Multifractal analysis and the variance of Gibbs measures
Transcript Of Multifractal analysis and the variance of Gibbs measures
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
T. Jordan and M. Pollicott Warwick University
Abstract. The multifractal decomposition of Gibbs measures for conformal iterated function system is well known. We look at a ﬁner decomposition which also takes into account the rate of convergence. This is motivated by work by Olsen in the selfsimilar case. Our study of this ﬁner decomposition involves investigation of the variance of Gibbs measures. This is a problem of independent interest.
0. Introduction
Hausdorﬀ dimension is one of the most useful and eﬀective tools in understanding
the nature of fractal sets. For example, given a Cantor set X in the real line we
can describe its “size” in terms of its Hausdorﬀ dimension [5]. Often we want to
consider subsets deﬁned in terms of measures. Let us denote by B(x, r) = {y ∈
X : d(x, y) < r} a ball of radius r > 0 about a point x ∈ X then given a reference
probability measure ν on a set X and we can associate its pointwise dimension at
x by
log ν(B(x, r))
dν(x) = lim
r→0
log r
,
when it exists. Multifractal analysis describes the dimension of the sets of points x for which the limit takes a given value [6], [13], [18]. A particularly successful theory can be developed in the context of dynamically deﬁned sets. Let us consider a dynamically deﬁned Cantor set X supporting a suitable probability measure ν. More precisely, let X be the limit set for a C2 iterated function scheme satisfying the strong separation condition and let ν be a selfsimilar measure ν, with respect to a Ho¨lder potential g : X → R. Let T : X → X be the associated expanding map. The multifractal spectrum of ν describes the set of points whose (symbolic) pointwise dimension function takes diﬀerent values. We can ﬁrst decompose the limit set X as
X = Xα ∪ X∞
α∈R
where Xα = {x : dν (x) = α}, for α ∈ R, and X∞ denotes the points for which the limit dν (x) doesn’t exist. The usual (symbolic) multifractal spectrum of the measure ν describes the Hausdorﬀ dimension F (α) = dimH (Xα) of these sets. This important function has been extensively studied by several authors (e.g., [14],
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T. JORDAN AND M. POLLICOTT
[9],[3],[11], [16]). In this paper we want to consider a ﬁner structure of these sets. More precisely, we deﬁne, for each γ ∈ R+,
log ν(B(x, (T n) (x)−1) − α log (T n) (x)−1
Xα,γ = x : lim sup
√
=γ .
n→+∞
n log log n
We can then write
Xα =
Xα,γ ∪ Xα,∞
γ∈R+
where Xα,∞ is the set of point such that the limit supremum isn’t ﬁnite. A more reﬁned multifractal spectrum is therefore given by F (α, γ) = dimH Xα,γ. Our main result is the following.
Theorem. Assume that g and α log T  ◦ π do not diﬀer by a coboundary and a constant. For suﬃciently large γ we have that F (α, γ) = F (α).
Our motivation for this result was an interesting paper by Olsen. In particular, Olsen [10] established a similar result in the special case of selfsimilar maps and Bernoulli measures. We present an alternative dynamical approach which has the advantage that it extends to C2 dynamically deﬁned sets X and Gibbs measures. In section 6 we consider other applications of these results. An integral part of our analysis is the study of the variance of Gibbs measures which is of independent interest.
1. Iterated function schemes
We recall the deﬁnition and basic facts about iterated function schemes. An iterated function scheme consists of a family T1, · · · , Tk : [0, 1] → [0, 1] of C2 contractions. The limit set X = X(T1, · · · , Tk) is the smallest closed set for which ∪ki=1Ti(X) = X. We assume that they satisfy the strong separation condition, i.e., the sets Ti(X) are pairwise disjoint and X will be a Cantor set. We use the following deﬁnitions.
Deﬁnition. Let T : X → X be the locally expanding map deﬁned by
T (x) = Ti−1 if x ∈ Ti(X).
Given 0 < α ≤ 1, for g : X → R we write
g(x) − g(y)
gα = sup
x=y
x − yα
.
For any compact set Y ⊂ R, the space of αHo¨lder continuous functions Cα(Y ) = {g : gα < +∞} is a Banach space with norm gα = gα + g∞.
Deﬁnition. Let pi : [0, 1] → R, i = 1, · · · , k, be Ho¨lder continuous functions such
that
k i=1
pi
(x)
=
1
and
0
≤
pi(x)
≤
1.
A
measure
ν
on
X
is
called
selfconformal
if there exists such functions pi such that
(p1T1∗µ + · · · pkTk∗ν) = ν,
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
3
i.e., (p1(x)w(T1x) + · · · + pk(x)w(Tkx))dν(x) = w(x)dν(x), for any continuous function w : X → R.
The selfconformal measures fall into a broad class of measures. Let g : X → R be a Ho¨lder continuous function. We deﬁne the pressure P (g) of g by
P (g) = sup{h(T, µ) + gdµ : µ = T invariant probability measure},
where h(T, µ) is the entropy of the measure µ with respect to the transformation T.
Deﬁnition. A Gibbs measure for g ∈ Cα(X, R) is an invariant probability measure
on T such that
1 ≤ µ(Tx0 ◦ · · · ◦ Txn (X)) ≤ C
C
egn(x)−nP (g)
(1.1)
where gn(x) = g(x) + g(T x) + · · · + g(T n−1x). The sets Tx0 ◦ · · · ◦ Txn (X) are called cylinders.
A Gibbs measure is always ergodic. If µ is a Gibbs measure for g then P (g) = h(T, µ) + gdµ. A selfconformal measure is necessarily a Gibbs measure with respect to the function g(x) = log px0 (T x), where x ∈ Tx0 (X).
Lemma 1.1. If we assume that
k i=1
eg (Ti x)
=
1
then
we
have
that
min{egn(x)} ≤ µ(Ti0 ◦ · · · ◦ Tin−1 (X)) ≤ max{egn(x)}.
x
e−nP (g)
x
Given a probability measure ν we deﬁne its Hausdorﬀ dimension to be the inﬁmum of the Hausdorﬀ dimensions of Borel sets of full measure.
Deﬁnition. We can deﬁne the (symbolic) pointwise dimension at x ∈ X by
log ν(B(x, (T n) (x)−1))
dν (x) = lim
n→+∞
log (T n) (x)−1
,
providing the limit exists.
In the case of Gibbs measures µ the Federer property holds, i.e., there exists λ > 1 and C1 > 1 such that for every x ∈ X and r > 0 we have that
ν(B(x, λr)) ≤ C1ν(B(x, r)).
There exists C2 > 0 such that x, y ∈ Ti0 ◦ · · · ◦ Tin−1 X we have that
C2−1 ≤ ((TT nn)) ((xy)) ≤ C2.
Thus for any x ∈ Ti1 ◦ · · · ◦ Tin X we have that
C1−1λ−N ≤ νν((BTi(0x◦, ·(·T·n◦)T(ixn)−1−X1))) ≤ C1λN ,
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T. JORDAN AND M. POLLICOTT
where N is chosen so that λN > C2, and
C−1 ≤ diam(Ti0 ◦ · · · ◦ Tin−1 X) ≤ C2.
2
(T n) (x)−1
Thus for ν the (symbolic) pointwise dimension coincides with
dν (x) = lim log ν(Ti0 ◦ · · · ◦ Tin−1 X) , n→+∞ log diam(Ti0 ◦ · · · ◦ Tin−1 X)
and with the usual pointwise dimension. Let Σ =
∞ 0
{1,
·
·
·
, k}
be
a
full
shift
space on ksymbols and let σ : Σ → Σ denote the shift given by (σx)n = xn+1. Let
π : Σ → X be the natural coding deﬁned by
π(x) = lim Tx0 · · · Txn (0).
n→+∞
Example. The simplest case is where there are linear contractions with rates 0 < r1, · · · , rk < 1 and a ﬁxed probability vector p = (p1, · · · , pk). In this case, ν = π∗µp
corresponds to the Bernoulli measure µp = (p1, · · · , pk)Z+ on Σ.
We can deﬁne the (symbolic) multifractal spectrum of the measure ν by
F (α) = dimH (Xα)
where Xα = {x : dν(x) = α}. This function has been extensively studied by various authors, notably Pesin and Weiss [14], Ledrappier [9], Cawley and Mauldin [3], Olsen [11]. In [14] the approach taken was to use Gibbs measures and thermodynamic formalism.
Let Φ : X → R be a Ho¨lder continuous function and ν the associated Gibbs measure. The multifractal spectrum of µ is described in [14]. We outline their approach which will be crucial in the rest of this paper. For q ∈ R we deﬁne
φq = −t(q) log T (x) + q(Φ + P (Φ))
where t(q) is chosen to be the unique value such that P (φq) = 0. Let νq be the Gibbs measure associated with φq. It can be shown that for νq almost all x, dν(x) = −t (q). Since t (q) is a strictly monotone function, we can associate each value of q to a value of α by α(q) = −t (q). Using this method [14] gives the following result.
Proposition 1.1. Let T1, . . . , Tk : [0, 1] → [0, 1] be a conformal C2 iterated function scheme satisfying the strong separation condition. Let X be the limit set and a conformal measure ν.
(1) F (α) is analytic and convex in a neighbourhood of α0 = dimH (X) which we will denote (αmin, αmax).
(2) For α ∈ (αmin, αmax) there exists a measure να such that να(Xα) = 1 and dν(x) = α, a.e. (να). Moreover, dim να = F (α).
(3) For α ∈ (αmin, αmax), F (α) is given by the Legendre transform of t(q). That is F (α) = inf{αq + t(q)}.
q
To analyze the ﬁner sets F (α, γ) we will use the statistical properties of Gibbs measures.
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2. Statistical Properties of Gibbs measures
We say that f ∈ C0(X, R) is cohomologous to a constant if there exists u ∈ C0(X, R) and c ∈ R such that f = u ◦ T − u + c. If µ is a Gibbs measure then the following analogues of well known statistical theorems hold.
Proposition 2.1 (Law of the Iterated Logarithm) [4]. Let f be a Ho¨lder continuous function with f dµ = 0. Assume that f is not cohomologous to a constant, then there exists γ > 0 such that for a.e. (µ) x we have that
√ f n(x) → n log log n
2γ, as n → +∞.
Proposition 2.2 (Central Limit Theorem)[1], [4]. Let f be a Ho¨lder continuous function with f dµ = 0. Assume that f is not cohomologous to a constant, then there exits γ > 0 such that
lim µ
n→+∞
x ∈ X : √1 f n(x) ≤ t n
= √1 2πγ
t
γy2
e− 2 dy.
−∞
We say that f is a lattice function if there exists u ∈ C0(X, R), c ∈ R and ψ ∈ C0(X, aZ), for some a > 0, such that f = u ◦ X − u + ψ + c. Generically the functions f n will be nonlattice.
Proposition 2.3 (Local Limit Theorem). Let f be a Ho¨lder continuous function with f dµ = 0.
(1) Assume that f is a nonlattice function then there exists γ > 0
µ {x ∈ X : a ≤ f n(x) ≤ b}
lim
n→∞
√1 (√b−a)
= 1,
n 2γ
[8], [2, Th 9.2]. (2) Assume that f is a lattice function then provided b − a is suﬃciently large
there exists C > 0 (depending on γ) such that
1 µ{x ∈ X : a < f n(x) < b}
C≤
√1
≤ C,
n
for all n suﬃciently large [2, Thm 9.6].
In these three propositions the value of γ is the same. All of these results are special cases of more general invariance principles [4]. The next lemma relates dynamical properties of the measure ν to the pointwise (symbolic) dimension of ν on X.
Lemma 2.1. If µ is a Gibbs measure for a Ho¨lder continuous function Φ : X → R and ν is an ergodic measure then
dµ(x) =
Φ(x)dν − P (Φ) for a.e (ν) x. log T (x)dν
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T. JORDAN AND M. POLLICOTT
Proof. We ﬁrst observe that log (T n)  =
n−1 i=0
log
T

and
then
we
can
write
lim 1 log (T n) (x) = n→+∞ n
log T dν, a.e. (ν) x.
(2.1)
The Gibbs property of µ and the Birkhoﬀ Ergodic Theorem gives that
lim 1 log µ[x0, . . . , xn−1] = lim 1 (Φn(x) − nP (Φ))
n→+∞ n
n→+∞ n
= Φdν − P (Φ) for a.e.(ν) x.
(2.2)
The proof follows from combining (2.1) and (2.2).
Deﬁnition. The variance of a Gibbs measure ν (with respect to the function g : X → R) and a Ho¨lder function F : X → R is deﬁned to be
γ(X, ν) := lim 1 n→+∞ n
n−1
F (T ix) −
i=0
2
F dν dν(x)
The proof of convergence of this limit, and alternative deﬁnitions, appear in [12] and [2]. The variance plays a key role in the following result which start are analysis of the ﬁner multifractal spectrum F (α, γ). It is a simple extension to the nonlinear case of Proposition 1.1 in [10].
Lemma 2.4. Let µ be a selfconformal measure corresponding to a Gibbs measure with potential Φ which is not cohomologous to a constant. Fix q ∈ R and α = √−t (q). If γα is the variance of f (x) = Φ(x)−P (Φ)−α log T (x) then F (α) = F (α, 2γα).
Proof. It can be deduced from the work in [15] that f dµ = 0. Thus by the Law of the Iterated Logarithm (Proposition 2.1) for να we can write
lim sup √ f n(x) = n→∞ n log log n
2γα, for a.e. (να).
This completes the proof.
We would like to replace γα by other values γ. In order to see this we would like to consider other Gibbs measures µ (associated to suitable Ho¨lder continuous functions g : X → R) which satisfy the following properties:
(1) the variance is γ = γ(µ ); and (2) the measure µ has the same limit
log ν (B(x, (T n) (x)−1))
α = lim
n→+∞
log (T n) (x)−1
=
f dµ − P (f ) log T dµ
(2.3)
for a.e. (ν ) x ∈ X. In particular we look for Gibbs measures µ such that f dµ = 0 for the function f deﬁned in Lemma 2.4. This property implies
(2.3).
In the next two sections we shall consider this problem in detail.
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
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3. Estimating the variance of Gibbs measures
The variance appears in a number of diﬀerent statistical properties, as we saw in the last section, and so estimating its value is a problem of independent interest. For example, the variance appears in a number of statistical properties of hyperbolic systems (e.g., Propositions 2.1, 2.2 and 2.3). In practice, we shall use a well known characterization of the variance in terms of the second derivative of the pressure [17],[12]. The diﬀerentiability of the pressure is most conveniently studied using a characterization in terms of transfer operators. In order to work at a fairly general level, assume that u, v : X → R are any two Ho¨lder continuous functions. Let Cα(X) denote the αHolder continuous functions. We can deﬁne a transfer operator Lu : Cα(X) → Cα(X) by
Luh(x) =
eu(y)h(y).
T y=x
We want to consider two simple normalization hypotheses.
Hypothesis I. Assume that Lu1 = 1. Hypothesis II. Assume that Luv = 0.
In order to show that these assumptions can be made without any signiﬁcant loss of generality, ﬁrst recall that the Gibbs measure and variance are unchanged by adding coboundaries and constants to functions.
Lemma 3.1. Given any u we can ﬁnd w ∈ Cα(X) such that u = u + w ◦ T − w − P (u) satisﬁes hypothesis I.
Proof. This is a standard result [12].
Let µ denote the unique Gibbs state associated to u. If Lu1 = 1 then L∗uµ = µ [12]. We require the following result on the spectrum of Lu : Cα(X) → Cα(X).
Lemma 3.2. The eigenvalue 1 for Lu is simple. Moreover, the spectral radius of the operator L − µ : Cα(X) → Cα(X) is strictly smaller than 1. More precisely,
there exist 0 < ρ < 1 and C > 0 such that Lnuvα ≤ Cρnvα, for all v ∈ Cα(X)and n ≥ 1.
Proof. The proof appears, for example, in [12]
We can now this lemma to prove the following.
Lemma 3.3. Assume hypothesis I. Then given any v with vdµ = 0 we can ﬁnd r ∈ Cα(X) such that v = v − rT + r satisﬁes hypothesis II.
Proof. We can deﬁne r =
∞ n=1
Lnu v .
This converges to a function in Cα(X),
because of the spectral properties of the operator L − µ described in Lemma 3.2.
Since Lu1 = 1 we have that LuUT = I, where UT v = v ◦ T . In particular, Luv =
Luv + Lu(r − UT r), but since Lu(r − UT r) = Lur − r = −Luv, by construction, we
see that Luv = 0.
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T. JORDAN AND M. POLLICOTT
Lemma 3.4. Under hypothesis I and hypothesis II we have that γ = v2dµ.
Proof. Let λ(u) be the maximal eigenvalue of the operator Lu. The variance γ is also characterized by γ = d2λ(dut2+tv) [12]. This essentially follows from perturbation theory on the eigenvalue equation Lu+tvh(t) = λ(t)h(t). The ﬁrst derivative of both sides of this identity gives
Lu+tv(vh(t) + h (t)) = λ (t)h(t) + λ(t)h (t).
When t = 0, we have λ (0) = 0 and λ(0) = 1 and so
Lu(h (0)) = Lu(v + h (0)) = h (0).
(3.1)
Since 1 is a simple eigenvalue for Lu, with constant eigenfunction, we deduce that h (0) is a constant function. The second derivative of both sides of the identity gives
Lu+tv(v2h(t) + 2vh (t) + h (t)) = λ (t)h(t) + 2λ (t)h (t) + λ(t)h (t).
We can evaluate this second expression at t = 0. We can then integrate both sides with respect to µ, and since µ = L∗uµ we have that
µ(v2h(0)) + 2µ(vh (0)) + µ(h (0)) = λ (0)µ(h(0)) + 2λ (0)µ(h (0)) + λ(0)µ(h (0)).
(3.2)
Since λ(0) = 1 we can cancel the last terms on each side. By hypothesis II, we know (by considering the expression (3.1)) that h (0) = 0, which eliminates an extra term on each side of (3.2) and leaves the identity
λ (0) = µ(v2h(0)) . µ(h(0))
However, by hypothesis I we have that h(0) = 1 and, by the usual normalization, µ(h(0)) = 1. This gives the result.
Unfortunately, we ﬁrst have to accept the following limitation if we consider only T invariant probabilities.
Corollary. Under hypotheses I and II we can bound
(inf v)2 ≤ γ = λ (0) ≤ (sup v)2.
If we don’t assume hypotheses I and II we get a similar result. Applying Lemmas 3.1 and 3.3, changes the function v by at most a coboundary and a constant. Furthermore, the variance is unchanged by adding a coboundary and a constant (since the pressure is unchanged by adding coboundaries to u and v).
Example. We can consider the special case of locally constant functions u and v which are constant on each inverse branch Ti(X) and a selfsimilar iterated function
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
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system. In this case the Gibbs measure µ for u is a projection of Bernoulli measure on the Bernoulli shift Σ associated to a vector p = (p1, · · · , pk), where
e−ui pi = e−u1 + · · · + e−uk
and ui is the value u(x) which takes on Ti(X). Assuming hypothesis II, the variance in this case can be calculated to be
k
γ = pivi2.
i=1
where vi is the value v(x) which takes on Ti(X).
Returning to the application we are interested in, the corollary suggests that we need to look at a broader class of measures. More precisely, for n ≥ 2 we shall consider the probability measures on X which are T ninvariant, rather than the more restrictive assumption of being T invariant.
4. Invariant measures for T n
Given α, γ > 0, we want to consider invariant measures να,γ for which the local dimension dν (x) at almost all points is the same with respect to either να or να,γ. The measure να,γ can be used to give a lower bound on dimH (Xα,γ). However, we see from the previous section that it is not enough to consider T invariant Gibbs measures and we need to consider T ninvariant Gibbs measures, for n ≥ 2.
Deﬁnition. Let Mn denote the space of T ninvariant Gibbs measures on X.
Clearly, the T invariant measures µ are also contained in Mn. The next lemma compares the variances for these two points of view.
Lemma 4.1. Let f be a Ho¨lder continuous function and let µ be a T invariant
Gibbs measure.
(1) h(T n, µ) = nh(T, µ)
(2) If γ(T n, µ, f n) is the variance for the function f n with respect to T n and µ
then
γ(T, µ, f )
=
1 n
γ
(T
n
,
µ,
f
n
)
Proof. The ﬁrst part is Abramov’s Theorem. For the second part, we observe that
γ(T, µ, f ) = var(φ, µ) : = lim 1 k→+∞ k
k−1
f (T ix) −
i=0
2
f dµ dµ(x)
= lim 1 k→+∞ nk
nk−1
f (T ix) −
i=0
2
f dµ dµ(x)
= 1 lim 1 n k→+∞ k
nk−1
f k(T nix) −
i=0
2
f dµ dµ(x)
= 1 γ(T n, µ, f n). n
This completes the proof.
Considering µ as an element of Mn leads to a similar formulation of the variance.
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T. JORDAN AND M. POLLICOTT
Lemma 4.2. Let µ ∈ M be a Gibbs measure with potential g satisfying hypothesis I (with respect to T ). Assume that f satisﬁes hypothesis II then it corresponds to a measure in M with potential gn satisfying hypothesis I (with respect to T n) and f n satisﬁes hypothesis II.
Proof. Let us denote by Lg,T and Lgn,T n the transfer operators in each case. Since Lg,T (h ◦ T ) = h and Lgn,T n = Lng,T we see that Lgn,T n 1 = 1 and Lgn,T n f n = 0, i.e., they satisfy hypothesis I and II (with respect to T n).
We can use the following estimates to arrange the variance to be higher. If i is a ﬁnite word and x ∈ X then we let Tix = Ti0 ◦ · · · ◦ Tin−1 (x).
Lemma 4.3. There exist D, E > 0, such that for all n ≥ 1 and x, y we can bound for i = n
f n(Tix) − f n(Tiy) ≤ D and gn(Tix) − gn(Tiy) ≤ E.
Proof. We can bound
n−1
f n(Tix) − f n(iy) ≤ f (T kTix) − f (T kTiy)
k=0 n−1
≤ Cθα(n−k)
k=0
≤ Cθ = D 1−θ
where θ = maxi{Ti ∞} < 1 and α, C > 0 are constants coming from the Ho¨lder continuity of f . Similarly, we can bound the expression for g.
From the deﬁnition of a Gibbs measure µ for g with P (g) = 0, we have the following result.
Corollary. For any cylinder of length n we can bound egn(Tix)−E ≤ µ(TiX ) ≤ egn(Tix)+E
The following lemma will prove useful later.
Lemma 4.4. Assume that f dµ = 0. If f is not a coboundary, then we can ﬁnd periodic points T mx = x and T mx = x such that f m(x) < 0 and f m(x ) > 0.
Proof. By Livsic’s theorem, we know that f m(x) = 0 whenever T mx = x is equivalent to f being a coboundary [12]. If f is not a coboundary, then there must be either a periodic point T mx = x such that T m(x) > 0 or a periodic point T mx = x such that T m(x ) < 0. However, it is easy to see that both cases must exist simultaneously. We then take the least common multiple to complete the proof.
We can assume without loss of generality that T x = x and T x = x are both ﬁxed points. Let us denote by δ = f (x) > 0 and δ = f (x ) < 0.
Notation. Given n ≥ 1, let j denote the word of length n corresponding to the cylinder containing x. Let k denote the word of length n corresponding to the cylinder containing x .
Deﬁnition. Let G (n) be the set of cylinders i containing at least one point y for which f n(y) ≤ .
T. Jordan and M. Pollicott Warwick University
Abstract. The multifractal decomposition of Gibbs measures for conformal iterated function system is well known. We look at a ﬁner decomposition which also takes into account the rate of convergence. This is motivated by work by Olsen in the selfsimilar case. Our study of this ﬁner decomposition involves investigation of the variance of Gibbs measures. This is a problem of independent interest.
0. Introduction
Hausdorﬀ dimension is one of the most useful and eﬀective tools in understanding
the nature of fractal sets. For example, given a Cantor set X in the real line we
can describe its “size” in terms of its Hausdorﬀ dimension [5]. Often we want to
consider subsets deﬁned in terms of measures. Let us denote by B(x, r) = {y ∈
X : d(x, y) < r} a ball of radius r > 0 about a point x ∈ X then given a reference
probability measure ν on a set X and we can associate its pointwise dimension at
x by
log ν(B(x, r))
dν(x) = lim
r→0
log r
,
when it exists. Multifractal analysis describes the dimension of the sets of points x for which the limit takes a given value [6], [13], [18]. A particularly successful theory can be developed in the context of dynamically deﬁned sets. Let us consider a dynamically deﬁned Cantor set X supporting a suitable probability measure ν. More precisely, let X be the limit set for a C2 iterated function scheme satisfying the strong separation condition and let ν be a selfsimilar measure ν, with respect to a Ho¨lder potential g : X → R. Let T : X → X be the associated expanding map. The multifractal spectrum of ν describes the set of points whose (symbolic) pointwise dimension function takes diﬀerent values. We can ﬁrst decompose the limit set X as
X = Xα ∪ X∞
α∈R
where Xα = {x : dν (x) = α}, for α ∈ R, and X∞ denotes the points for which the limit dν (x) doesn’t exist. The usual (symbolic) multifractal spectrum of the measure ν describes the Hausdorﬀ dimension F (α) = dimH (Xα) of these sets. This important function has been extensively studied by several authors (e.g., [14],
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T. JORDAN AND M. POLLICOTT
[9],[3],[11], [16]). In this paper we want to consider a ﬁner structure of these sets. More precisely, we deﬁne, for each γ ∈ R+,
log ν(B(x, (T n) (x)−1) − α log (T n) (x)−1
Xα,γ = x : lim sup
√
=γ .
n→+∞
n log log n
We can then write
Xα =
Xα,γ ∪ Xα,∞
γ∈R+
where Xα,∞ is the set of point such that the limit supremum isn’t ﬁnite. A more reﬁned multifractal spectrum is therefore given by F (α, γ) = dimH Xα,γ. Our main result is the following.
Theorem. Assume that g and α log T  ◦ π do not diﬀer by a coboundary and a constant. For suﬃciently large γ we have that F (α, γ) = F (α).
Our motivation for this result was an interesting paper by Olsen. In particular, Olsen [10] established a similar result in the special case of selfsimilar maps and Bernoulli measures. We present an alternative dynamical approach which has the advantage that it extends to C2 dynamically deﬁned sets X and Gibbs measures. In section 6 we consider other applications of these results. An integral part of our analysis is the study of the variance of Gibbs measures which is of independent interest.
1. Iterated function schemes
We recall the deﬁnition and basic facts about iterated function schemes. An iterated function scheme consists of a family T1, · · · , Tk : [0, 1] → [0, 1] of C2 contractions. The limit set X = X(T1, · · · , Tk) is the smallest closed set for which ∪ki=1Ti(X) = X. We assume that they satisfy the strong separation condition, i.e., the sets Ti(X) are pairwise disjoint and X will be a Cantor set. We use the following deﬁnitions.
Deﬁnition. Let T : X → X be the locally expanding map deﬁned by
T (x) = Ti−1 if x ∈ Ti(X).
Given 0 < α ≤ 1, for g : X → R we write
g(x) − g(y)
gα = sup
x=y
x − yα
.
For any compact set Y ⊂ R, the space of αHo¨lder continuous functions Cα(Y ) = {g : gα < +∞} is a Banach space with norm gα = gα + g∞.
Deﬁnition. Let pi : [0, 1] → R, i = 1, · · · , k, be Ho¨lder continuous functions such
that
k i=1
pi
(x)
=
1
and
0
≤
pi(x)
≤
1.
A
measure
ν
on
X
is
called
selfconformal
if there exists such functions pi such that
(p1T1∗µ + · · · pkTk∗ν) = ν,
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
3
i.e., (p1(x)w(T1x) + · · · + pk(x)w(Tkx))dν(x) = w(x)dν(x), for any continuous function w : X → R.
The selfconformal measures fall into a broad class of measures. Let g : X → R be a Ho¨lder continuous function. We deﬁne the pressure P (g) of g by
P (g) = sup{h(T, µ) + gdµ : µ = T invariant probability measure},
where h(T, µ) is the entropy of the measure µ with respect to the transformation T.
Deﬁnition. A Gibbs measure for g ∈ Cα(X, R) is an invariant probability measure
on T such that
1 ≤ µ(Tx0 ◦ · · · ◦ Txn (X)) ≤ C
C
egn(x)−nP (g)
(1.1)
where gn(x) = g(x) + g(T x) + · · · + g(T n−1x). The sets Tx0 ◦ · · · ◦ Txn (X) are called cylinders.
A Gibbs measure is always ergodic. If µ is a Gibbs measure for g then P (g) = h(T, µ) + gdµ. A selfconformal measure is necessarily a Gibbs measure with respect to the function g(x) = log px0 (T x), where x ∈ Tx0 (X).
Lemma 1.1. If we assume that
k i=1
eg (Ti x)
=
1
then
we
have
that
min{egn(x)} ≤ µ(Ti0 ◦ · · · ◦ Tin−1 (X)) ≤ max{egn(x)}.
x
e−nP (g)
x
Given a probability measure ν we deﬁne its Hausdorﬀ dimension to be the inﬁmum of the Hausdorﬀ dimensions of Borel sets of full measure.
Deﬁnition. We can deﬁne the (symbolic) pointwise dimension at x ∈ X by
log ν(B(x, (T n) (x)−1))
dν (x) = lim
n→+∞
log (T n) (x)−1
,
providing the limit exists.
In the case of Gibbs measures µ the Federer property holds, i.e., there exists λ > 1 and C1 > 1 such that for every x ∈ X and r > 0 we have that
ν(B(x, λr)) ≤ C1ν(B(x, r)).
There exists C2 > 0 such that x, y ∈ Ti0 ◦ · · · ◦ Tin−1 X we have that
C2−1 ≤ ((TT nn)) ((xy)) ≤ C2.
Thus for any x ∈ Ti1 ◦ · · · ◦ Tin X we have that
C1−1λ−N ≤ νν((BTi(0x◦, ·(·T·n◦)T(ixn)−1−X1))) ≤ C1λN ,
4
T. JORDAN AND M. POLLICOTT
where N is chosen so that λN > C2, and
C−1 ≤ diam(Ti0 ◦ · · · ◦ Tin−1 X) ≤ C2.
2
(T n) (x)−1
Thus for ν the (symbolic) pointwise dimension coincides with
dν (x) = lim log ν(Ti0 ◦ · · · ◦ Tin−1 X) , n→+∞ log diam(Ti0 ◦ · · · ◦ Tin−1 X)
and with the usual pointwise dimension. Let Σ =
∞ 0
{1,
·
·
·
, k}
be
a
full
shift
space on ksymbols and let σ : Σ → Σ denote the shift given by (σx)n = xn+1. Let
π : Σ → X be the natural coding deﬁned by
π(x) = lim Tx0 · · · Txn (0).
n→+∞
Example. The simplest case is where there are linear contractions with rates 0 < r1, · · · , rk < 1 and a ﬁxed probability vector p = (p1, · · · , pk). In this case, ν = π∗µp
corresponds to the Bernoulli measure µp = (p1, · · · , pk)Z+ on Σ.
We can deﬁne the (symbolic) multifractal spectrum of the measure ν by
F (α) = dimH (Xα)
where Xα = {x : dν(x) = α}. This function has been extensively studied by various authors, notably Pesin and Weiss [14], Ledrappier [9], Cawley and Mauldin [3], Olsen [11]. In [14] the approach taken was to use Gibbs measures and thermodynamic formalism.
Let Φ : X → R be a Ho¨lder continuous function and ν the associated Gibbs measure. The multifractal spectrum of µ is described in [14]. We outline their approach which will be crucial in the rest of this paper. For q ∈ R we deﬁne
φq = −t(q) log T (x) + q(Φ + P (Φ))
where t(q) is chosen to be the unique value such that P (φq) = 0. Let νq be the Gibbs measure associated with φq. It can be shown that for νq almost all x, dν(x) = −t (q). Since t (q) is a strictly monotone function, we can associate each value of q to a value of α by α(q) = −t (q). Using this method [14] gives the following result.
Proposition 1.1. Let T1, . . . , Tk : [0, 1] → [0, 1] be a conformal C2 iterated function scheme satisfying the strong separation condition. Let X be the limit set and a conformal measure ν.
(1) F (α) is analytic and convex in a neighbourhood of α0 = dimH (X) which we will denote (αmin, αmax).
(2) For α ∈ (αmin, αmax) there exists a measure να such that να(Xα) = 1 and dν(x) = α, a.e. (να). Moreover, dim να = F (α).
(3) For α ∈ (αmin, αmax), F (α) is given by the Legendre transform of t(q). That is F (α) = inf{αq + t(q)}.
q
To analyze the ﬁner sets F (α, γ) we will use the statistical properties of Gibbs measures.
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
5
2. Statistical Properties of Gibbs measures
We say that f ∈ C0(X, R) is cohomologous to a constant if there exists u ∈ C0(X, R) and c ∈ R such that f = u ◦ T − u + c. If µ is a Gibbs measure then the following analogues of well known statistical theorems hold.
Proposition 2.1 (Law of the Iterated Logarithm) [4]. Let f be a Ho¨lder continuous function with f dµ = 0. Assume that f is not cohomologous to a constant, then there exists γ > 0 such that for a.e. (µ) x we have that
√ f n(x) → n log log n
2γ, as n → +∞.
Proposition 2.2 (Central Limit Theorem)[1], [4]. Let f be a Ho¨lder continuous function with f dµ = 0. Assume that f is not cohomologous to a constant, then there exits γ > 0 such that
lim µ
n→+∞
x ∈ X : √1 f n(x) ≤ t n
= √1 2πγ
t
γy2
e− 2 dy.
−∞
We say that f is a lattice function if there exists u ∈ C0(X, R), c ∈ R and ψ ∈ C0(X, aZ), for some a > 0, such that f = u ◦ X − u + ψ + c. Generically the functions f n will be nonlattice.
Proposition 2.3 (Local Limit Theorem). Let f be a Ho¨lder continuous function with f dµ = 0.
(1) Assume that f is a nonlattice function then there exists γ > 0
µ {x ∈ X : a ≤ f n(x) ≤ b}
lim
n→∞
√1 (√b−a)
= 1,
n 2γ
[8], [2, Th 9.2]. (2) Assume that f is a lattice function then provided b − a is suﬃciently large
there exists C > 0 (depending on γ) such that
1 µ{x ∈ X : a < f n(x) < b}
C≤
√1
≤ C,
n
for all n suﬃciently large [2, Thm 9.6].
In these three propositions the value of γ is the same. All of these results are special cases of more general invariance principles [4]. The next lemma relates dynamical properties of the measure ν to the pointwise (symbolic) dimension of ν on X.
Lemma 2.1. If µ is a Gibbs measure for a Ho¨lder continuous function Φ : X → R and ν is an ergodic measure then
dµ(x) =
Φ(x)dν − P (Φ) for a.e (ν) x. log T (x)dν
6
T. JORDAN AND M. POLLICOTT
Proof. We ﬁrst observe that log (T n)  =
n−1 i=0
log
T

and
then
we
can
write
lim 1 log (T n) (x) = n→+∞ n
log T dν, a.e. (ν) x.
(2.1)
The Gibbs property of µ and the Birkhoﬀ Ergodic Theorem gives that
lim 1 log µ[x0, . . . , xn−1] = lim 1 (Φn(x) − nP (Φ))
n→+∞ n
n→+∞ n
= Φdν − P (Φ) for a.e.(ν) x.
(2.2)
The proof follows from combining (2.1) and (2.2).
Deﬁnition. The variance of a Gibbs measure ν (with respect to the function g : X → R) and a Ho¨lder function F : X → R is deﬁned to be
γ(X, ν) := lim 1 n→+∞ n
n−1
F (T ix) −
i=0
2
F dν dν(x)
The proof of convergence of this limit, and alternative deﬁnitions, appear in [12] and [2]. The variance plays a key role in the following result which start are analysis of the ﬁner multifractal spectrum F (α, γ). It is a simple extension to the nonlinear case of Proposition 1.1 in [10].
Lemma 2.4. Let µ be a selfconformal measure corresponding to a Gibbs measure with potential Φ which is not cohomologous to a constant. Fix q ∈ R and α = √−t (q). If γα is the variance of f (x) = Φ(x)−P (Φ)−α log T (x) then F (α) = F (α, 2γα).
Proof. It can be deduced from the work in [15] that f dµ = 0. Thus by the Law of the Iterated Logarithm (Proposition 2.1) for να we can write
lim sup √ f n(x) = n→∞ n log log n
2γα, for a.e. (να).
This completes the proof.
We would like to replace γα by other values γ. In order to see this we would like to consider other Gibbs measures µ (associated to suitable Ho¨lder continuous functions g : X → R) which satisfy the following properties:
(1) the variance is γ = γ(µ ); and (2) the measure µ has the same limit
log ν (B(x, (T n) (x)−1))
α = lim
n→+∞
log (T n) (x)−1
=
f dµ − P (f ) log T dµ
(2.3)
for a.e. (ν ) x ∈ X. In particular we look for Gibbs measures µ such that f dµ = 0 for the function f deﬁned in Lemma 2.4. This property implies
(2.3).
In the next two sections we shall consider this problem in detail.
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
7
3. Estimating the variance of Gibbs measures
The variance appears in a number of diﬀerent statistical properties, as we saw in the last section, and so estimating its value is a problem of independent interest. For example, the variance appears in a number of statistical properties of hyperbolic systems (e.g., Propositions 2.1, 2.2 and 2.3). In practice, we shall use a well known characterization of the variance in terms of the second derivative of the pressure [17],[12]. The diﬀerentiability of the pressure is most conveniently studied using a characterization in terms of transfer operators. In order to work at a fairly general level, assume that u, v : X → R are any two Ho¨lder continuous functions. Let Cα(X) denote the αHolder continuous functions. We can deﬁne a transfer operator Lu : Cα(X) → Cα(X) by
Luh(x) =
eu(y)h(y).
T y=x
We want to consider two simple normalization hypotheses.
Hypothesis I. Assume that Lu1 = 1. Hypothesis II. Assume that Luv = 0.
In order to show that these assumptions can be made without any signiﬁcant loss of generality, ﬁrst recall that the Gibbs measure and variance are unchanged by adding coboundaries and constants to functions.
Lemma 3.1. Given any u we can ﬁnd w ∈ Cα(X) such that u = u + w ◦ T − w − P (u) satisﬁes hypothesis I.
Proof. This is a standard result [12].
Let µ denote the unique Gibbs state associated to u. If Lu1 = 1 then L∗uµ = µ [12]. We require the following result on the spectrum of Lu : Cα(X) → Cα(X).
Lemma 3.2. The eigenvalue 1 for Lu is simple. Moreover, the spectral radius of the operator L − µ : Cα(X) → Cα(X) is strictly smaller than 1. More precisely,
there exist 0 < ρ < 1 and C > 0 such that Lnuvα ≤ Cρnvα, for all v ∈ Cα(X)and n ≥ 1.
Proof. The proof appears, for example, in [12]
We can now this lemma to prove the following.
Lemma 3.3. Assume hypothesis I. Then given any v with vdµ = 0 we can ﬁnd r ∈ Cα(X) such that v = v − rT + r satisﬁes hypothesis II.
Proof. We can deﬁne r =
∞ n=1
Lnu v .
This converges to a function in Cα(X),
because of the spectral properties of the operator L − µ described in Lemma 3.2.
Since Lu1 = 1 we have that LuUT = I, where UT v = v ◦ T . In particular, Luv =
Luv + Lu(r − UT r), but since Lu(r − UT r) = Lur − r = −Luv, by construction, we
see that Luv = 0.
8
T. JORDAN AND M. POLLICOTT
Lemma 3.4. Under hypothesis I and hypothesis II we have that γ = v2dµ.
Proof. Let λ(u) be the maximal eigenvalue of the operator Lu. The variance γ is also characterized by γ = d2λ(dut2+tv) [12]. This essentially follows from perturbation theory on the eigenvalue equation Lu+tvh(t) = λ(t)h(t). The ﬁrst derivative of both sides of this identity gives
Lu+tv(vh(t) + h (t)) = λ (t)h(t) + λ(t)h (t).
When t = 0, we have λ (0) = 0 and λ(0) = 1 and so
Lu(h (0)) = Lu(v + h (0)) = h (0).
(3.1)
Since 1 is a simple eigenvalue for Lu, with constant eigenfunction, we deduce that h (0) is a constant function. The second derivative of both sides of the identity gives
Lu+tv(v2h(t) + 2vh (t) + h (t)) = λ (t)h(t) + 2λ (t)h (t) + λ(t)h (t).
We can evaluate this second expression at t = 0. We can then integrate both sides with respect to µ, and since µ = L∗uµ we have that
µ(v2h(0)) + 2µ(vh (0)) + µ(h (0)) = λ (0)µ(h(0)) + 2λ (0)µ(h (0)) + λ(0)µ(h (0)).
(3.2)
Since λ(0) = 1 we can cancel the last terms on each side. By hypothesis II, we know (by considering the expression (3.1)) that h (0) = 0, which eliminates an extra term on each side of (3.2) and leaves the identity
λ (0) = µ(v2h(0)) . µ(h(0))
However, by hypothesis I we have that h(0) = 1 and, by the usual normalization, µ(h(0)) = 1. This gives the result.
Unfortunately, we ﬁrst have to accept the following limitation if we consider only T invariant probabilities.
Corollary. Under hypotheses I and II we can bound
(inf v)2 ≤ γ = λ (0) ≤ (sup v)2.
If we don’t assume hypotheses I and II we get a similar result. Applying Lemmas 3.1 and 3.3, changes the function v by at most a coboundary and a constant. Furthermore, the variance is unchanged by adding a coboundary and a constant (since the pressure is unchanged by adding coboundaries to u and v).
Example. We can consider the special case of locally constant functions u and v which are constant on each inverse branch Ti(X) and a selfsimilar iterated function
MULTIFRACTAL ANALYSIS AND THE VARIANCE OF GIBBS MEASURES
9
system. In this case the Gibbs measure µ for u is a projection of Bernoulli measure on the Bernoulli shift Σ associated to a vector p = (p1, · · · , pk), where
e−ui pi = e−u1 + · · · + e−uk
and ui is the value u(x) which takes on Ti(X). Assuming hypothesis II, the variance in this case can be calculated to be
k
γ = pivi2.
i=1
where vi is the value v(x) which takes on Ti(X).
Returning to the application we are interested in, the corollary suggests that we need to look at a broader class of measures. More precisely, for n ≥ 2 we shall consider the probability measures on X which are T ninvariant, rather than the more restrictive assumption of being T invariant.
4. Invariant measures for T n
Given α, γ > 0, we want to consider invariant measures να,γ for which the local dimension dν (x) at almost all points is the same with respect to either να or να,γ. The measure να,γ can be used to give a lower bound on dimH (Xα,γ). However, we see from the previous section that it is not enough to consider T invariant Gibbs measures and we need to consider T ninvariant Gibbs measures, for n ≥ 2.
Deﬁnition. Let Mn denote the space of T ninvariant Gibbs measures on X.
Clearly, the T invariant measures µ are also contained in Mn. The next lemma compares the variances for these two points of view.
Lemma 4.1. Let f be a Ho¨lder continuous function and let µ be a T invariant
Gibbs measure.
(1) h(T n, µ) = nh(T, µ)
(2) If γ(T n, µ, f n) is the variance for the function f n with respect to T n and µ
then
γ(T, µ, f )
=
1 n
γ
(T
n
,
µ,
f
n
)
Proof. The ﬁrst part is Abramov’s Theorem. For the second part, we observe that
γ(T, µ, f ) = var(φ, µ) : = lim 1 k→+∞ k
k−1
f (T ix) −
i=0
2
f dµ dµ(x)
= lim 1 k→+∞ nk
nk−1
f (T ix) −
i=0
2
f dµ dµ(x)
= 1 lim 1 n k→+∞ k
nk−1
f k(T nix) −
i=0
2
f dµ dµ(x)
= 1 γ(T n, µ, f n). n
This completes the proof.
Considering µ as an element of Mn leads to a similar formulation of the variance.
10
T. JORDAN AND M. POLLICOTT
Lemma 4.2. Let µ ∈ M be a Gibbs measure with potential g satisfying hypothesis I (with respect to T ). Assume that f satisﬁes hypothesis II then it corresponds to a measure in M with potential gn satisfying hypothesis I (with respect to T n) and f n satisﬁes hypothesis II.
Proof. Let us denote by Lg,T and Lgn,T n the transfer operators in each case. Since Lg,T (h ◦ T ) = h and Lgn,T n = Lng,T we see that Lgn,T n 1 = 1 and Lgn,T n f n = 0, i.e., they satisfy hypothesis I and II (with respect to T n).
We can use the following estimates to arrange the variance to be higher. If i is a ﬁnite word and x ∈ X then we let Tix = Ti0 ◦ · · · ◦ Tin−1 (x).
Lemma 4.3. There exist D, E > 0, such that for all n ≥ 1 and x, y we can bound for i = n
f n(Tix) − f n(Tiy) ≤ D and gn(Tix) − gn(Tiy) ≤ E.
Proof. We can bound
n−1
f n(Tix) − f n(iy) ≤ f (T kTix) − f (T kTiy)
k=0 n−1
≤ Cθα(n−k)
k=0
≤ Cθ = D 1−θ
where θ = maxi{Ti ∞} < 1 and α, C > 0 are constants coming from the Ho¨lder continuity of f . Similarly, we can bound the expression for g.
From the deﬁnition of a Gibbs measure µ for g with P (g) = 0, we have the following result.
Corollary. For any cylinder of length n we can bound egn(Tix)−E ≤ µ(TiX ) ≤ egn(Tix)+E
The following lemma will prove useful later.
Lemma 4.4. Assume that f dµ = 0. If f is not a coboundary, then we can ﬁnd periodic points T mx = x and T mx = x such that f m(x) < 0 and f m(x ) > 0.
Proof. By Livsic’s theorem, we know that f m(x) = 0 whenever T mx = x is equivalent to f being a coboundary [12]. If f is not a coboundary, then there must be either a periodic point T mx = x such that T m(x) > 0 or a periodic point T mx = x such that T m(x ) < 0. However, it is easy to see that both cases must exist simultaneously. We then take the least common multiple to complete the proof.
We can assume without loss of generality that T x = x and T x = x are both ﬁxed points. Let us denote by δ = f (x) > 0 and δ = f (x ) < 0.
Notation. Given n ≥ 1, let j denote the word of length n corresponding to the cylinder containing x. Let k denote the word of length n corresponding to the cylinder containing x .
Deﬁnition. Let G (n) be the set of cylinders i containing at least one point y for which f n(y) ≤ .