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Hausdorff Dimensions for Graph-Directed Measures Driven By HAUSDORFF DIMENSIONS FOR GRAPH-DIRECTED MEASURES DRIVEN BY INFINITE ROOTED TREES KAZUKI OKAMURA Abstract. We give upper and lower bounds for the Hausdorff dimen- sions for a class of graph-directed measures when its underlying directed graph is the infinite N-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains sev- eral measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierp´ınski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham’s func- tional equations driven by linear fractional transformations. Contents 1. Introduction and Main results 1 2. Proof of Theorem 1.1 5 3. Examples 11 4. Open problems 29 References 30 1. Introduction and Main results Let N 2. Let ΣN := 0, 1,...,N 1 . Let Y be a set and Z be a measurable≥ space. For i {Σ , let G −: Y} [0, 1] and H : Y Y be ∈ N i → i → maps and Fi : Z Z be measurable maps. Then we consider the following equation for a family→ of probability measures µ on Z: { y}y∈Y µ = G (y)µ F −1, (1.1) y i Hi(y) ◦ i i∈ΣN arXiv:1703.03934v3 [math.CA] 6 Sep 2019 X under the assumption that Gi(y) = 1. (1.2) i∈XΣN Equations for probability measures similar to (1.1) appear in the Markov type measures in Edgar-Mauldin [EM92], a self-similar family of measures in Strichartz [St93, Definition 2.2] and the graph directed self-similar measures in Olsen [Ol94, Section 1.1]. Let (V, E) be a directed graph where multi- edges and self-loops are allowed. [EM92], [St93] and [Ol94] focus on the case 2010 Mathematics Subject Classification. 26A27, 26A30, 28A78, 28A80, 60G30. Key words and phrases. Hausdorff dimension for measure, graph-directed measures, sin- gularity problem, de Rham’s functional equations. 1 2 KAZUKI OKAMURA that V is finite. Several arguments in these references such as the Perron- Frobenius theorem and ergodic theorem for finite Markov chains depend on the fact that V is finite. Here we consider the case that (V, E) is the infinite N-ary tree on which each edge is equipped with a direction from a vertex closer to the root to its descendants, and that Fi i is an iterated function system and Z is its attractor. { } If V = Y (y)= y (See (1.4) and Remark 1.2 (ii) below for the definition of Y (y).), then, { } µ = G (y)µ F −1. y i y ◦ i iX∈ΣN If Fi i is a family of contractions on a complete metric space, then, µy is the{ invariant} measure of the iterated function system (Z, F ) equipped { i}i∈ΣN with probability weights (Gi(y))i∈ΣN . However, this paper rather focuses on the case that µy is not an invariant measure of an iterated function system. Our class contains several measures which are not not necessarily invariant or Gibbs measures and appear in fractals and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional standard Sierp´ınski gasket, the Kusuoka energy measures on it, and furthermore measures defined by solutions of de Rham’s functional equations driven by linear fractional transformations. We give upper and lower bounds for the Hausdorff dimensions for µy y∈Y under certain regularity conditions for the functions G and H{ } { i}i∈ΣN { i}i∈ΣN on Y and the assumption that Fi i∈ΣN is an iterated function system on a complete metric space satisfying{ certain} conditions. We now state some ap- plications of our results. We extend the main result of [ADF14] considering restrictions of non-constant harmonic functions on the two-dimensional stan- dard Sierp´ınski gasket. Our proof gives an alternative proof of singularity of the Kusuoka energy measures on the standard 2-dimensional Sierp´ınski gas- ket [Ku89]. Furthermore, de Rham’s functional equations driven by linear fractional transformations considered in [Ok14] are also generalized. Specifi- cally, we deal with the case that an equation is driven by N linear fractional transformations which are weak contractions. [Ok14] deals with the case that an equation is driven by only two linear fractional transformations which are contractions. We also discuss singularity of µy with respect to self-similar measures of iterated function systems equipped with probability weights which are not a canonical measure on Z. 1.1. Framework and main results. Now we describe our framework and main results. Let ΣN := 0, 1,...,N 1 ,N 2. Let Si, i ΣN , be contractive maps on a complete{ metric space− (}M, d),≥ and K be the∈ attractor of the iterated function system Si i∈ΣN . We put the Borel σ-algebra on K induced by the metric d on M.{ For}x K and r > 0, let B(x, r) = y K : d(x,y) r . ∈ { ∈ ≤ } For m 1, let Si1···im := Si1 Sim and Ki1···im := Si1···im (K). For each i Σ ≥, let r be the Lipschitz◦···◦ constant of S , in other words, ∈ N i i d(Si(x),Si(y)) ri := sup . x,y;x6=y d(x,y) 3 Assume that 0 < ri < 1 for each i ΣN . Let s be a positive real number such that ∈ s ri = 1. (1.3) i∈XΣN Let Y be a set. Let H : Y Y, i Σ , be maps. For each y Y , let i → ∈ N ∈ Y (y) := H H (y) Y i ,...,i Σ , l 1 . (1.4) { il ◦···◦ i1 ∈ | 1 l ∈ N ≥ } Let G : Y [0, 1], i Σ , be maps satisfying that for each y Y , i → ∈ N ∈ G (y) = 1. We remark that this sum is taken with respect to i Σ , i ∈ N i∈ΣN notX to y Y . Let dim∈ A be the Hausdorff dimension of A K. Let the Hausdorff H ⊂ dimension of a measure µ be dimH µ := inf dimH B µ(B) = 1 . The following is our main result. { | } Theorem 1.1. Under the above setup, a family of probability measures µ on the attractor K satisfying that { y}y∈Y µ = G (y)µ S−1, y Y, y i Hi(y) ◦ i ∈ iX∈ΣN exists and is unique. Furthermore we have the following two statements for every y Y : (i) Let s∈be the positive number in (1.3). Assume that there exist a constant s ǫ0 0, min ri and an integer l 1 such that ∈ i∈ΣN ≥ Y (y) G−1 ([rs ǫ , rs + ǫ ]) ∩ i i − 0 i 0 i∈\ΣN (G H H )−1 rs ǫ , rs + ǫ = (1.5) ∩ j ◦ il ◦···◦ i1 j − 0 j 0 ∅ j∈Σ \N for every i ,...,i Σ . Then, 1 l ∈ N dimH µy < s. (1.6) (ii) Assume that there exists constants r (0, 1),c> 0 and D> 0 such that for every x K and m 1, ∈ ∈ ≥ (i , , i ) (Σ )m B(x, crm) K = D, |{ 1 · · · m ∈ N | ∩ i1···im 6 ∅}| ≤ and furthermore it holds that c inf Gi(w) sup Gi(w) 1 c (1.7) ≤ w∈Y (y) ≤ w∈Y (y) ≤ − for some i Σ and c> 0. Then, ∈ N dimH µy > 0. Remark 1.2. (i) If (M, d) is a Euclid space and S , i Σ , are similitudes i ∈ N and satisfy the open set condition, then, it holds that dimH K = s for the attractor K and the positive number s in (1.3) ([F14, Theorem 9.3]). (ii) The subset Y (y) of Y in (1.4) corresponds to the infinite N-ary tree with root y, specifically, the set of vertices is Y (y), and the set of directed edges 4 KAZUKI OKAMURA is given by H H (y), H H H (y) i ,...,i , i Σ , l 1 , il ◦···◦ i1 il+1 ◦ il ◦···◦ i1 | 1 l l+1 ∈ N ≥ where we let Hil Hi1 (y) := y if l = 0. (iii) In the setup◦···◦ above, we regard Y simply as a set, but we will often put a metric structure and a linear structure on Y . Features of our setup are that Y is only a set and irrelevant to K, Y contains countably many points, Gi and Hi, i ΣN , are not constant functions, and furthermore, µy is not an invariant measure∈ of a certain iterated function system equipped with probability weights. (iv) The assumption for (K, Si i) in Theorem 1.1 (ii) is the same as in assumption (2) in [Kig95, Corollary{ } 1.3]. It is known by Bandt and Graf [BG92] that if this holds then the open set condition for S holds. It { i}i is satisfied if (M, d) is the Euclid space and Si, i ΣN , are similitudes and satisfy the open set condition (See the proof of∈ [F14, Theorem 9.3]). In examples which are dealt with later, (1.6) implies that µy is singular with respect to a “canonical” probability measure on the attractor K whose Hausdorff dimension is the positive number s satisfying (1.3). In the case of homogeneous self-similar sets, it holds that s = log N/ log(1/r). As an outline of our proof, we follow [Ok14, Theorem 1.2]. However, our method is more transparent than [Ok14]. Specifically, we do not use four kinds of random subsets of natural numbers as in [Ok14, Lemma 3.3]. See Lemma 2.6 below for an alternative way.
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