Random cutout sets with spatially inhomogeneous intensities

arXiv:1504.03447v1 [math.DS] 14 Apr 2015

RANDOM CUTOUT SETS WITH SPATIALLY INHOMOGENEOUS INTENSITIES TUOMO OJALA, VILLE SUOMALA, AND MENG WU Abstract. We study the Hausdorff dimension of Poissonian cutout sets defined via inhomogeneous intensity measures on Q-regular metric spaces. We obtain formulas for the Hausdorff dimension of such cutouts in self-similar and selfconformal spaces using the multifractal decomposition of the average densities for the natural measures.

1. Introduction Given a metric space X and a sequence of open balls B(xn , rn ) ⊂ X, we define the cutout set corresponding to the sequence (xn , rn ) ∈ X × (0, 1) as [ E = X \ B(xn , rn ) . n

That is, E is the set left uncovered by the union of the balls B(xn , rn ). If the centres of these cutouts are dynamically defined (e.g. if xn+1 = T (xn ) for a given dynamics T : X → X) or if xn are randomly distributed, it is of interest to investigate whether E 6= ∅ and to determine its structure and size such as Hausdorff dimension. This problem arises from Diophantine approximation and versions of the Dvoretzky covering problem as well as in the study of renewal sets (see e.g. [19, 15, 17]). In this paper, we consider only the case in which the xn are random variables. We refer to [9, 18] for recent accounts and further references in the dynamical setting (see also [14]). We shall next describe our model in detail. Let X = (X, H, d) be a bounded metric space endowed with a measure H, which is (Ahlfors-David) Q-regular for some 0 < Q < ∞: there are constants 0 < c0 < C0 < ∞ such that c0 r Q ≤ H (B(x, r)) ≤ C0 r Q , for all x ∈ X, 0 < r < diam(X). (Throughout the paper, a measure will refer to a locally finite Borel regular outer measure.) 2010 Mathematics Subject Classification. Primary 60D05; Secondary 28A80, 37D35, 37C45. 1

2

TUOMO OJALA, VILLE SUOMALA, AND MENG WU

For each 0 < γ < +∞, let Y be a Poisson point process on X × (0, 1) with dr on (0, 1). Thus, intensity γH × ρ, where ρ is the measure defined by ρ(dr) = rQ+1 Y is a random collection of pairs (x, r) ∈ X × (0, 1) such that (1) For each Borel set A ⊂ X × (0, 1), the random variable ♯(A ∩ Y) is Poisson distributed with mean γH × ρ(A). (2) For disjoint Ai , the random variables ♯(Ai ∩ Y) are independent. In particular, Y is a.s. countably infinite. We consider the random cutout set: [ E=X\ B(x, r) . (x,r)∈Y

Note that the intensity of Y and the induced probability P crucially depend on γ. Let γ0 := sup{γ > 0 : P(E 6= ∅) > 0} . A central problem is to determine the exact value of γ0 (0 < γ0 < ∞ always holds, see Remark 2.5). Further, when 0 < γ < γ0 , we would like to determine the a.s. Hausdorff dimension of E. Since for any γ > 0, there is a positive probability for extinction (E = ∅), we follow [30] and define the essential dimension of the random set E as P − esssup dimH (E). This is the unique value s ≥ 0 such that dimH (E) ≤ s a.s. and for all t < s, there is a positive probability that dimH (E) > t. The case when X is a subdomain of some Euclidean space Rd and H = Ld is the d-dimensional Lebesgue measure is well understood. In particular, γ0 = γ0 (d) = d/α(d), where α(d) = Ld (B(0, 1)) and for 0 < γ ≤ d/α(d), the essential Hausdorff dimension (and also the box-dimension) equals d − γα(d), see [5, 30, 27, 20, 26]. In this case, the point process Y is translation invariant in an obvious way, but it possesses also strong scale invariance: If I, λI ⊂ (0, 1) for λ > 0, then it is equally likely that a point x ∈ X is covered by a ball B(xn , rn ) ∈ Y for rn ∈ I as it is for rn ∈ λI. There are many works (e.g. [19, 11, 30, 24, 4]) in which this condition has been relaxed by replacing the measure ρ(dr) by a more general measure of the dr form h(r) . For such generalizations, it is still possible to get results on the size of E and the range of γ, for which E 6= ∅ with positive probability. However, it turns out that the model is much more sensitive for the changes in the spatial component H and in essentially all the works we are aware of, only the case in which H = L has been considered. The papers [12] and [30] are notable exceptions. In these papers, various estimates for the dimension of the cutout sets are obtained in the context of a general metric space. However, when it comes to determining the value of the essential dimension, it is assumed in [30] that H = L and also in [12], there is a strong homogeneity assumption on H (implying in particular that supx,y∈X H(B(x,r)) −→ 1 as r ↓ 0). H(B(y,r))

RANDOM CUTOUT SETS WITH SPATIALLY INHOMOGENEOUS INTENSITIES

3

We note that in many of the references given above, the model is actually a one where rn is a deterministic sequence and xn are independent and uniformly distributed. However, in the case of translation invariant intensity the methods in the case of deterministic radii and iid centres are essentially the same as in the Poissonian case described above. Further, in many of the cited works, a significant part of attention has been given to the study of the random covering set \ [ F = B(xn , rn ) , k∈N n≥k

consisting of the points covered infinitely often by the balls B(xn , rn ). However, under the present assumptions and for any choice of γ, it follows from Fubini’s theorem that a.s. H(X \ F ) = 0 so that the dimensional properties of F are uninteresting. Further, for the case of deterministic radii, as well as for more general Poissonian dr intensities H(dx) × h(r) , the dimensional properties of the associated random covering set in the setting of Q-regular spaces are analogous to the Euclidean situation (where H is the Lebesgue measure). For instance, the proof of [13, Proposition 4.7] adapts easily to the case of Q-regular metric spaces. These observations indicate that changing the spatial component of the intensity measure does not affect the fractal properties of the random covering sets, as opposed to the “dual problem” of determining the dimension of the cutout set E. Before going further, let us provide a simple example to get an idea why the lack of homogeneity in H is a subtle issue for the cutouts. Suppose X = X1 ∪ X2 where, say, X1 and X2 are disjoint subintervals of [0, 1]. Let µ = aLX1 + bLX2 and suppose 0 < a < b < 12 . Now, conditional on E ∩ X1 = ∅, we know from the above discussion that a.s. dimH (E) ≤ 1 − 2b while on E ∩ X1 6= ∅, there is a positive probability that dimH (E) = 1 − 2a. This shows that one cannot expect any a.s. constancy result for the Hausdorff dimension of E. Of course, it still holds that the essential dimension is 1 − 2a (see also Remark S 3.4). For each 0 < t < 1, let Et = X \ (x,r)∈Y ,r>t B(x, r) and for x ∈ X denote Z p(x, t) = P (x ∈ Et ) = exp −γ

1

H(B(x, r))r

r=t

−Q−1

dr

.

This formula suggests an intimate connection to the lower and upper (Q-)average densities of H defined at x ∈ X as A(H, x) = lim inf A(H, x, t) ,

(1.1)

A(H, x) = lim sup A(H, x, t) ,

(1.2)

t→0

t→0

4


where A(H, x, t) =

R1

r=t

H(B(x, r))r −Q−1 dr . − log t

If A(H, x) and A(H, x) coincide, we denote the common value by A(H, x). Observe that p(x, t) = tγ A(H,x,t) , (1.3) so that the expected measure of Et equals Z E(H(Et )) = tγ A(H,x,t) dH(x) .

(1.4)

x∈X

It is well known that for fractal H, the density limr↓0 H(B(x,r)) fails to exist at HrQ almost all points. However, for many important Q-regular measures (see [3, 6, 7, 28, 29]), the average density A(H, x) is known to exist and take a constant value α at H-almost all points of X = suppH. Recalling (1.4), a first naive guess would be to predict that in such a case the essential dimension of the random set E would equal Q − γα. However, it turns out that in most cases of interest, the dimension of E is affected by the zero measure set, where A(H, x) 6= α and a finer analysis of the multifractal properties of the average densities is needed in order to catch the correct dimension of the cutout set E. The structure of the paper is as follows. In Section 2, using the familiar first and second moment methods, we present some tools to estimate the dimension of the intersections of E with certain sub- and superlevel sets of the average densities A(H, ·), A(H, ·). This part applies to any Q-regular measure and can be used directly to obtain some (coarse) estimates on the value of γ0 and on the essential dimension of E. In Section 3, we present the main result of the paper; We consider the case when X is self-similar, or more generally self-conformal, and satisfies the strong separation condition. Using tools from thermodynamical formalism and expressing the average densities as ergodic averages, we examine their multifractal spectrum. This enables us to obtain a formula for γ0 and for the essential dimension of E when 0 < γ < γ0 . 2. Auxiliary dimension estimates In this section, we provide some useful upper and lower estimates for the Hausdorff dimension of E ∩ {α < A(H, x) < β} when α and β vary. Our standing assumption is that H is a Q-regular measure on the metric space X. Further, the parameter γ > 0 that determines (together with H) the intensity of Y is fixed throughout the section. For F ⊂ X and t > 0, we denote by F (t) = {y ∈ X : d(y, F ) ≤ t}, the closed t-neighbourhood of the set F .


5

2.1. Dimension upper bound. For each 0 < α < β < ∞ and 0 < r < 1, we denote X(α, β, r) = {x ∈ X | α < A(H, x, r) < β}. Lemma 2.1. (i) There exists C < ∞, independent of t, such that P(x ∈ E(t)) ≤ CP(x ∈ Et ). (ii) If 0 ≤ α′ < α < β < β ′ < ∞, there exists r0 > 0 such that X(α, β, r)(r) ⊂ X(α′ , β ′ , r) for all 0 < r < r0 . Proof. (i) Observe that by definition of E(t) and elementary geometry, we have [ E(t) ⊂ X \ B(x, r − t) =: Et′ . (x,r)∈Y,r>t

Et′ ).

So P(x ∈ E(t)) ≤ P(x ∈ Thus, we only need to show that P(x ∈ Et′ ) ≤ CP(x ∈ Et ) for some C < ∞ independent of t. Since x ∈ Et′ if and only if A ∩ Y = ∅ for A = {(y, r) : r > t, y ∈ B(x, r − t)} , we deduce that P(x ∈ Now, we have Z 1 t

Et′ )

Z ≤ exp −γ

H(B(x, r − t))

1

H(B(x, r − t))

t

dr r Q+1

≥ =

Z

Z

dr r Q+1

1

H(B(x, r − t))

2t 1

t

.

(2.1)

dr

r Q+1 dr . H(B(x, r)) Q+1 r (1 + t/r)Q+1

(2.2)

An elementary calculation shows that 1 ≥ 1 − (Q + 1)y for all y ∈ [0, 1]. (1 + y)Q+1 Applying this to y = t/r in (2.2), we get Z 1 Z 1 Z 1 dr dr dr H(B(x, r − t)) Q+1 ≥ H(B(x, r)) Q+1 − (Q + 1)t H(B(x, r)) Q+2 r r r t t t R 1 dr < +∞, substituting the above inSince C ′ = sup0 0 there exists r0 > 0 such that R1 H(B(x, r + t) \ B(x, r − t)) t ≤ε − log t for every 0 < t < r0 . Since for every x ∈ X(α, β, t)(t), there exists y ∈ X(α, β, t) such that d(x, y) < t, we deduce that R1 H(B(x, r + t) \ B(x, r − t)) < β′ , D(H, x, t) ≤ D(H, y, t) + t − log t ′ when ε < β − β. The lower bound follows by a similar calculation. Lemma 2.2. Let 0 < α′ < α < β < β ′ < ∞ and C, η ≥ 0. Suppose that H(X(α′ , β ′, r)) ≤ Cr η for all 0 < r < 1. Then a.s. dimH E ∩ lim sup X(α, β, r) ≤ Q − γα′ − η , r↓0

if Q − γα′ − η ≥ 0 while E ∩ lim supr↓0 X(α, β, r) = ∅ if Q − γα′ − η < 0. ′

Proof. Observe that by (1.3), P(x ∈ Er ) ≤ r γα , for x ∈ X(α′ , β ′ , r). Pick α′ < α e< ′ ′ e α, β < β < β . Using Lemma 2.1, we have for each θ < γα + η that ! X e 2−n ) ∩ E (2−n ) E 2θn H X(e α, β, n∈N

≤C1

X n

≤C1

X

2

θn

Z

P (x ∈ E2−n ) dH(x)

X(α′ ,β ′ ,2−n ) ′

2nθ H(X(α′ , β ′, 2−n ))2−nγα

n

≤C2

X

′

2n(θ−γα −η) < ∞ .

n

In particular, we see that almost surely, θn −n −n e lim 2 H X(e α, β, 2 ) ∩ E (2 ) = 0 . n→∞


7

Since H is Q-regular this implies a.s the existence of N0 ∈ N such that for all e 2−n ) ∩ E) is covered by a union of balls n ≥ N0 , the set (X(e α, β, B xn,1 , 2−n , . . . , B xn,mn , 2−n with mn ≤ 2n(Q−θ) . Since

lim sup X(α, β, r) ⊂ r↓0

for all N ≥ N0 , and

∞ [ mn [

n=N i=1

X

mn 2−n(Q−θ+ε) ≤

n≥N

X

B xn,i , 2−n , 2−nε −→ 0 ,

n≥N

for any ε > 0, this implies the claim. Note that if Q − γα′ < 0, we have mn = 0 e 2−n ) ∩ E)(2−n ) = ∅ for all n ≥ N0 . and thus (X(e α, β,

2.2. A lower estimate. Let µ be a measure on X. For each t > 0, we define a measure νt by (2.3) dνt (x) = p(x, t)−1 1Et (x) dµ(x) . Then (νt )t>0 is a T -martingale in the sense of Kahane [16] and it is easy to check that a.s νt is weakly convergent to a random limit measure ν. Let 0 < s < ∞ be such that Z Z d(x, y)−s dµ(x) dµ(y) < ∞ , (2.4) X

X

and define a Kernel K : X × X → [0, ∞[ by

K(x, y) = d(x, y)−sP(x ∈ Ed(x,y) ) . Lemma 2.3. E

Z Z

K(x, y) dν(x) dν(y)

(2.5)

< ∞.

Proof. It suffices to show that for all 0 < t < 1, Z Z E K(x, y) dνt(x) dνt (y) < C < ∞ ,

(2.6)

where C is independent of t. Indeed, using that x 7→ A(H, x, r) is continuous (this follows e.g. from the calculation in the proof of Lemma 2.1) and recalling (1.3) allows to express K(x, y) as a limit of increasing continuous functions, so that (2.6) yields the claim. We first claim that for all 0 < δ < 1, P(x, y ∈ Eδ ) ≤ CP(x ∈ Eδ )P(y ∈ Eδ )/P(x ∈ Ed(x,y) ) ,

(2.7)

8


where C is independent of δ, d(x, y). Indeed, this is a result of direct calculation (we assume that δ < d(x, y)/2 as otherwise (2.7) follows directly from (1.3)): Z 1 H(B(x, s) ∪ B(y, s))s−Q−1ds δ

≥ ≥

Z

Z

1

H(B(y, s))s

−Q−1

δ 1 −Q−1

ds +

Z

Z

d(x,y)/2

H(B(x, s))s−Q−1 ds δ 1

H(B(y, s))s ds + H(B(x, s))s−Q−1 ds δ δ Z 1 − H(B(x, s))s−Q−1 ds − C1 , d(x,y)

R d(x,y) where C1 is a constant such that d(x,y)/2 H(B(x, s))s−Q−1 ds ≤ C1 and thus only depends on the Q-regularity data of the measure H. The claim (2.7) now follows by multiplying the inequality by −γ and taking the exponential. Combining (2.7), Fubini’s theorem, and (2.4) we calculate, Z Z E K(x, y) dνt (x) dνt (y) Z Z P(x, y ∈ Et )P(x ∈ Ed(x,y) )d(x, y)−s dµ(x) dµ(y) = P(x ∈ Et )P(y ∈ Et ) X X Z Z ≤C d(x, y)−s dµ(x)dµ(y) < ∞ . Since this upper bound is independent of t, we are done.

The following lemma employs the standard connection between capacity and dimension in the situation at hand. Recall that the lower local dimension of a measure ν at x ∈ X is defined as dimloc (ν, x) = lim inf r↓0

log ν(B(x, r)) . log r

Lemma 2.4. Suppose that s − γα > 0. If for µ-almost all x ∈ X, A(H, x) < α, then ν(X) > 0 with positive probability and almost surely, dimloc (ν, x) ≥ s − γα , for ν-almost all x ∈ X.


9

Proof. We first observe that if N ⊂ X is µ-null, then it is almost surely ν-null. Indeed, for each ε > 0, there is an open set Uε ⊃ N, such that µ(Uε ) < ε. Thus Fatou’s lemma gives E(ν(N)) ≤ E(ν(Uε )) ≤ E(lim inf νt (Uε )) ≤ lim inf Eνt (Uε ) = µ(Uε ) < ε . t↓0

t↓0

Whence E(ν(N)) = 0, or in other words, ν(N) = 0 almost surely. Let Z K(x, y) dν < M} . FM = {x ∈ X | A(H, x) < α and y∈X

Then, by the above and Lemma 2.3, it follows that a.s.

ν(X \ FM ) −→ 0 as M −→ ∞ . On the other hand, for all x ∈ FM , and all small enough 0 < r < 1, (2.5) and (1.3) give K(x, y) ≥ dist(x, y)γα−s ≥ r γα−s for y ∈ B(x, r) and whence Z γα−s r ν(B(x, r)) ≤ K(x, y) dν < M , y∈B(x,r)

implying ν(B(x, r)) ≤ Mr s−γα . The second claim of the Lemma now follows by taking logarithms, letting r ↓ 0 and finally letting M −→ ∞. To prove that ν(X) > 0 is an event of positive probability, we first pick so small r0 > 0 that µ(F ) > 0, where F = {x ∈ X | A(H, x, r) < α for all 0 < r < r0 }. Calculating as in the proof of Lemma 2.3 yields Z Z 2 d(x, y)−γ A(H,x,d(x,y)) dµ(x) dµ(y) E νt (F ) ≤ C x∈F y∈F Z Z ≤C d(x, y)−s dµ(x)dµ(y) < ∞ .

In other words, νt (F ) is an L2 -bounded martingale with nonzero expectation (since µ(F ) > 0). Whence, ν(X) ≥ ν(F ) > 0 with positive probability.

Remarks 2.5. (i) Lemmas 2.2 and 2.4 can be used directly to obtain upper and lower estimates on γ0 and on the dimension of E. Let d0 = inf x∈X A(H, x), D0 = supx∈X A(H, x) (note that c0 ≤ d0 ≤ D0 ≤ C0 ). Applying Lemma 2.2 with η = 0, implies γ0 ≤ Q/d0 and dimB (E) ≤ Q − γd0 a.s, if 0 < γ ≤ γ0 . In turn, Lemma 2.4 applied for µ = H and s = Q, gives the estimate γ0 ≥ Q/D0 and provided 0 < γ < Q/D0 , implies that dimH (E) ≥ Q − γD0 with positive probability. (ii) As will be seen in Proposition 2.6 below, even if A(H, x) = A(H, x) = α for H-almost every x, these estimates are usually far from being sharp. Actually, as will be seen in the Section 3, the dimension of E depends intimately on the multifractal properties of the average density of H.

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(iii) As of curiosity, we mention that if X = Td is the d-dimensional torus and Q = d, then γ0 = d/d0 and for γ ≤ γ0 , the essential dimension is d − γd0 . Indeed, for each c > d0 , there is a point x ∈ Td and r > 0 such that H(B(x, r)) ≤ cr d . An application of the Lebesgue density theorem yields a Borel set B ⊂ B(x, r) such that H(B) > 0 and A(H, x) < c for all x ∈ B. Lemma 2.4 then yields that dimH (E) ≥ Q − γc is an event of positive probability. For α ≥ 0, denote X(α) = {x ∈ X : D(H, x) = α}. Let αmin = inf{α : X(α) 6= ∅}, αmax = sup{α : X(α) 6= ∅}. Further, let f (α) = dimH (X(α)). Note that f obtains its maximum at some αmin ≤ α0 ≤ αmax and f (α0 ) = Q. By Frostman’s lemma, for each αmin < α < αmax with X(α) 6= ∅, there exists a probability measure µα on X such that µα (X(α)) = 1 and further Z Z d(x, y)ε−f (α) dµα (x)dµα (y) < C < ∞ (2.8)

for all ε > 0. Let

m(γ) =

sup

f (α) − γα.

αmin ≤α≤αmax

The following proposition is a consequence of Lemmas 2.2 and 2.4, which says that if f (α) is continuous on (αmin , αmax ) and the quantity D(H, x, r) satisfies a large deviation principle then the dimension of E is given by m(γ). Proposition 2.6. Suppose that f (α) is continuous on (αmin, αmax ) and H(X(α, β, r)) = O(r Q−f (β)−ε ) (2.9) for all 0 < α < β ≤ α0 and all ε > 0. If m ≥ 0, then almost surely dimH (E) ≤ m and dimH (E) ≥ m with positive probability. If m < 0, then E = ∅ almost surely. Proof. Suppose that m ≥ 0. We first consider the upper bound. Since trivially E(H(X(α0 , +∞, r)) ≤ H(X) = C < ∞, Lemma 2.2 implies that almost surely, dimH E ∩ {x | A(H, x)) ≥ α0 } ≤ max{0, Q − γα0 } = max{0, f (α0 ) − γα0 } ≤ m .

Next, let 0 < α < β < α0 . Combining Lemma 2.2 and (2.9), gives for all small ε > 0 that dimH E ∩ {x | α ≤ A(H, x)) ≤ β} ≤ max{0, f (β + ε) − γα + ε} . Letting ε ↓ 0 and using the continuity of f on ]αmin , αmax [ implies 1 1 − γα} dimH E ∩ {x | α ≤ A(H, x)) ≤ α + } ≤ max{0, f α + n n


11

for each αmin ≤ α = k/n < αmax , k ∈ N. Since there are only finitely many such values of α, we get γk k+1 − . (2.10) max dimH E ∩ {x | A(H, x)) ≤ α0 } ≤ f k n n k∈N,αmin < ≤α0 n

Letting n −→ ∞ and using the (uniform) continuity of f on ]αmin , αmax [ once more, finally yields the almost sure upper bound dimH (E) ≤ m . If m < 0, a straightforward modification of the argument using the latter claim of Lemma 2.2 implies E = ∅ almost surely. To prove the lower bound, we pick α such that m=

max

αmin ≤α≤α0

f (α) − γα > 0

and consider the measure µα provided by Frostman’s Lemma as in (2.8). Consider νt as in (2.3) and ν such that νt ⇀ ν. Lemma 2.4 implies that with positive probability ν(X) > 0 and further (applying the lemma with α + ε and letting ε ↓ 0) a.s. dimloc (ν, x) ≥ m , for ν-almost all x ∈ X. Since supp(ν) ⊂ E, this shows in particular that dimH (E) ≥ m with positive probability. Remark 2.7. The method presented above works for more general gauge functions h : (0, 1) → (0, +∞) and measures H so that C −1 < H(B(x, r))/h(r) < C for dr some C < ∞. In this case the Poisson intensity is γH(dx) × rh(r) . In the above, Q we have considered the case h(r) = r , for simplicity of notation and because the self-conformal measures in Section 3 are Q-regular. 3. Application to self-conformal spaces Let M be a d-dimensional Riemann manifold and G = {gi }ℓi=1 a conformal iterated function system (IFS) of class C 1+ε on M, i.e., gi are conformal contractions with tangent maps satisfying a Hölder condition of exponent ε. Let X ⊂ M be the self-conformal S set corresponding to G, that is, X is the unique compact set satisfying X = ℓi=1 gi (X). We suppose that the IFS G satisfies the strong separation condition, i.e., gi (X) ∩ gj (X) = ∅ for i 6= j. Let S : X → X be the inverse map of G on X, that is, the restriction of S on gi (X) is gi−1. Then (X, S) becomes a dynamical system. It is well known that (see e.g. [7, Chapter 5]) there exists a unique probability measure H on X, called the natural measure, which is S-invariant, ergodic and Q-regular, Q being the dimension of X.

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We will apply the auxiliary results from the previous section to determine the essential dimension of E. Instead of considering the continuous sequence {A(H, x, r), r > 0}, we will use the discrete one {A(H, x, |DS n (x)|−1 ), n ∈ N}, where DS n is the tangent map of S n . Since |DS n+1(x)|/|DS n (x)| = |DS(S n (x))| ∈ (1, maxx |DS(x)|] for all n ≥ 1, the limit behavior of A(H, x, r) when r → 0 is the same as that of A(H, x, |DS n (x)|−1 ) when n → ∞. We write n−1

X 1 fk (x), A(H, x, |DS (x)| ) = log |DS n (x)| k=0 n

−1

where fk (x) =

Z

|DS k (x)|−1

H(B(x, t))t−Q−1 dt . |DS k+1 (x)|−1

In our context, it is known that (see e.g. [6, Proposition 4.1], [7, Chapter 6.2]) the functions {fk }k satisfy for all k ≥ 0 and all x ∈ X |fn (S k x) − fn+k (x)| < εn

(3.1)

where εn → 0. Recall that X(α) = {x ∈ X : A(H, x) = α} and f (α) = dimH (X(α)). We will make use of the multifractal properties of X(α) that we present now. First, we introduce some notions and results. For simplicity of presentation, we express these results in the context of self-conformal sets/measures, although they are valid in a much more general setting. Notations. Let Λ = {1, · · · , ℓ}. Recall that gi , for i ∈ Λ, are conformal contractions. S For u = u1 · · · uk ∈ Λk we write gu = gu1 ◦ · · · ◦ guk . Let Xu = gu (X). Denote Λ∗ = n≥1 Λn and for u ∈ Λ∗ , let [u] = {(vn )n≥1 ∈ Λ∞ : v1 = u1 , . . . , vn = un }. For any x ∈ X, there exists (un )n≥1 ∈ Λ∞ such that {x} = limn gun1 (X) =: gu∞ (X) 1 where we write un1 = u1 · · · un . The transformation S can be defined as {S n (x)} = g u∞ (X). n A sequence Φ = {ϕn } of functions ϕn : X → R is called asymptotically additive if for each ε > 0 there exists a continuous function ϕ : X → R such that lim sup n→∞

where An ϕ =

Pn−1 k=0

1 sup |ϕn (x) − An ϕ(x)| < ε n x∈X

ϕ ◦ S k . If ϕn = An ϕ for all n, then Φ is called additive.

(3.2)


13

Pn−1 As a consequence of (3.1), the sequence { k=0 fk }n is asymptotically additive. Indeed, for any ε > 0, there exists N ≥ 1 such that εN < ε, then by (3.1) we have n−1 X 1 fk (x) − An fN (x)| < εN . lim sup sup | n→∞ n x∈X k=0

Now, we introduce the notion of pressure function. Let Φ = {ϕn }n be a sequence of continuous function ϕn : X → R. The pressure function associated to Φ is defined by X 1 (3.3) sup exp(ϕn (x)). P (Φ) = lim sup log n→∞ n x∈Xu n u∈Λ

Actually, when Φ is asymptotically additive, we can replace limsup by lim in the definition of P (Φ). Indeed, from the asymptotically additivity of ϕn , we deduce that for any ε > 0 there exists ϕ : X → R such that sup |ϕn (x) − An ϕ(x)| ≤ nε, for n ≫ 1.

(3.4)

x∈X

P P nε ± So, Bn := v∈Λn supx∈Xv exp(ϕn (x)) = (Ce ) v∈Λn supx∈Xv exp(An ϕ(x)) for ± some constant C > 0, whereP the notation A = C B means that C −1 B ≤ A ≤ CB. ñ := Since the sequence B v∈Λn supx∈Xv exp(An ϕ(x)) is sub-additive, the limit 1 ñ exists. So we have limn n log B | lim inf n

1 1 log Bn − lim sup log Bn | ≤ ε. n n n

Letting ε → 0 shows that the limit limn n1 log Bn exists. Let M(X, S) be the set of all S-invariant probability measures on X. For µ ∈ M(X, S) and an asymptotically additive sequence Φ = {ϕn }, define Z ϕn (x) Φ∗ (µ) := lim dµ(x). n→∞ X n

By (3.2), the limit in the above definition exists. Note that since µ is S-invariant, R R dµ(x) = we have X An ϕ(x) ϕ dµ for all n. (If µ is ergodic, then by Birkhoff’s n X ergodic theorem we deduce that Φ∗ (µ) is the µ-almost sure limit of ϕnn(x) as n → ∞). Further, it is known (see [10, Lemma A.4.], [2, Proposition 4]) that the map µ 7→ Φ∗ (µ) is continuous in the weak-star topology. Pn−1 fk }n and log DS = {log |DS n |}n . Let us return to the set X(α). Denote F = { k=0 Then F is asymptotically additive and DS is additive. Let F∗ (µ) Ω= : µ ∈ M(X, T ) . log DS∗ (µ)

14


We will use the following multifractal properties (Proposition 3.1) of X(α), most of them are from [2, Theorem 1] (see also [10]). Before presenting those properties, we need to introduce the notion of u-dimension. We will present this notion in our setting of self-conformal sets/measures. Let u : X → R+ be a continuous function. For each word v ∈ Λn , we write ( n−1 ) X u(v) = sup u(S k x) : x ∈ Xv . k=0

Given a set F ⊂ X and α ∈ R, we define

N(F, α, u) = lim inf n→∞ Γ

X

exp(−αu(v))

v∈Γ

where the infimum is taken over all countable collections Γ ∈ ∪k≥n Λk such that F ∈ ∪v∈Γ Xv . The u-dimension of F with respect to S is defined by dimu (F ) = inf {α ∈ R : N(F, α, u) = 0} . Note that if u = log |DS|, then the u-dimension dimu (F ) coincides with the Hausdorff dimension dimH (F ). This follows immediately from the existence of constants c1 , c2 > 0 such that c1 (diamXv )α ≤ exp(−αu(v)) ≤ c2 (diamXv )α . Proposition 3.1. The following statements hold: (1) The set Ω is a closed interval. (2) We have X(α) 6= ∅ if and only if α ∈ Ω and if α ∈ Ω, then F∗ (µ) hµ (S) =α . : µ ∈ M(X, T ) and dimu (X(α)) = max R log DS∗ (µ) u dµ X In particular, f (α) = max R

F∗ (µ) hµ (S) : µ ∈ M(X, T ) and =α . log DS∗ (µ) log |DS| dµ X

Here, hµ (S) denotes the measure-theoretic entropy of µ with respect to S. (3) The function f obtains its maximum at some αmin < α0 < αmax and f (α0 ) = Q. (4) A(H, x) = α0 for H-almost all x ∈ X. (5) The function f : int(Ω) → R is continuous. (6) If α ∈ Ω, then inf P (q(F − α log DS) − f (α) log DS) = 0.

q∈R

(7) H(X(0, β, r)) = O(r Q−f (β)−ε ) for all 0 < β ≤ α0 and all ε > 0.


15

Proof. The statements (2), (5) and (6) can be found in [2, Theorem 1]. Note that the definition of pressure function given in [2] is different from ours, but these two definitions actually give the same pressure function (see [1, Sections 2.2 and 4.2.2], [22, Proposition 3]). The statements (3) and (4) can be deduced from [6]: in Proposition 4.1 of [6] it is proved that there exists a constant α0 > 0 such that D(H, x) = α0 for H-a.e. x, so f (α0 ) = dim(H) = dimH (X) = Q which is the maximum of f . ∗ (µ) is continuous and M(X, T ) is For the statement (1), since the map µ 7→ logFDS ∗ (µ) a compact and convex set, we only need to notice that a subset of R, which is the image of a compact convex set under a continuous map, must be a closed interval. We give a proof for (7) in Appendix A, see Lemma A.1. As a consequence of (5), (7) and Proposition 2.6, we have: Theorem 3.2. If m = maxαmin ≤α≤α0 f (α) −γα ≥ 0, then almost surely dimH (E) ≤ m and dimH (E) ≥ m with positive probability. If m < 0, then E = ∅ almost surely. Example 3.3. Suppose that X is a self-similar set with equal contraction ratios (e.g. the classical ternary Cantor set), that is, there is constant 0 < a < 1 such that |gi′ | = a for all i, j ∈ Λ. Then, in this case, DS is constant on X and F is an additive sequence (see [7, Chapter 6.2]). Moreover F is Hölder continuous. It is well known that (see e.g. [8, 25, 23] ) the multifractal spectrum f (α) is analytical, strictly convex on Ω and for any α ∈ Ω we have (3.5) f (α) = inf P˜ (q) − αq q∈R

where P˜ (q) =

P (qF ) . − log a

We make two remarks:

(1) Observe that since f ′ (α0 ) = 0, we have m > Q − γα0 . Thus, the almost sure dimension of E is not due to the H-almost sure value of A(H, x) but is affected by the multifractal behaviour of the average densities. ) . This means that the (2) From (3.5), one can show that m(γ) = P˜ (−γ) = P−(−γF log a critical value (about the parameter γ) for the emptiness (or for the positivity of the Hausdorff dimension) of E is the unique zero of the pressure function (the pressure function in our case is strictly monotone). Remark 3.4. (ii) It seems plausible that in Theorem 3.2, dimH (E) is equal to the essential dimension a.s. conditioned on E 6= ∅. In other words, P(E 6= ∅ and dimH (E) < m) = 0. However, the proof only implies dimH (E) = m a.s. on ν(X) > 0, where ν is the random measure as in Lemma 2.4 corresponding to the value of α so that m = f (α) − γα. We expect that P(ν(X) = 0 and E 6= ∅), but haven’t been able to prove this. As pointed out in [26], this problem is open also in the case of X = [0, 1], H = L.

16


Appendix A. In this Appendix, we give the proof of the following lemma which is the statement (7) of Proposition 3.1. Lemma A.1. Under the setting of Proposition 3.1, we have H(X(0, β, r)) = O(r Q−f (β)−ε ) for all 0 < β ≤ α0 and all ε > 0. Notations and classical estimates. For u ∈ Λ∗ , let u˜ be the word obtained by erasing the last letter. For 0 < τ < 1, consider the “cut-set” Wτ = {u ∈ Λ∗ : diam(gu (X)) ≤ τ and diam(gu˜ (X)) > τ }. F It is clear that for any 0 < τ < 1, Λ∞ = u∈Wτ [u] and the IFS {gu }u∈Wτ generates the same attractor X, moreover H is the natural measure associated to {gu }u∈Wτ . For any x ∈ X, there exists (vn )n≥1 ∈ Wτ∞ such that {x} = limn gv1n (X) =: gv1∞ (X). We denote the inverse map corresponding to the IFS {gu }u∈Wτ by Sτ , so that we have {Sτn (x)} = gvn∞ (X). A well known calculation (see e.g. [21]) shows that a C 1+ε conformal iterated function system satisfies the bounded distortion principle: there exists L > 1 such that kg ′ (x)k L−1 ≤ u′ ≤ L for all u ∈ Λ∗ , x, y ∈ X. kgu (y)k ′ Let λ0 = min{||gi (x)|| : i ≤ ℓ, x ∈ X} > 0. Then for any u = u1 · · · un ∈ Λ∗ and y ∈ X, kgu′ (y)k = kgu′˜ (gun (y))kkgu′ n (y)k ≥ L−1 λ0 max kgu′˜ (z)k. z∈X

Now let u ∈ Wτ . Then ′ τ ≤ diam(gu˜ (X)) ≤ max kgu′˜ (z)kdiam(X) ≤ Lλ−1 0 diam(X) min kgu (z)k. z∈X

z∈X

On the other hand, X = gu−1 (gu (X)) so we have diam(X) ≤ max k(gu′ )−1 (z)kdiam(gu (X)) ≤ max k(gu′ )−1 (z)k · τ z∈X

z∈X

and max kgu′ (z)k z∈X

=

min k(gu′ )−1 (z)k z∈X

−1

≤ Lτ diam(X)−1 .

So there exists a constant C > 1 such that for any 0 < τ < 1 and any u ∈ Wτ , we have τ C −1 ≤ min kgu′ (z)k, max kgu′ (y)k ≤ τ C. (A.1) z∈X

y∈X


17

From (A.1), we deduce that τ −n C −n ≤ min kDSτn (z)k, max kDSτn (y)k ≤ τ −n C n . z∈X

(A.2)

y∈X

Now we can give the proof of Lemma A.1. proof of Lemma A.1. Fix x ∈ X and a small 0 < r < 1. Let n = n(x, r) ∈ N be such that |DSτn+1(x)|−1 ≤ r ≤ |DSτn (x)|−1 . r r ≤ n ≤ log τlog . Here and in the rest of the From (A.2), we know that log τlog −log C +log C −1 proof, we always take a τ < C so that log τ + log C < 0. Then we have R1 R1 −Q−1 H(B(x, t))t−Q−1 dr H(B(x, t))t dt |DSτn (x)|−1 r ≥ . − log r log |DSτn+1(x)|

So we get {x ∈ X : A(H, x, r) ≤ β} ⊂

(

x∈X :

R1

|DSτn (x)|−1

H(B(x, t))t−Q−1 dr

log |DSτn+1(x)|

≤β

)

=: Aτ,n .

Thus we have log H(Aτ,n ) log H(Aτ,n ) log H(Aτ,n ) log H(X(0, β, r)) ≤ ≤ ≤ . n+1 − log r − log r log |DSτ (x)| (n + 1)(− log τ + log C) For proving the claim of the lemma we only need to show that lim sup lim sup τ →0+

n→∞

log H(Aτ,n ) ≤ Q − f (β). −n log τ

Recall that we can rewrite Aτ,n as ( Aτ,n =

x∈X :

Pn−1

τ k=0 fk (x) log |DSτn+1(x)|

≤β

)

R |DSτk (x)|−1 H(B(x, t))t−Q−1 dr is asymptotically additive for the where fkτ (x) = |DS k+1 (x)|−1 τ system (X, Sτ ). By Chebyshev’s inequality, for any λ ≥ 0 !! Z n−1 X fkτ (x) dH(x) H(Aτ,n ) ≤ exp λ β log |DSτn+1(x)| − X

≤

X

v1n ∈Wτn

k=0

H(gv1n (X))

sup x∈gvn (X) 1

exp λ β log |DSτn+1 (x)| −

n−1 X k=0

fkτ (x)

!!

.

18


Since H is the natural measure of the IFS (gu )u∈Wτ , we have that H(gv1n (X)) ≍ exp(−Q log |DSτn (x)|) for any x ∈ gv1n (X). Whence H(Aτ,n ) . X

sup

v1n ∈Wτn x∈gv1n (X)

exp λ β log |DSτn+1 (x)| −

n−1 X

fkτ (x)

k=0

!

− Q log |DSτn (x)|

!

,

(A.3)

whenever λ ≥ 0. Note that | log |DSτn+1 (x)| − log |DSτn (x)|| ≤ maxz∈X log |DSτ (z)|. Taking logarithms and dividing both sides of (A.3) by n and then taking limsup we get lim sup n

log H(Aτ,n ) ≤ Pτ (λ(β log DSτ − Fτ ) − Q log DSτ ), n

λ ≥ 0.

where Pτ (λ(β log DSτ − Fτ ) − Q log DSτ ) is the pressure function (of the system (X, Sτ )) associated to the sequence of functions ( ! ) n−1 X λ β log |DSτn (x)| − fkτ (x) − Q log |DSτn (x)| . k=0

We now show that the above inequality holds also when λ < 0. For this, we only need to show that Q log DSτ )≥ 0 for λ < 0. Fix λ < 0. R Pτ (λ(β log DSτ − Fτ ) −P n−1 τ fk (x) dH(x). We are going to Denote Bn = X exp λ β log |DSτn (x)| − k=0 log Bn prove that lim supn n ≥ 0, which will imply Pτ (λ(β log DSτ −Fτ )−Q log DSτ ) ≥ 0. By Jensen’s inequality we have ! ! Z n−1 X fkτ (x) dH(x) Bn ≥ exp λ β log |DSτn (x)| − X

So

log Bn ≥ n We know that

Z

k=0

! n−1 log |DSτn (x)| 1 X τ f (x) dH(x). λ β − n n k=0 k X Pn−1

fkτ (x) = α0 , H-a.e. x n→∞ log |DSτn (x)| lim

k=0

Since α0 ≥ β and λ < 0, in view of (A.4), we get lim sup n

log Bn ≥ 0. n

(A.4)


19

So we have proved that lim sup n

log H(Aτ,n ) ≤ inf Pτ (λ(β log DSτ − Fτ ) − Q log DSτ ). λ∈R n

For completing the proof, we only need to show that lim sup τ →0+

inf λ∈R Pτ (λ(β log DSτ − Fτ ) − Q log DSτ ) ≤ Q − f (β). − log τ

(A.5)

From the definition of the pressure function Pτ and the fact n(− log τ − log C) ≤ log |DSτn (x)| ≤ n(− log τ + log C), we deduce that |Pτ (λ(β log DSτ − Fτ ) − Q log DSτ ) − Pτ (λ(β log DSτ − Fτ ) − f (β) log DSτ ) − (Q − f (β)) log τ | ≤ 2 log C . So for proving (A.5), it is sufficient to show that inf Pτ (λ(β log DSτ − Fτ ) − f (β) log DSτ ) = 0,

λ∈R

but this is exactly the statement (6) of Proposition 3.1 for the system (X, Sτ ). This ends the proof of the lemma. References [1] Luis Barreira. Thermodynamic formalism and applications to dimension theory, volume 294 of Progress in Mathematics. Birkhäuser/Springer Basel AG, Basel, 2011. [2] Luis Barreira, Yongluo Cao, and Juan Wang. Multifractal analysis of asymptotically additive sequences. J. Stat. Phys., 153(5):888–910, 2013. [3] Tim Bedford and Albert M. Fisher. Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3), 64(1):95–124, 1992. [4] Hermine Biermé and Anne Estrade. Covering the whole space with Poisson random balls. ALEA Lat. Am. J. Probab. Math. Stat., 9:213–229, 2012. [5] Youssef El Helou. Recouvrement du tore Tq par des ouverts aléatoires et dimension de Hausdorff de l’ensemble non recouvert, volume 9 of Publications Mathématiques d’Orsay 78. Université de Paris-Sud, Département de Mathématique, Orsay, 1978. [6] K. J. Falconer. Wavelet transforms and order-two densities of fractals. J. Statist. Phys., 67(34):781–793, 1992. [7] Kenneth Falconer. Techniques in fractal geometry. John Wiley & Sons, Ltd., Chichester, 1997. [8] Ai Hua Fan. Sur les dimensions de mesures. Studia Math., 111(1):1–17, 1994. [9] Ai-Hua Fan, Jörg Schmeling, and Serge Troubetzkoy. A multifractal mass transference principle for Gibbs measures with applications to dynamical Diophantine approximation. Proc. Lond. Math. Soc. (3), 107(5):1173–1219, 2013. [10] De-Jun Feng and Wen Huang. Lyapunov spectrum of asymptotically sub-additive potentials. Comm. Math. Phys., 297(1):1–43, 2010. [11] P. J. Fitzsimmons, Bert Fristedt, and L. A. Shepp. The set of real numbers left uncovered by random covering intervals. Z. Wahrsch. Verw. Gebiete, 70(2):175–189, 1985.

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¨skyla ¨, Finland Department of Mathematics and Statistics, University of Jyva E-mail address: [email protected] Department of Mathematical Sciences, University of Oulu, Finland E-mail address: [email protected] Department of Mathematical Sciences, University of Oulu, Finland E-mail address: [email protected]

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