We also show that assumption
C holds for µ-a.e. point.
The Banach spaces will be given by L 1 , defined with respect to the Lebesgue
measure on Rn , and a quasi-Hölder space with properties analogous to BV which we
define below. A key property of the quasi-Hölder space is that characteristic functions
of balls have bounded norm (as in the BV norm) which turns out to be a very useful
property.
The maps are defined on compact sets Z ∈ RN . Denote by dist(·, ·) the usual
metric in RN and for ε > 0 let Bε (x) = {y ∈ RN : dist(x, y) < ε} be the ε-ball
centred at x. Let Bε (A) = {y ∈ RN : dist(y, A) ≤ ε} and write Z ◦ for the interior of
Z and Z its closure.
A map T : Z → Z is said to be a multidimensional piecewise expanding map, if
S
there exists a family of finitely many disjoint open sets {Zi } such that Leb(Z\ i Zi ) =
0 and there exist open sets Zei ⊃ Zi and C 1+α maps Ti : Zei → RN (for some 0 < α ≤ 1)
and some sufficiently small real number ε1 > 0 such that for all i,
• (H1) Ti (Zei ) ⊃ Bε1 (T (Zi )) and Ti |Zi = T |Zi ;
• (H2) For x, y ∈ T (Zi ) with dist(x, y) ≤ ε1 ,
| det DTi−1 (x) − det DTi−1 (y)| ≤ c| det DTi−1 (x)|dist(x, y)α ;
• (H3) There exists s = s(T ) < 1 such that ∀x, y ∈ T (Zei ) with dist(x, y) ≤ ε1 ,
we have
dist(Ti−1 x, Ti−1 y) ≤ s dist(x, y).
23
• (H4) Let G(ε) := supx G(x, ε, ε1 ) where
G(x, ε, ε1 ) :=
X Leb(Ti−1 Bε (∂T Zi ) ∪ B(1−s)ε1 (x))
Leb(B(1−s)ε1 (x))
i
(6.1)
and assume that
sup sα + 2 sup
ε≤δ
δ≤ε1
G(ε) α δ < 11
εα
(6.2)
We now introduce the Banach space of quasi-Hölder functions in which the spectrum of the Perron-Frobenius operator P is investigated. Given a Borel set Γ ⊂ Z,
we define the oscillation of ϕ ∈ L 1 (Leb) over Γ as
osc(ϕ, Γ) := ess sup ϕ − ess inf ϕ.
Γ
Γ
The function x 7→ osc(ϕ, Bε (x)) is measurable (see [23, Proposition 3.1]) For
0 < α ≤ 1 and ε0 > 0, we define the α-seminorm of ϕ as
Z
−α
|ϕ|α = sup ε
osc(ϕ, Bε (x)) dLeb(x).
0<ε≤ε0
RN
Let us consider the space of functions with bounded α-seminorm
Vα = {ϕ ∈ L 1 (Leb) : |ϕ|α < ∞},
and endow Vα with the norm
k · kα = k · k1 + | · |α
1
This condition could be greatly simplified as follows. Suppose the boundaries of Zi are C 1
codimension one embedded compact submanifold, then define the quantity:
η0 (T ) := sα +
4s
1−s
Y (T )
ξN −1
ξN
where
Y (T ) = sup
x
X
# {smooth pieces intersecting ∂Zi containing x} ,
i
is the maximal number of smooth components of the boundaries that can meet in one point and
ξN =
π N/2
(N/2)! ,
the N -volume of the N -dimensional unit ball of RN . We require that η0 (T ) < 1, and
this may replace the condition (6.2) above.
24
which makes it into a Banach space. We note that Vα is independent of the choice
of ε0 and that Vα is continuously injected in L ∞ (Leb). According to [34, Theorem
5.1], there exists an absolutely continuous invariant probability measure (a.c.i.p.) µ,
with density bounded above, and bounded below from zero, which has exponential
decay of correlations against L 1 observables on the finitely many mixing components
of Vα : in view of the next Theorem 6.4 we will from now restrict ourselves to one of
those components, by taking a mixing iterate of T . More precisely, if the map T is as
defined above and if µ is the mixing a.c.i.p., then there exist constants C < ∞ and
ϑ < 1 such that
Z
n
ψ ◦ T h dµ −
Z
Z
ψdµ
hdµ ≤ CkψkL 1 khkα ϑn
(6.3)
Z
for all ψ ∈ L 1 and for all h ∈ Vα . Moreover kP n φkα ≤ Ckφkα for all φ ∈ Vα and
thus equation 5.1 holds.
We now show that characteristic functions of balls are bounded in the k · kα norm.
Lemma 6.1 Let Bi (p) be a nested sequence of balls about a point p ∈ X, then there
exists a constant D(α) such that
k1Bi kα ≤ D(α)
for all i.
Proof: Take any set A with a rectifiable boundary. If p is not in a 2 neighborhood
of the boundary of A, then the oscillation is zero, otherwise it is 1. Therefore we have
R
osc(1A , B (p)) dLeb(p) ≤ c1 . Then we must divide by α . As α ≤ 1 we have the
ratio bounded by c1 ∗ 1−α .
The boundedness of the characteristic functions in the k·kα -norm allows us to proceed
as in Proposition 5.1 (see also [6]) and to obtain the following result.
Proposition 6.2 Assume a piecewise expanding map T on a compact set Z ⊂ Rn
satisfies conditions (H1)–(H4) and is mixing with respect to its absolutely continuous
invariant measure µ. Let Bi := B(p, ri ) be balls of radius ri about a point p such
25
µ(Bi ) = ∞ and µ(Bi ) ≤ Ciγ2 for some constants C2 and γ ∈ (0, 1]. Then the
P
variance a2n := E[( nj=1 (1Bi ◦ T i − 1i ))2 ] satisfies
that
P
i
an
lim sup √
≥1
log n
n→∞
and
n
1 X
1
(1Bi ◦ T i − ) → N (0, 1)
√
an j=1
i
in distribution.
Proof: The SBC property (I) is immediate from the decay of correlations, Equation 6.3 and the bound k1Bi kα ≤ D(α) by the proof of Proposition 6.1. The growth
estimate follows from Lemma 2.4.
We now make an additional assumption. Suppose that we have M domains of local
W
−i
injectivity for the map T ; if we take the join Z j := j−1
i=0 T Z, where Z denotes
the partition, mod-0, into the closed sets Zi , i = 1, · · · , M , then on each element
(j)
Zl , l = 1, · · · , |Z j |, each of which is the closure of its interior, the map T j is injective
(j)
(j)
and of class C 1+α on an open neighborhood of Zl : we call Z̃l
such an extension.
In order to prove condition (C) we require a further assumption which is also called
the finite range structure. We assume:
(j)
(j)
• (H5) Let U (j) := {T j Zl , ∀l = 1, · · · , |Z j |}, and put U = ∪∞
. Then U
j=1 U
consists of only finitely many subsets of Z with positive Lebesgue measure,
hence Um = inf U ∈U m(U ) is bounded below.
Lemma 6.3 Under the assumptions (H1)–(H5) Assumption (C) is satisfied.
Proof. Denote
Ek (ε) := {x; dist(T k x, x) ≤ ε}.
By Lemma 8.2 (see Appendix) it is enough to prove that there exists C > 0, δ > 0
such that for all k and ε,
µ(Ek (ε))) < Cεδ .
26
(j)
We now fix j and consider the cylinder, say, Zl . Let us suppose that {zk }k≥1 is
(j)
a sequence of points in Zl
(j)
converging to x ∈ Zl , namely dist(zk , x) → 0 when
k → ∞, and that dist(T j (zk ), x) → 0 for k → ∞. With abuse of definition we say
that such a point x is fixed. If there are points in the sequence {zk }k≥1 which are
(j)
(j)
on the boundary of Zl , we think of T j as its C 1+α extension on Z̃l . We want
(j)
to show that in Z̃l
there is only one fixed point x. By contradiction, suppose y
is another fixed point and {wk }k a sequence converging to y and whose T j images
(j)
converge to y as well. Suppose that Z̃l
[x, y] is contained in
(j)
Z̃l 2 .
is a convex set in such a way the segment
We now fix η small enough and take k big enough and
such that dist(x, zk ), dist(x, T j (zk )), dist(y, wk ), dist(y, T j (wk )), are all smaller than
η. We also put Dm,j := inf{||DT j (x)||} > 1, where the inf is taken over the points
x where the derivative is defined. The norm is the operator norm, which is strictly
larger than 1 since the map is uniformly expanding. Then we have
dist(x, y) ≥ dist(T j (zk ), T j (wk )) − dist(x, T j (zk )) − dist(y, T j (wk ))
and by applying Taylor’s formula
dist(x, y) ≥ Dm,j dist(zk , wk ) − 2η ≥ Dm,j [dist(x, y) − 2η] − 2η
which gives a contradiction, since Dm,j > 1, by sending η to 0. Hence x is the only
fixed point.
(j)
Let us now take a measurable set V ⊂ Z̃l
(j)
containing the fixed point x ∈ Z̃l .
We require that the diameter of the image T j (V ) be at most ε; such an image will
therefore be contained in the ball of center T j (x) and of radius ε. The Lebesgue
measure of this ball will be equal to ξN εN , where the factor ξN was defined in the
preceding footnote. Then we have
Leb(Bε (x)) = ξN εN ≥ Leb(T j (V )) ≥ | det(DT j (κ))|Leb(V )
2
(j)
If not we could join x and y with a chain of segments contained each in Z̃l : the argument will
work again since the sum of the lengths of those segments is larger than the distance between x and
y and this is what we need in bounding from below.
27
for a suitable point κ ∈ Z̃lj , where in the last inequality we used a local change of
variable and the continuity of DT j . By distortion, we could replace this point by
(j)
(j)
another one, say ι such that Leb(T j (Zl ) = | det(DT j (ι))|Leb(Zl ); we call B the
distortion constant satisfying
| det(DT j (ι))|
| det(DT j (κ))|
≤ B. We therefore get
(j)
Leb(V ) ≤
ξN εN B
ξN εN B Leb(Zl )
≤
| det(DT j (ι))|
Um
Since the density of the absolutely continuous invariant measure µ is bounded from
(j)
above (remember it is in L ∞ (Leb)), by, say, hM , and since each Zl
will contribute
with at most one fixed point, by taking the sum over the l we will equivalently get an
upper bound on the total measure of the balls including the T j (V ); hence we finally
get
µ{x; dist(T j x, x)} ≤
ξN hM εN B
.
Um
and this bound is independent of j.
As a consequence of Lemma 2.8 we have,
Theorem 6.4 Assume a piecewise expanding map T on a compact set Z ⊂ Rn
satisfies conditions (H1)–(H5) and is mixing with respect to its absolutely continuous invariant measure µ. For µ a.e. p if Bi (p) are nested balls about p such that
P
C2
i µ(Bi ) = ∞ and µ(Bi ) ≤ iγ for some constants C2 > 0 and γ ∈ (0, 1]. Then
a2n
n
X
1
= E[( (1Bi ◦ T i − ))2 ] = log n + O(1)
i
j=1
and
n
√
1 X
1
(1Bi ◦ T i − ) → N (0, 1)
i
log n j=1
in distribution.
7
Discussion.
There are several natural questions remaining unanswered. In particular can the
CLT for shrinking targets be proved for Anosov systems or non-uniformly hyperbolic
28
diffeomorphisms? Chernov and Kleinbock have proved the SBC property for balls in
Anosov systems [4] but the SBC property is unknown for non-uniformly hyperbolic
diffeomorphisms. More generally can a limit theory be developed for the statistics
of non-stationary stochastic processes arising as observations (which change in time)
on deterministic dynamical systems which may also may evolve in time, such as
sequential dynamical systems?
8
8.1
Appendices
Gal-Koksma Theorem.
We recall the following result of Gal and Koksma as formulated by W. Schmidt [35, 36]
and stated by Sprindzuk [38]:
Proposition 8.1 Let (Ω, B, µ) be a probability space and let fk (ω), (k = 1, 2, . . .) be
a sequence of non-negative µ measurable functions and gk , hk be sequences of real
numbers such that 0 ≤ gk ≤ hk ≤ 1, (k = 1, 2, . . . , ). Suppose there exists C > 0 such
that
Z
!2
X
(fk (ω) − gk )
X
dµ ≤ C
hk
(∗)
m<k≤n
m<k≤n
for arbitrary integers m < n. Then for any > 0
X
fk (ω) =
1≤k≤n
gk + O(Θ1/2 (n) log3/2+ Θ(n))
1≤k≤n
for µ a.e. ω ∈ Ω, where Θ(n) =
8.2
X
P
1≤k≤n
hk .
Assumption (C) for expanding systems
In this appendix we show that if the invariant measure µ has a density ρ(x) with
respect to Lebesgue measure m then Assumption (C) is valid. Recall we define
Ek () := {x : d(T k x, x) ≤ }
29
Lemma 8.2 Let Bi (x) denote a decreasing sequence of balls with center x and suppose µ(Bi (x)) ≤
C2
iγ
for some constants C1 , C2 and 0 < γ ≤ 1. Suppose µ has a
density ρ with respect to Lebesgue measure m with support X and there exists C7 > 0,
δ > 0 such that for all k, ,
µ(Ek ()) < C7 δ
Then for µ a.e. p ∈ X there exists η(p) ∈ (0, 1) and κ > 1 such that for all i
sufficiently large
µ(Bi (p) ∩ T −r Bi (p)) ≤ µ(Bi (p))1+η
for all r = 1, . . . , logκ i.
Proof. Let ρ(x) =
dµ
(x)
dm
be the density of µ with respect to Lebesgue measure m.
Let σ ≥ 1 and γ > σ. We choose k so that for all x a ball of radius k about x,
denoted B(x, k ), satisfies c1 /k σ ≤ m(B(x, k )) ≤ c2 /k σ , so k ' k −σ/D where D is
the dimension of X and c1 , c2 > 0.
Let κ > 1 and define Ak := {x : d(T j x, x) ≤ k for some 1 ≤ j ≤ logκ k}.
S κk
τ
Evidently Ak ⊂ log
j=1 Ej . By the estimate on Ek () for all large k, µ(Ak ) ≤ c3 k
where τ < δ. Let
Fk := {x : µ(B(x, k ) ∩ Ak ) ≥ 1/k γ }
and define the Hardy-Littlewood maximal function Mk for φ(x) = 1Ak (x)ρ(x) by
Z
1
Mk (x) := sup
1Ak (y)ρ(y) dm(y).
a>0 m(Ba (x)) Ba (x)
σ−γ
If x ∈ Fk then Mk > c−1
.
2 k
A theorem of Hardy and Littlewood ([10] Theorem 3.17) states that
m(|Mk | > C) ≤ c4
k1Ak ρk1
C
for some constant c4 , where k · k1 is the L 1 norm with respect to m. Hence
σ−γ
m(Fk ) ≤ m(Mk > c−1
) ≤ c4 µ(Ak )c2 k γ−σ ≤ k γ−σ(1+τ̃ )
2 k
where 0 < τ̃ < τ /D. We need to alter τ /D to τ̃ to take into account the fact that a
ball of radius has measure roughly D .
30
Choosing σ large enough that στ̃ > 1 and then taking σ < γ < σ − 1 + στ̃ the
P
series k m(Fk ) converges.
So for m a.e. x0 there exists an N (x0 ) such that x0 6∈ Fk for all k > N (x0 ). Hence
for k > N (x0 ), µ(B(x, k ) ∩ Ak ) ≤ 1/k γ , thus µ(B(x, k ) ∩ Ak ) ≤ m(B(x, k ))1+η for
some η > 0 (recall m(B(x, k )) '
1
kσ
and γ > σ).
Furthermore by the Lebesgue differentiation theorem for m a.e. x
Z
1
lim
ρ(y)dm = ρ(x)
→0 m(B (x)) B (x)
and for µ a.e. x , ρ(x) > 0 as X is the support of µ. Hence for m a.e. x0 there
exists an Ñ (x0 ) and η̃ > 0 such that for all k > Ñ (x0 ) we have µ(B(x, k ) ∩ Ak ) ≤
µ(B(x, k ))1+η̃ .
As κ was arbitrary by interpolating between the sequence k we have that for µ
a.e. x ∈ X there exists η 0 > 0 , κ0 > 1 such that
µ(Bi (x) ∩ T −r Bi (x)) ≤ µ(Bi (x))1+η
0
0
for 1 ≤ r ≤ logκ i. This is Assumption (C).
Acknowledgements
We would like to thank the anonymous referees whose careful reading and perceptive
comments helped us improve the final version of this work. MN and LZ were partially
supported by the National Science Foundation under Grant Number DMS-1101315.
Part of this work was done while MN was visiting the Centre de Physique Théorique
in Marseille with French CNRS support. SV was supported by the CNRS-PEPS
Mathematical Methods of Climate Theory, by the ANR-Project Perturbations and by
the PICS (Projet International de Coopération Scientifique), Propriétés statistiques
des systèmes dynamiques detérministes et aléatoires, with the University of Houston, n. PICS05968. Part of this work was done while he was visiting the Centro de
Modelamiento Matemático, U MI2807, in Santiago de Chile with a CNRS support
(délégation).
31
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