GEOMETRY OF SELF-SIMILAR MEASURES ON INTERVALS WITH OVERLAPS AND APPLICATIONS TO SUB-GAUSSIAN HEAT KERNEL ESTIMATES

. We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS’s) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the inﬁnite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities deﬁned by auxiliary IFS’s to compute measures of cells on diﬀerent levels. These auxiliary IFS’s do satisfy the OSC and are used to deﬁne new metrics. As an application, we obtain sub-Gaussian heat kernel es- timates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians.


(Communicated by Camil Muscalu)
Abstract. We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than 2 and are closely related to the spectral dimension of fractal Laplacians.

QINGSONG GU, JIAXIN HU AND SZE-MAN NGAI
Bass [1,2] for the Sierpiński carpets, and by Kigami [17,19] for time changes of self-similar diffusions on self-similar sets. Equivalent conditions for two-sided estimates of heat kernels for local Dirichlet forms on metric measure spaces are given by Grigor'yan, Lau and the second author [10,11], Grigor'yan and Telcs [13], and others (see also [3] and [15] for certain classes of resistance forms).
The most typical measures on fractals appearing in such studies are s-dimensional Hausdorff measures for some number s > 0, and they are equivalent to self-similar measures generated by iterated function systems (IFS's) satisfying the open set condition (OSC). This paper studies self-similar measures generated by IFS's that do not satisfy the OSC. These measures are not Ahlfors-regular but still possess the doubling property with respect to suitable metrics. Although the self-similar sets themselves are intervals and hence Dirichlet forms can be defined easily, the associated self-similar measures exhibit complicated fractal behavior, and therefore heat kernel estimates become much more awkward.
Let a, b ∈ R, a < b, and set K := [a, b]. Let Consider the following form E with domain F defined as Note that for any u ∈ F and any x, y ∈ K, |u(x) − u(y)| 2 ≤ E(u)|x − y|, Let µ be a Radon measure on K with full support supp(µ) = K = [a, b] (that is, µ (I) > 0 for any nonempty open interval I in K). Clearly, the form (E, F) is densely defined, non-negative definite, symmetric, bilinear and Markovian in L 2 (µ) := L 2 (K, µ). By using (1.3), it is not hard to see that H 1 (a, b) is complete under the norm E(u) + u 2 L 2 (µ) . The form (E, F) given by (1.2) and (1.1) is thus a Dirichlet form in L 2 (µ) (cf. [9]). Moreover, (E, F) is regular, conservative and strongly local in L 2 (µ) (since 1 ∈ F and E(1) = 0, the form (E, F) is conservative). This paper first investigates the geometric properties of certain self-similar measures with overlaps, and then obtains two-sided estimates of the heat kernel of the form (E, F) for these measures by using existing machinery. We call a map S : R n → R n a contractive similitude on R n if S(x) = rAx + b, where r ∈ (0, 1), b ∈ R n and A is an n × n orthogonal matrix. Let {S i } N i=0 be contractive similitudes on R such that K = where 0 < ρ i < 1 for each i and N i=0 ρ i = 1. It is easy to see that µ(A) = 0 (1.6) if A is a singleton 1 .
Let {T j } m j=0 be an auxiliary IFS of contractive similitudes: |T j (x) − T j (y)| = r j |x − y| for any x, y ∈ K, (1.7) where 0 < r j < 1 for each j, such that {T j (K) : j = 0, 1, . . . , m} forms a partition of K, in the sense that T j (K), (1.8) and the intervals T i (K) and T j (K) can only intersect at their end-points if i = j.
We call K ω an n-cell, and write K ω ∼ K τ if two words ω, τ satisfy K ω ∩K τ = ∅. We say that two words ω, τ having the same length, or the corresponding cells K ω , K τ are neighbors or neighboring if K ω ∼ K τ and ω = τ . Note that {T 0 , T 1 , . . . , T m }, not {S 0 , S 1 , . . . , S N }, is used to define K ω . For each n ∈ N, let J = {0, 1, . . . , m} and let J n := {0, 1, . . . , m} n , be respectively the sets of words with length n and those with finite length. Here J 0 is defined to be the singleton {∅} of the empty word ∅, and we use the convention that ω∅ = ∅ω = ω for any word ω.
For two finite words ω and τ , we say ω < τ if there exists a non-empty word γ such that τ = ωγ. We write ω ≤ τ (and call ω a father of τ ) if ω < τ or ω = τ . Let d * be a metric on K which produces the same topology as the Euclidean metric, and let denotes the open ball with center x and radius r under the metric d * . Throughout this paper, the sign f g means that C −1 g ≤ f ≤ Cg for some universal constant C > 0 independent of the arguments f and g. Fix some number β > 1. We introduce the following conditions that may or may not be satisfied: (1) comparability of neighboring cells: if τ and σ are neighbors, then (1.10)

QINGSONG GU, JIAXIN HU AND SZE-MAN NGAI
(2) generalized mid-point property: for any points x, y, z ∈ K with x < y < z, (1.14) Note that we can take β = 2 in the above conditions if µ is the Lebesgue measure and d * is the Euclidean metric. In this paper we are interested in the situation where β is strictly greater than 2 with respect to suitable µ and d * on K.
Note that condition (1.11) implies the mid-point property, which in turn implies the chain condition, see for example [12,Definition 3.4]. (A distance d on a nonempty set X is said to have the mid-point property if for any x, y ∈ X, there exists some z ∈ X such that d(x, z) = d(z, y) = d(x, y)/2.) Let µ be a Radon measure on K = [a, b] with full support, and let (E, F) be defined by (1.2) and (1.1). Let {T j } m j=0 be an auxiliary IFS defined by (1.7) such that (1.8) holds. When the metric d * on K has been constructed such that conditions (1.10)-(1.14) are all satisfied for d * , then by applying known equivalent conditions for heat kernel estimates given in [18,Theorem 15.10] with g(r) = r β , one can conclude that the jointly continuous heat kernel p t (x, y) of (E, F) exists and satisfies the upper estimate and the lower estimate for all t ∈ (0, 1) and all x, y ∈ K. We consider two classes of specific Radon measures µ and introduce a new metric d * accordingly. The first class consists only of the infinite Bernoulli convolution associated with the golden ratio. Let 15) and let µ be the self-similar measure with supp(µ) = [0, 1] satisfying: The metric d * and the constant α ∈ 0, 1 2 in the following theorem will be given in Section 2. Theorem 1.1. Let µ be defined by (1.16), {T j } 2 j=0 be defined by (2.1) and d * by (2.16) below. Let α ∈ 0, 1 2 be defined by (2.17) and β := 1/α > 2. Then all the conditions (1.10)-(1.14) are satisfied. Consequently, the heat kernel p t (x, y) of (E, F) exists and satisfies the two-sided estimates (UE) and (LE) with this parameter β. Theorem 1.1 will be proved in Section 2. The second class is a family of convolutions of Cantor-type measures. Let where m ≥ 3 is an integer. The attractor of this IFS is a symmetric Cantor-type set. Let ν m be the self-similar measure defined by the IFS (1.17) with probability weights p 0 = p 1 = 1/2. The m-fold convolution µ m = ν * m m of ν m is the self-similar measure defined by the following IFS with overlaps (see [25]): together with probability weights That is, The metric d * and the constant α ∈ 0, 1 2 in the following theorem will be given in Section 3. . Let d * be a metric defined by (3.13) below, let α ∈ 0, 1 2 be a constant defined by (3.14) and β := 1/α > 2. Then the same conclusions as those of Theorem 1.1 hold with this value of β.
We will prove Theorem 1.2 in Section 3. We remark that we cannot deal with the case m = 2 for technical reasons. For m ≥ 3, we can separate the set J = {0, 1, . . . , m − 1} into two subsets J 0 = {0, m − 1} and J 1 = {1, . . . , m − 2}, so that the equation (3.15) below makes sense as J 1 is not empty. However, this technique does not apply when m = 2 since the set J 1 is empty.
Our main efforts in this paper are in constructing the metric d * on K for the two classes of self-similar measures with overlaps and proving the generalized midpoint property (1.11) of d * , which is crucial in obtaining the full off-diagonal lower estimate of the heat kernel.
The issue of constructing a metric (not necessarily geodesic) that is suitable for describing heat kernel behavior for a given symmetric regular Dirichlet space has been extensively studied by a series of works [17,18,19] by Kigami. In particular, for the time change of the reflecting Brownian motion on a compact interval by a measure, he has shown in [17,19] that, in order to obtain the sub-Gaussian heat kernel estimate in Euclidean spaces, it suffices to verify that the measure is volume doubling with respect to the Euclidean metric only, see [17, ] (or [18,Corollary 15.12]) where the existence of a metric d * with respect to which µ is volume doubling, off-diagonal upper bound and near-diagonal lower bound of the heat kernel were obtained. What is new in this paper is that we have explicitly constructed a metric that is also geodesic, and have specified the exact value of the walk dimension that is strictly greater than 2. This topic is highly non-trivial for an IFS with overlaps.
We outline the proof of Theorems 1.1 and 1.2 here. First, we use second-order identities and the structure of the associated matrices to show that the measures of two neighboring cells are comparable (Lemmas 2.1 and 3.1) and consequently that µ is doubling with respect to the Euclidean metric. Then, we construct the abovementioned metric d * and use the spectral dimension formula to prove the generalized mid-point property (Propositions 2.4 and 3.6). Next, we use the spectral dimension formula to show that the resistance metric (or equivalently, the Euclidean metric), the metric d * , and the measure µ "match well" (Lemmas 2.7 and 3.8). Finally, we use the above estimates to verify that conditions (1.10)-(1.14) hold.
Our results also have applications in the study of the wave propagation speed problem on fractals (see [27,6]). In fact, using results in this paper and a generalization of a theorem of Y.-T. Lee [24], Ngai et al. [26] proved that waves defined by the Laplacians in Theorems 1.1 and 1.2 have infinite propagation speed.
2. Infinite Bernoulli convolution associated with the golden ratio. Let K = [0, 1] and µ be given by (1.16) and (1.15). In this section we introduce a new metric d * on K, and show that conditions (1.10)-(1.14) are all satisfied.
It is shown in [28] that by introducing the auxiliary IFS {T 0 , T 1 , T 2 }: Figure 1), one can obtain the following second-order identities: For all Borel subsets A ⊂ [0, 1], (see [28, p.109]). Let be the contraction ratios of the auxiliary IFS {T 0 , T 1 , T 2 }. We will introduce a new metric d * on K below. Before this, we show condition (1.10). We use the following notation. For each n ∈ N, let where J 0 and J 0 0 are defined to be the singleton {∅} of the empty word ∅. For a symbol ξ ∈ {0, 1, 2}, let e ξ be the row matrix defined as For ω = (ω 1 , . . . , ω n ), with |ω| ≥ 1, by applying (2.2) repeatedly, we obtain Lemma 2.1. For any two neighboring words ω and τ , we have Consequently, condition (1.10) is satisfied.
Proof. Without loss of generality, we assume that K ω is on the left of K τ . Then either of the following relationships holds for such ω and τ : or ω = θ12 · · · 2 and τ = θ20 · · · 0 , (2.9) where θ is some finite word (possibly the empty word) and ≥ 0 is some integer.
For any 0 ≤ x < y ≤ 1, we define a set W(x, y) of finite words as follows: , and ω is a father , (2.14) where the notion "ω = ω 1 · · · ω n is a father" means that no proper ancestor ω 1 · · · ω k (k < |ω|) of ω satisfies both of the following conditions: In other words, a word ωξ with ξ = ∅ cannot belong to W(x, y) if ω ∈ W(x, y), and hence ω has shortest length among all words satisfying the conditions ω n = 1 and , a singleton. Note that the word "11" does not belong to W(ρ 2 , ρ) since "1" is its father. As another example, let [x, y] = [0, ρ 2 ] = K 0 . In this case, is in W(0, ρ 2 ), although each word ω in this set ends with the symbol "1", and The reason is that all of them are offspring of the word "01", which is in W(0, ρ 2 ).
Note that W(x, y) = ∅ for any x, y with 0 ≤ x < y ≤ 1. Define a function d * : with c J given by for any index J = j 1 · · · j k ∈ J k 0 := {0, 2} k and any integer k ≥ 0. Here we use the convention that M ω := I, the identity matrix, if ω is the empty word.
Remark 2.2. Let ∆ µ be the Laplacian defined by µ (see [5,16]). Then where dim s (µ) is the spectral dimension of the corresponding Dirichlet and Neumann Laplacians −∆ µ (see [25]). In fact (see [25, p.654 (the value of α is close to but strictly less than 0.5). This is in sharp contrast with the classical case where α = 0.5 for the Euclidean metric and the Lebesgue measure.
for any 0 ≤ x < y < z ≤ 1. Moreover, granted that d * (K) < ∞, a fact to be proved in Proposition 2.5, d * is a metric on K and consequently, condition (1.11) is satisfied.
To do this, we first claim that for any ω = ω 1 · · · ω n with ω n = 1, we have The left-hand side of (2.24) is (r ω µ(K ω )) α = r α ω · µ(K ω ) α , and in view of (2.5), the right-hand side of (2.24) is: Thus we only need to show that (2.25) To do this, we first consider the case |ω| ≥ 2. We use (2.6) to get where a b c = e ω1 M ω2 · · · M ωn−1 is some row vector with nonnegative entries. Similarly, we have We note that the conclusions of equations (2.26) and (2.27) are valid also for ω with |ω| = 1 as long as a b c is chosen to satisfy a + b + c = 4 (even though the computations in these equations do not make sense literally for such ω). Therefore, by (2.17), (2.26) and (2.27), we obtain (2.25), proving (2.24).
and that Let τ ∈ W(x, z) and k ∈ N. Then by using (2.24) k times, we have ..,J k ∈J * 0 , and in particular at most one of these cells can contain y. Therefore, by classifying all the other terms on the right-hand side of (2.30) according to its unique prefix in W(x, y) ∪ W(y, z) and using (2.24) , we obtain ω∈W(x,y)∪W(y,z)
For any word ω, define the diameter d * (K ω ) of the cell K ω under the metric d * by (1)) .
Proof. We first claim that for any finite word ω, the following holds: Hence, We first show the '≤' part of the first line in (2.33). We first consider the case |ω| ≥ 2. Indeed, by (2.20), and using (2.6), we see that where a b c := e ω1 M ω2 · · · M ωn and a, b, c are nonnegative numbers. Thus we have Again, combining the definition of d * , (2.6), and (2.21), we get and We note that the above computations and estimates of µ(K ω1 ) and µ(K ω0J1 ) are valid also for ω with |ω| = 1 as long as a b c is chosen to satisfy a + b + c = 4 (even though the arguments in this proof do not make sense literally for such ω). Thus, the right-hand side of (2.35) satisfies (definition of α) Using this and (2.34), we obtain thus proving the second '≤' in the first line in (2.33).
Combining this with Proposition 2.4 proves that d * is a metric on K.
We need the following proposition.
Proposition 2.6. Let x, y ∈ K, x < y and let ω be a shortest word such that K ω ⊆ [x, y]. Set |ω| = n. Then there are at most four n-cells between the points x and y.
Note that x, y cannot be separated apart by any (n − 1)-cell; otherwise, ω would not be the shortest. In other words, both x and y must lie in the union of two neighboring (n − 1)-cells.
We first consider the case ω = ω 0. The point x must lie to the left of K ω , since Let τ be the left neighboring (n − 1)-word of ω (even though the case ω = 0 · · · 0 does not admit τ , this case can still be treated similarly). Then Figure 2). Clearly, there are at most four n-cells between the points x and y: thus proving our conclusion.
Proof. It suffices to show that there exists a constant C > 1 (depending only on ρ) such that for any 0 ≤ x < y < z ≤ 1 with d * (x, y) = d * (y, z), we have (2.48) Choose any two shortest words ω and τ such that We claim that there exists a universal integer k ≥ 0 (depending only on ρ) such that |ω| − |τ | ≤ k. (2.49) Indeed, without loss of generality, assume that |ω| ≥ |τ | ≥ 1 , and let ω ≤ ω be such that |ω | = |τ |, ω = ω θ for some word θ (possibly θ = ∅). Then by applying Proposition 2.6, we see that the number of words with the same length |τ | and lying between y and z is at most 4. See Figure 3 for the worst case when ω = ω. More precisely, the cell K ω can be connected to the cell K τ by at most nine |τ |-cells. Thus, by repeatedly using Lemma 2.1, we have Combining this with (2.50) and the inclusion K ω ⊆ K ω , we see that there exists some C 0 > 0 such that and after dividing by µ(K τ ), we get Combining this with (2.51), we have where we have used the following fact from (2.5): This shows that |ω| − |τ | is bounded by a universal integer k, proving our claim. Note that by (2.36) and Lemma 2.1, for any word σ and any i ∈ {0, 1, 2}. From this and using (2.49), we have which together with (2.50) gives Finally, from this and using (2.41), we see that thus proving (2.48). The lemma follows.
Proof. For any x ∈ [0, 1], any small 0 < r < d * (K)/2 and any integer ≥ 2, choose z and y in [0, 1] such that either x < y < z or z < y < x, and such that From this and using (1.12), we have Thus, in order to prove (1.14), it suffices to show that where the convergence is uniform with respect to x, r and all possible choices of y , z. We may assume that x < y < z; the other case z < y < x is similar. Choose a shortest word ω := ω( ) such that Then by (2.42) we have |x − y | r ω .
(2.56) Consider a chain of k + 1 neighboring words starting from ω and with length |ω|. By Proposition 2.4 and Lemma 2.5, there exists a constant C 0 > 1 such that the sum of d * distances of these cells is not more than Choose k to be the largest integer such that  Figure 4).
However, it follows from earlier discussions that there are k + 1 = 3 N +1 + 1 cells with length |ω| inside [x, z]. Thus, after summing up, the total number of |ω|-cells in K is at least a contradiction, since the number of |ω|-cells on K is 3 |ω| . This proves the claim.
Proof of Theorem 1.1. For example, for m = 3, In general,
For ω = ω 1 · · · ω ∈ J , we use the notation as before. For i ∈ J , let e i denote the unit vector in R m whose (i + 1)-st coordinate is 1. Applying (3.4) repeatedly yields Throughout this section we let M t , u t denote the transposes of a matrix M and a vector u respectively. Proof. Assume, without loss of generality, that K ω is on the left of K τ . Then exactly one of the following relationships holds for i = 0, 1, . . . , m − 2 (θ can be empty): ω = θi(m − 1) · · · (m − 1) k and τ = θ(i + 1)0 · · · 0 k .
Recall that for −(m − 1) ≤ k ≤ m − 1, the k-diagonal of an m × m matrix A = (a j ) consists of the entries a j with j = + k. The main diagonal is the 0diagonal. We say that A is of Type 0 (or Type m − 1) if all its k -diagonals are zero, except possibly for k = 0 or k = ±(m − 1), and that it is of Finally, (c) follows from (b) by taking transpose and using (a).
We now study products of such matrices. (2) the nonzero entries corresponding to multiplying A with the (−j )-diagonal of B are (m − j + , + 1), 0 ≤ ≤ j, which belong to the (−j )-diagonal, and (1, j + 1), which belongs to the j-diagonal.
Hence AB is of Type j. Next, we assume 1 ≤ i ≤ m − 2 and j = 0. Then AB = (B t A t ) t . By Proposition 3.2(a), B t is of Types 0 and A t is of Type i . By what we have just proved above, B t A t is of Type i . By Proposition 3.2(a) again, we see that AB is of Type i. + 1, 2), . . . , (m, i + 1); the j-diagonal of B is: (1, j + 1), (2, j + 2), . . . , (m − j, m); the (−j )-diagonal of B is: (m − j, 1), (m − j + 1, 2), . . . , (m, j + 1). By checking the product of these diagonals, we see that if i + j ≤ m − 1, then the nonzero entries of AB are along the (i + j)-diagonal or the −(i + j) -diagonal. If i + j ≥ m , then the nonzero entries of AB are along the (i + j − m + 1) -diagonal or the −(i + j − m + 1) -diagonal. In the former case, AB is of Type i + j; in the latter case, it is of Type i + j − m + 1. This completes the proof. For i ∈ {0, m − 1}, let M i be the matrix formed from M i by keeping its first and last rows and assigning 0 to all other entries. For i ∈ J 1 , let M i denote the matrix formed from M i by keeping its (m − i)−th row and assigning 0 to all other entries. For J = (j 1 , . . . , j k ) ∈ J * 0 , where k ≥ 0, define (see [23])  ω = ω 1 · · · ω n ∈ J * \ {∅} : where the notion "ω = ω 1 · · · ω n is a father" means that none of the proper ancestors (or prefixes) ω 1 · · · ω k (k < n) satisfies all of the following conditions: Define a set of words S by where α is the unique solution of the equation where c i,J is given by (3.11). We refer to [25, Theorem 1.3, Lemma 6.12] for the existence of α and [25, Corollary 1.4] for the estimate α < 1/2. We remark that 2α is the spectral dimension of the Laplacian −∆ µ defined by µ [25]. For example if m = 3, then α ≈0.4985 < 0.5 (this value is close to but strictly less than 0.5).
After summing over {σ : σ ∈ S, |σ| = k − 1} and using (3.28), we obtain (3.27), thus proving our claim. where we have used the fact that (1/m α ) m−1 j=1 w α j < 1 by (3.14). By taking θ = ∅ and using the fact that r σ = 1/m |σ| , we see that the left-hand side of (3.26) is On the other hand, using (3.30) with θ = J σ and summing up the equalities, we can write the right-hand side of (3.26) as which is true by using the definition (3.14) of α.