arXiv:1703.03934v3 [math.CA] 6 Sep 2019 de n eflosaealwd E9] S9]ad[Ol94] and [St93] ( [EM92], allowed. Let are self-loops 1.1]. and Section edges sel [Ol94, directed graph Olsen the in and 2.2] fami Definition self-similar [St93, a Strichartz [EM92], Edgar-Mauldin in measures type ne h supinthat assumption the under asand maps esrbesae For space. measurable qainfrafml fpoaiiymeasures probability of family a for equation 2010 uaiypolm eRa’ ucinlequations. functional Rham’s de problem, gularity phrases. and words Key qain o rbblt esrssmlrt 11 appear (1.1) to similar measures probability for Equations Let .Oe problems References Open 4. Examples 1.1 Theorem of 3. Proof results Main and 2. Introduction 1. ASOF IESOSFRGRAPH-DIRECTED FOR DIMENSIONS HAUSDORFF ESRSDIE YIFNT OTDTREES ROOTED INFINITE BY DRIVEN MEASURES ahmtc ujc Classification. Subject Mathematics N esrso t n,maue endb ouin fd Rham’ de transformati of fractional solutions linear by by driven defined equations measures tional and, it, on K the measures Sierp´ınski non-const gasket, two-dimensional of the functional conta restrictions on and functions by class geometry defined our fractal measures However in specifically, appearing measures. measures Gibbs eral necessarily not dire finite are by driven measures self-similar graph-directed in o ls fgahdrce esrswe t underl its the when is measures graph-directed graph of class a for sions Abstract. ≥ F i .LtΣ Let 2. : Z → 1. egv pe n oe onsfrteHudr dimen- Hausdorff the for bounds lower and upper give We infinite Z nrdcinadMi results Main and Introduction asoffdmninfrmaue rp-ietdmeasures, graph-directed measure, for dimension Hausdorff N emaual as hnw osdrtefollowing the consider we Then maps. measurable be µ i := y ∈ N = ayte.Teemaue r ieetfrom different are measures These tree. -ary { AUIOKAMURA KAZUKI Σ i 0 ∈ X i N , Σ ∈ X 1 N let , Σ Contents N , . . . , N G G ,E V, 62,2A0 87,2A0 60G30. 28A80, 28A78, 26A30, 26A27, i ( i y G 1 ( ) y i µ eadrce rp hr multi- where graph directed a be ) 1 = ) − H : i Y ( 1 y } ) . Let . → ◦ { µ F y [0 i − } , 1 y ]and 1] , Y ∈ Y eastand set a be tdgah and graphs cted ons. on soaenergy usuoka n harmonic ant igdirected ying -iia measures f-similar Z ou ntecase the on focus H yo esrsin measures of ly equations, : i nteMarkov the in n sev- ins func- s : Y → Z Y (1.2) (1.1) ea be sin- be 30 29 11 5 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. We emphasize that not only Theorem 1.1 is applicable to the models described above, but also there is potential for applications to different models. Indeed, we deal with some special examples other than the models described above. 1.2. Comparison with related results. Our purpose is to know whether µy is absolutely continuous or singular with respect to a natural canonical probability measure on Z. In several examples, we deal with the case that Z = K = [0, 1]. If µy is singular, then, the distribution function of µy has derivative zero at almost every point. The singularity problem proposed by Kaimanovich [Kai03] is a little similar to our motivation. [Kai03] considers whether the harmonic measure on a boundary of Markov chain is singular with respect to a “canonical” measure on the boundary. It is also interesting to consider whether not only the harmonic measure but also other measures on the boundary defined in natural ways are singular or not with respect to the canonical measure. Indeed, in [Kai03, p.180], investigating dynamical properties for the Kusuoka energy measure is proposed, and recently [JOP17] considers them. The Kusuoka energy measure is not a Gibbs measure, and therefore, techniques of the thermodynamic formalism are not suitable to apply. See [JOP17, Section 3] for more details. If the Hausdorff dimension of µy is strictly smaller than the Hausdorff dimension of a canonical measure, then, µy is singular with respect to the canonical measure. So, it is desirable to know an exact value or a good upper bound of the Hausdorff dimension of µy. In general, for iterated func- tion systems with place-dependent probability weights, deriving a dimension formula for invariant measures is valuable. For a class of iterated function systems which are driven by non-linear weak contractions and equipped 5 with place-dependent probability weights, Jaroszewska-Rams [JR08] gave an upper bound for the Hausdorff dimension of an invariant measure in the form of the entropy divided by the Lyapunov exponent. For a large class of iterated function systems driven by similitudes with considerable over- laps, Hochman [Ho14] obtained the exact dimension formula. In order to give an upper bound for the Hausdorff dimension, we need to estimate the entropy and the Lyapunov exponent, both of which are the values of in- tegrands with respect to µy, which might be singular with respect to the canonical measure on K. However, intricate calculations are actually re- quired for estimating the Hausdorff dimensions of invariant measures of an iterated function system, in particular when the probability weights of the iterated function system are place-dependent or the iterated function system is driven by non-similitudes. See B´ar´any-Pollicott-Simon [BPS12, Sections 7 and 8] and B´ar´any [Ba15, Section 5] for example. In the case that Y consists of only one point and furthermore all F , i i ∈ ΣN , are similitudes on a complete metric space satisfying assumption (2) in [Kig95, Corollary 1.3], then, by [BG92], the open set condition holds for Fi i and µy is a self-similar measure where the corresponding probabilities {G } are not place-dependent. However, we mainly focus on the case that { i}i Y contains at least countably many points and the case that µy might not be a Gibbs measure or an invariant measure of an iterated function system. In the case that Y contains countably many points and has a linear struc- ture, the values of G (y) and H (y), i Σ , can be non-linear functions i i ∈ N on Y . So, even if we have a form of dimension formula for µy expressed by G , H , and µ , it is difficult for estimating dim µ from possible { i}i { i}i y H y dimension formulae. In this paper, we focus on the issue whether µy is sin- gular or not with respect to the canonical measure on K whose Hausdorff dimension is the positive number s satisfying (1.3) above, rather than pur- suing a form of dimension formula for µy. We give upper and lower bounds for the Hausdorff dimensions of µy without deriving any forms of dimension formulae.
The rest of the paper is organized as follows. Section 2 is devoted to the proof of Theorem 1.1. Section 3 is devoted to deal with various examples including the restriction of harmonic functions, the energy measures on the Sierp´ınski gasket and de Rham’s functional equations. In Section 4, we state open problems.
2. Proof of Theorem 1.1 Let N := 1, 2,... . Henceforth we put the product σ-algebra on a sym- { N } N bolic space (ΣN ) . Let π : (ΣN ) K be a surjective map such that → N π(ix)= S (π(x)) for every i Σ and x (Σ ) . (2.1) i ∈ N ∈ N π is uniquely determined and is called the natural projection. N For n N, let Xn(x) be the projection of x (ΣN ) to n-th coordinate. ∈ ∈ N Let n := σ(X1,...,Xn). This is a σ-algebra on (ΣN ) . For n 1, let a cylinderF set ≥ N I(i ,...,i ) := w (Σ ) X (w)= i , 1 k n , i ,...,i Σ . 1 n ∈ N k k ≤ ≤ 1 n ∈ N n o
6 KAZUKI OKAMURA
N Consider (1.1) in the case that Z = (ΣN ) and Fi(x)= ix. By (1.2) and the Kolmogorov extension theorem, there exists a unique N probability measure νy on (ΣN ) such that n ν (I(i ,...,i )) = G H H (y). y 1 n ik ◦ ik−1 ◦···◦ i1 kY=1 Then, νy y∈Y is a family of solutions of (1.1). Then, by (2.1), a family of { } −1 the push-forward measures νy π y∈Y is a family of solutions of (1.1) for the case that Z = K and F{= ◦S , i } Σ . i i ∈ N Since each Si is contractive, then, by following the proof of [F97, Theorem 2.8], we can show that a family of solutions of (1.1) for Z = K and Fi = Si is unique. It holds that for every y Y , ∈ ν π−1 = µ . (2.2) y ◦ y Recall the definition of the positive number s in (1.3). For every y Y , x (Σ )N and n N, let ∈ ∈ N ∈ ν (I(X (x),...,X (x))) R (x) := y 1 n . y,n (r r )s X1(x) · · · Xn(x) Then it follows from induction in n that Lemma 2.1. For every y Y , x (Σ )N and n N, it holds that ∈ ∈ N ∈ R (x)= R (x)G H H (y)(r )s. y,n+1 y,n Xn+1(x) ◦ Xn(x) ◦···◦ X1(x) Xn+1(x) Let
P := (p ,...,p ) [0, 1]N p = 1 . N 0 N−1 ∈ | i iX∈ΣN R s Define a relative entropy sN : PN with respect to (ri )i∈ΣN by → s ri sN (p0,...,pN−1) := pi log , pi iX∈ΣN where we put 0 log 0 = 0. We remark that sN (p0,...,pN−1) 0 and s s ≤ sN (p0,...,pN−1) = 0 if and only if (p0,...,pN−1)= r0, . . . , rN−1 . It can occur that R (x) = 0, however, it holds that y,n−1 N ν ( x (Σ ) R (x) = 0 ) = 0 y { ∈ N | y,n−1 } for every n 1. Hence if we say “ν -a.s.x”, then, we can assume that ≥ y Ry,n−1(x) > 0 for every n. Then, by Lemma 2.1, Lemma 2.2. For every y Y , it holds that ∈ νy Ry,n E log n−1 (x)= sN Gj HX −1(x) HX1(x)(y) , νy-a.s.x. − R F ◦ n ◦···◦ j∈ΣN y,n−1 Let My,0 = 0. For n 1, let ≥ R R M M := log y,n Eνy log y,n . y,n − y,n−1 − R − − R Fn−1 y,n−1 y,n−1
(If Ry,n−1(x) = 0, then, we let (My,n My,n−1)(x) := 0. but such x does − not affect integrations with respect to νy.) Then, M , is a martingale under ν and we have that { y,n Fn}n≥0 y 7
Lemma 2.3. For every y Y , it holds that ∈ My,n lim = 0, νy-a.s. n→∞ n Proof. This part will be shown in the same manner as [Ok14, Lemma 2.3 (2)] by using Jensen’s inequality and Doob’s submartingale inequality. Let 2 C := sup pi( log pi) < + . (pi)i∈PN − ∞ i∈XΣN Then, for every n 1, ≥ R 2 C Eνy log y,n+1 . R ≤ (min r )2s " y,n # i i By this and Jensen’s inequality,
νy 2 4C sup E (My,n+1 My,n) 2s . n≥0 − ≤ (mini ri) h i By Doob’s submartingale inequality, we have that for every ǫ > 0 and every n 1, ≥ νy 2 n E (My,2n ) νy max My,k ǫ4 1≤k≤2n | |≥ ≤ ǫ4n νy 2 n E (M M ) k≤2 y,k − y,k−1 C = n 2s n−1 . P h ǫ4 i ≤ (mini ri) ǫ4 Therefore we have My,n lim sup | | √ǫ, νy-a.s. n→∞ n ≤
For i 1, y Y and x (Σ )N, let ≥ ∈ ∈ N h (y; x) := H H (y), i Xi(x) ◦···◦ X1(x) and
pi(y; x) := (Gj hi(y,x)) = Gj HX (x) HX1(x)(y) . ◦ j∈ΣN ◦ i ◦···◦ j∈ΣN By Lemmas 2.2 and 2.3, we have that
Proposition 2.4. For every y Y , it holds that ∈ n log Ry,n(x) 1 lim sup − = limsup sN (pi−1(y; x)) , νy-a.s.x n→∞ n n→∞ n Xi=1 and, n log Ry,n(x) 1 lim inf − = lim inf sN (pi−1(y; x)) , νy-a.s.x. n→∞ n n→∞ n Xi=1 2.1. Proof of (i). Throughout this subsection, we assume the assumption of Theorem 1.1 (i). 8 KAZUKI OKAMURA
Lemma 2.5. Let y Y . If for some positive constant ǫ , ∈ 1 log Ry,n lim sup − ǫ1, νy-a.s., n→∞ n ≤− then, there exists a Borel subset B K such that µ (B ) = 1 and 0 ⊂ y 0 dimH (B0) < s.
Proof. For 0 µ π A = 1. y n,1/k k\≥1 m[≥1 n\≥m Now we will show that dim π A < s. (2.3) H n,1/k k\≥1 m[≥1 n\≥m For 0 π A π (I(i ,...,i )) , n,1/k ⊂ 1 n n\≥m (i1,...,in)[∈A(n,1/k) and, for everyu (0,s), ∈ diam (π (I(i ,...,i )))u cu (r r )u 1 n ≤ 1 i1 · · · in (i1,...,inX)∈A(n,1/k) (i1,...,inX)∈A(n,1/k) cu (r r )s(r r )u−s ≤ 1 i1 · · · in i1 · · · in (i1,...,inX)∈A(n,1/k) (s−u)n u 1 1 c1 max exp n ǫ1 , ≤ i r − − k i 9 where in the final inequality we have used (2.4) and the assumption that ri > 0. Hence, if k is sufficiently large and u is sufficiently close to but strictly smaller than s, π A = 0, Hu n,1/k n\≥m where we let be the s-dimensional Hausdorff measure on M. Hence, Hs dim π A u The following is different from a part of the proof of [Ok14, Theorem 1.2 (ii)], specifically, [Ok14, Lemma 3.3]. Lemma 2.6. Let y Y . Assume that (1.5) holds for (ik)1≤k≤l. Then, for ∈ N every i N and x (Σ ) satisfying that X (x)= i , 1 k l, ∈ ∈ N i+k k ≤ ≤ l s (p (y; x)) sup s ((p ) ) p rs >ǫ . N i+k ≤ N j j∈ΣN j − j 0 k=1 j X X s If there exist a constant ǫ0 (0, min ri ), and an integer l 1 such that ∈ i ≥ (1.5) holds for every i1,...,il ΣN , then this inequality holds without the constraint that X (x)= i , 1∈ k l. i+k k ≤ ≤ 1 Rl Proof. In this proof, denotes the ℓ -norm on . We recall that sN ((pj)j∈ΣN )= k·ks 0 if and only if pi = ri for every i. Case 1. Since p (y; x) rs, , rs >ǫ , i − 0 · · · N−1 0 it holds that s (p (y; x)) sup s ((p ) ) p rs >ǫ . N i ≤ N j j∈ΣN j − j 0 j∈ΣN X Therefore, l s (p (y; x)) < sup s ((p ) ) p rs >ǫ . N i+k N j j∈ΣN j − j 0 k=1 j∈ΣN X X Case 2. If p (y; x) rs, , rs ǫ , i − 0 · · · N−1 ≤ 0 and moreover Xi+k(x )= ik, 1 k l, then, by (1.5), ≤ ≤ p (y; x) rs, , rs G (h (y; x)) rs >ǫ , i+l − 0 · · · N−1 ≥ i+l i+l − il 0 and hence, s (p (y; x)) sup s ((p ) ) p rs >ǫ . N i+l ≤ N j j∈ΣN j − j 0 j∈ΣN X 10 KAZUKIOKAMURA Therefore, l s (p (y; x)) < sup s ((p ) ) p rs >ǫ . N i+k N j j∈ΣN j − j 0 k=1 j∈ΣN X X Thus we have the assertion. Lemma 2.7. Let y Y . Then, for every i ,...,i Σ , there exists a ∈ 1 l ∈ N non-random constant c1 > 0 such that for νy-a.s.x, there exists a random subset I(x) N such that ⊂ I(x) 1,...,n lim inf | ∩ { }| c1, n→∞ n ≥ and furthermore it holds that Xi+k(x) = ik for every i I(x) and every 1 k l. ∈ ≤ ≤ Proof. Fix i ,...,i Σ . Let 1 l ∈ N c = c(y) := inf Gi(z). (2.5) i∈ΣN ,z∈Y (y) By (2.6), we have that c(y) > 0. Let C(n) := X = i , 1 k l . e e (n−1)l+k k ≤ ≤ Let M := 0 and for n 1, M M := 1 cl. Then, M M 2, 0 ≥ n− n−1 C(n)− | n− n−1|≤ and, M , is a submartingale.e Then, by Azuma’s inequality [A67], { n Fln}n f n f f e f f ncl ncl nc2l fν 1 < = ν M < exp . y C(k) 2 y n − 2 ≤ − 32 k=1 ! X e e e Hence by the Borel-Cantelli lemma, f n 1 cl lim inf 1C(k) , νy-a.s. n→∞ n ≥ 2 k=1 X e Proof of Theorem 1.1 (i). Let y Y . By Lemmas 2.6 and 2.7, there exists ∈ ǫ1 > 0 such that n 1 N lim sup sN (pk(y; x)) ǫ1, νy-a.s.x (ΣN ) . n→∞ n ≤− ∈ Xk=1 By this and Proposition 2.4, log Ry,n lim sup − ǫ1, νy-a.s. n→∞ n ≤− By this and Lemma 2.5, dimH µy < s. Remark 2.8. The conclusion also holds if we replace our assumptions with the conditions that 0 < inf Gi(w) sup Gi(w) < 1, (2.6) i∈ΣN ,w∈Y (y) ≤ i∈ΣN ,w∈Y (y) and that (1.5) holds for some ǫ (0, 1/N), l 1 and i ,...,i Σ . 0 ∈ ≥ 1 l ∈ N 11 2.2. Proof of (ii). For y Y , let ∈ Ry,n(x) := νy (I(X1(x),...,Xn(x))) . (2.7) Lemma 2.9. Let y Y . Assume that e∈ log Ry,n lim inf − a, νy-a.s. n→∞ n ≥ Then, it holds that µy(K1) = 0 for everye Borel subset K1 of K such that a dim (K ) < . (2.8) H 1 log(1/r) Proof. Let n 1 and δ > 0. Assume that (2.8) holds. Then, we can take open sets U ≥ in K such that for every l 1, { n,l}l ≥ diam(U ) crn, n,l ≤ K U , (2.9) 1 ⊂ n,l l[≥1 and a/ log(1/r) diam(Un,l) < δ. (2.10) Xl≥1 Let k(n, l) be an integer such that crk(n,l) diam(U ) < crk(n,l)−1. ≤ n,l Then, by k(n, l) n, we have that ≥ ν I (x) exp( k(n, l)a) diam(U )a/ log(1/r) y k(n,l) ≤ − ≤ n,l for every x k≥n log Ry,k ka . By this and the assumption of (ii), we have that∈ {− ≥ } T e ν π−1(U ) log R ka D diam (U )a/ log(1/r) . y n,l ∩ {− y,k ≥ } ≤ n,l k\≥n By this and (2.9) and (2.10),e we have that ν π−1(K ) log R ka Dδ. y 1 ∩ {− y,k ≥ } ≤ k\≥n e By this, (2.2) and the assumption, it holds that µy(K1) = 0. By Lemma 2.9, Proposition 2.4, and the assumption of (ii), c log c (1 c) log(1 c) dim µ − − − − > 0, H y ≥ log(1/r) where c is the constant in (1.7). This completes the proof of Theorem 1.1 (ii). 3. Examples This section is devoted to state various examples. All examples considered d here satisfy that (M, d) is the Euclid space R and Si i are contractive d { } similitude on R satisfying the open set condition, and Z = K and Fi = 12 KAZUKIOKAMURA Si, i ΣN , where K is the attractor of Si i. We recall that dimH K = s for s∈satisfying that { } s Lip(Si) = 1. iX∈ΣN Unless otherwise stated, we deal with the homogeneous case, i.e. all Lip- schitz constants ri, i ΣN are equal to each other and furthermore Si i satisfies the open set∈ condition. In the final subsection, we deal with{ the} non-homogeneous case. As we can see in [Kig95, Corollary 1.3] and its subsequent remarks, The- orem 1.1 is applicable to the case that (K, Si i) defines a self-similar set satisfying the open set condition such as the Sierp´ınski{ } gasket, the Sierp´ınski carpet, the Koch curve. Theorem 1.1 (ii) is applicable to the case that it is difficult to check whether S satisfies the open set condition such as { i}i the L´evy curve. See [Kig95, Appendix] for more details. Let V0 be the set of fixed points of S , i Σ . Let V := S (V ). Let i ∈ N m ∪i1,...,im∈ΣN i1,...,im 0 r0 := max Lip(Si) (0, 1). i∈ΣN ∈ Lemma 3.1. Assume that there exists D > 1 such that for every large n, n sup Vn B(x, 2r0 diam(K)) D. (3.1) x∈Vn | ∩ |≤ Then, there exist two constants 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 | 0 ∩ i1···im 6 ∅}| ≤ n Proof. Assume diam(U) diam(K)r0 . Let x,y U. Then there exist ≤ ∈ n points xn,yn Vn such that max d(x,xn), d(y,yn) r0 diam(K). Then ∈n { } ≤ d(xn,yn) 2r0 diam(K). By the assumption, there are at most D sets of ≤ forms Si1,...,in (K) covering U. Example 3.2. We consider the Euclid metric. (i) If S (z) = (z + i)/N, z R, then, K = [0, 1], and, (3.1) holds for i ∈ r0 = 1/N. (ii) If N = 4, S (z) = (z + q )/2, z R2, where i i ∈ q = (x,y) Z2 0 x,y 1 , { i}i { ∈ | ≤ ≤ } then, K = [0, 1]2, and, (3.1) holds for r = 1/2. 2 (iii) If N = 3, Si(z) = (z + qi)/2, z R , where qi i forms an equilateral triangle, then, K is a 2-dimensional∈ Sierp´ınski gasket,{ } and, (3.1) holds for r = 1/2. (iv) If N = 8, S (z) = (z + q )/3, z R2, where i i ∈ q = (x,y) Z2 0 x,y 2 (1, 1) , { i}i { ∈ | ≤ ≤ } \ { } then, K is a 2-dimensional Sierp´ınski carpet, and, (3.1) holds for r = 1/3. Proof. Let L := a q : a Z . This is a discrete subset of [0, 1] or i i i ∈ ( i ) 2 X −n R . By the definition of Si, it holds that for every n, Vn N L. Hence, sup d(x,y) cN −n, and, (3.1) holds for some D. ⊂ x,y∈Vn, x6=y ≥ 13 If Y is a one-point set, then, (1.1) does not depend on y, so we drop the notation, and then, the solution µ of (1.1) is an invariant measure of the iterated function system (K, S ) with weights G . { i}i∈ΣN { i}i∈ΣN Hereafter, if (G0,...,GN−1) = (p0,...,pN−1), then, we call ν and µ the N (p0,...,pN−1)-Bernoulli measure on Z = (ΣN ) and the (p0,...,pN−1)-self- similar measure on Z = K, respectively. We denote them by ν(p0,...,pN−1) and µ(p0,...,pN−1), respectively. 3.1. Singularity with respect to self-similar measures. Proposition 3.3 (Singularity with respect to self-similar measures). Let p ,...,p be positive numbers satisfying that p = 1. Assume that 0 N−1 i∈ΣN i for some i1,...,il, P Y (y) G−1 ([p ǫ ,p + ǫ ]) ∩ i i − 0 i 0 i∈\ΣN (G H H )−1 ([p ǫ ,p + ǫ ]) = . (3.2) ∩ j ◦ il ◦···◦ i1 j − 0 j 0 ∅ j∈\ΣN Then, (i) νy is singular with respect to ν(p0,...,pN−1). (ii) If moreover π−1(π(A)) A is at most countable for every subset A of N \ (ΣN ) , µy is singular with respect to µ(p0,...,pN−1). Proof. By Azuma’s inequality, νp0,...,pN−1 -a.s.x, there are infinitely many i such that Xi+k(x)= ik for every 1 k l. By (3.2), νp0,...,pN−1 -a.s.x, there are infinitely many i such that ≤ ≤ p (y; x) (p ,...,p ) ǫ . k i − 0 N−1 k≥ 0 Now assertion (i) follows from this and [Hi04, Theorem 4.1]. Assertion (ii) follows from assertion (i), (2.2) and the assumption. 3.2. Energy measures on Sierpi´nski gaskets. [Ku89] shows that energy measures for canonical Dirichlet forms on Sierpi´nski gaskets are singular with respect to the Hausdorff measure on them. It is generalized by [BST99], [Hi04]. Recently, [JOP17] considers the Kusuoka measure from an ergodic theoretic viewpoint. Their framework covers a general class of measures that can be defined by products of matrices. Let V0 := q0,q1,q2 be the set of vertices of an equilateral triangle in R2. Let K be{ a 2-dimensional} Sierpi´nski gasket, that is, the attractor of K = S (K), where we let S (z) := (z + q )/2, z R2. V is the set of ∪i=0,1,2 i i i ∈ 0 fixed points of Si, i = 0, 1, 2. Let ai K, i = 0, 1, 2, be unique fixed points of Fi, i = 0, 1, 2, respectively. Let ∈ 3/5 0 3/10 √3/10 3/10 √3/10 A0 := , A1 := , A2 := − . 0 1/5 √3/10 1/2 √3/10 1/2 − They are regular matrices and define linear transformation of Y . Let be the Euclid norm on R2. Let k · k Y := S1 = x R2 : x = 1 . { ∈ k k } 14 KAZUKIOKAMURA We regard Y as a topological space with respect to the Euclid distance on Y R2. For y Y and i = 0, 1, 2, let ⊂ ∈ 5 A y G (y) := A y 2, and, H (y) := i . i 3k i k i A y k i k Lemma 3.4. (i) 3 A2 + A2 + A2 = I , 0 1 2 5 2 where I2 denotes the identity matrix. In particular, (1.2) holds. (ii) For every y, 0 < inf Gi(z) sup Gi(z) < 1. i∈ΣN ,z∈Y (y) ≤ i∈ΣN ,z∈Y (y) Proof. (i) is immediately seen. (ii) The set of eigenvalues of A0, A1, A2 are 1/5, 3/5 . Hence, for every i and y, { } 1 3 G (y) . 15 ≤ i ≤ 5 Lemma 3.5. (i) If G0(y) = G0 H0(y), then, y ( 1, 0), (0, 1) and furthermore ◦ ∈ { ± ± } 1 3 G (y)= G H (y) , . 0 0 ◦ 0 ∈ 15 5 In particular, if G0(y) = 1/3, then, 1 G H (y) = . 0 ◦ 0 6 3 (ii) For every y Y , there exist ǫ0 (0, 1/N), l 1 and i1,...,il ΣN such that (1.5) holds.∈ ∈ ≥ ∈ Proof. Assertion (i) is easy to see. By using assertion (i) and that fact that Gi and Hi are continuous on Y and Y is compact, assertion (ii) follows. Lemma 3.6. Assume Gi(y)= pi (0, 1), i = 0, 1, 2. Then, (i) If y / ( 1, 0), (0, 1) , then, ∈ ∈ { ± ± } G (H (y)) = p . 0 0 6 0 (ii) If y ( 1, 0), (0, 1) , then, ∈ { ± ± } G (H (y)) = p . 0 1 6 0 Now we can apply Theorem 1.1 and Proposition 3.3 to this case, (N = 3, l = 2) Proposition 3.7. It holds that log 3 0 < dim µ < . (3.3) H y log 2 Furthermore, µy is singular with respect to every (p0,p1,p2)-self-similar mea- sure on K. Remark 3.8. This result could be deduced from the dimension theory of dynamical systems. By [Ku89, Remark 1.6, Corollary 2.12, and Example 1], 15 N we have that on the symbolic space (ΣN ) the Kusuoka measure is invari- ant and ergodic with respect to the canonical shift. See [Kaj12, Theorem 6.8] for a simple proof. By the uniqueness of measures of maximal dimen- sion stated in [MU03, Theorem 4.4.7] and mutual singularity of two distinct ergodic measures (This is an easy consequence of the Birkhoff ergodic theo- rem. See [PU10, Theorem 2.2.6] for example), we have the upper bound of (3.3). [JOP17] investigates the Kusuoka measure from an ergodic theoretic viewpoint, which could be seen as a generalization of Bernoulli measures. Let f be a harmonic function on K. Let h1 and h2 be the harmonic functions on K such that (h1(q0), h1(q1), h1(q2)) = (0, √2, √2) and (h (q ), h (q ), h (q )) = (0, 2/3, 2/3). 2 0 2 1 2 2 − Let v be the components of f in (h , h ). Let y = v/ v . Then, the energy 1 2 p pk k measure associated with f is µy. (Cf. [BST99].) Remark 3.9. In a formal level, the framework adopted in [Hi04] is inter- preted as follows. See [Hi04, Section 2] for details of Dirichlet forms. Let (K, ΣN , ψi i∈ΣN ) be a self-similar structure and µ be the invariant mea- sure. Let{ ( }, ) be a regular Dirichlet form on L2(K,µ). Assume (A1)-(A6) in [Hi04, SectionE F 2]. Y := , F (f ψ ,f ψ ) G (f) := s E ◦ i ◦ i , if (f,f) > 0. i i (f,f) E E 1 G (f) := , if (f,f) = 0. i N E H (f) := f ψ . i ◦ i Then, for f Y , µ is the normalized energy measure. ∈ f 3.3. Restriction of harmonic function on Sierpi´nski gasket. [ADF14] considers the restriction on [0, 1] of harmonic functions on the Sierpi´nski gas- ket, and shows that they are singular functions1 whenever they are mono- tone. The restrictions are among a wider class of functions containing several functions such as Lebesgue singular functions. [ADF14, Notation 6] is interpreted as follows in our framework: Let Y = [0, 1], (3.4) 2+ y 3 y G (y)= , G (y) = 1 G (y)= − , (3.5) 0 5 1 − 0 5 and 1 + 2y y H (y)= ,H (y)= . (3.6) 0 2+ y 1 3 y − Then, Sy in [ADF14, Subsection 2.1] is the distribution function of µy. [ADF14, Theorem 3] is their main theorem, and, [ADF14, Lemma 24 and Theorem 25] restates it in a more general framework. 1A singular function is a continuous, increasing function on [0, 1] whose derivative is zero almost surely with respect to the Lebesgue measure. 16 KAZUKIOKAMURA This is not of de Rham type in the subsection below, even if we exchange H0 with H1. Theorem 1.1 (i) gives the following improvements for [ADF14, Theorem 25]. (2.6) in Remark 2.8 corresponds to [ADF14, assumption (a) s in Lemma 24]. The assumption that (1.5) holds for ǫ0 (0, (mini ri) ), l 1 in Theorem 1.1 (i) corresponds to [ADF14, assumption∈ (b) in Lemma 24].≥ We loosen the assumptions in two ways and simultaneously obtain a stronger conclusion. The first way for weakening the assumption is adopt- ing the assumption that (1.5) holds for some ǫ (0, (min r )s), l 1 and 0 ∈ i i ≥ i1,...,il ΣN , which is strictly weaker than [ADF14, assumption (b) in Lemma 24].∈ The second way is the case that (2.6) fails but (1.5) holds for ǫ (0, (min r )s), l 1. 0 ∈ i i ≥ Theorem 3.10 (Application to singularity for real functions). Let Y = Z = [0, 1] and S (z) = (z + i)/N, z [0, 1], i Σ . Let y [0, 1]. Assume µ has i ∈ ∈ N ∈ y no atoms. Let ϕy be the distribution function of µy. Then, (i) If (2.6) holds and (1.5) holds for some ǫ (0, 1/N), l 1 and i ,...,i 0 ∈ ≥ 1 l ∈ ΣN , then, dimH µy < 1. (ii) If (1.5) holds for some ǫ (0, 1/N), l 1 and i ,...,i Σ , then, 0 ∈ ≥ 1 l ∈ N ϕy does not have non-zero derivative at almost every point with respect to every (p0,...,pN−1)-self-similar measure. (iii) If (1.5) holds for ǫ (0, 1/N), l 1, then, ϕ does not have non-zero 0 ∈ ≥ y derivative at every point, and, dimH µy < 1. Proof. Assume that ϕ has non-zero derivative at π(x) (0, 1). Then, y ∈ 1 lim GX (x) HX −1(x) HX1(x)(y)= . (3.7) n→∞ n ◦ n ◦···◦ 2 Assume that (1.5) holds for some ǫ0 (0, 1/N), l 1 and i1,...,il ∈ N≥ ∈ ΣN . Let µp0,...,pN−1 be a Bernoulli measure on 0, 1 . Then by Azuma’s inequality, { } l µ X = i = 1. p0,p1 { m+k k} n\≥1 m[≥n k\=1 N By this, (3.7) fails for µp0,...,pN−1 -a.e.x 0, 1 . Thus (ii) follows. The ∈ { } N assumption in assertion (iii) implies that (3.7) fails for every x 0, 1 . ∈ { } Remark 3.11. If we see the proof, assertion (ii) above holds even if we re- place an arbitrarily Bernoulli measure with an arbitrarily measure satisfying that there exists c (0, 1) such that for every m, k 1, ∈ ≥ k 1 k k 1 2k 1 k 1 k cµ − , µ − , − (1 c)µ − , . 2m 2m ≤ 2m 2m+1 ≤ − 2m 2m In the same manner as in the proof of assertion (ii) of Proposition 3.22 below, Proposition 3.12 (Singularity with respect to self-similar measures). If (3.4), (3.5) and (3.6) hold, then, µy is singular with respect to every (p0,p1)- self-similar measure. Remark 3.13. Our main results might be applicable to different models of Sierp´ınski gaskets such as level 3 Sierp´ınski gaskets. [ADF14] deals with the two-dimensional standard Sierp´ınski gaskets. 17 Remark 3.14. The restrictions of harmonic functions on the attractors of different iterated function systems have also been considered. [ES16, Theorem 3.2] states that the restrictions of harmonic functions on the Hata tree is singular. Assume the framework of [ES16, Theorem 3.2]. Denote a restriction by ϕ, and the measure whose distribution function is ϕ by µϕ. Then, 1 z ϕ 0 z α 2, h 2 α 2 ≤ ≤ | | ϕ(z)= (3.8) | | 1| | z α 2 1 1 ϕ − | | + α 2 z 1, − h 2 1 α 2 h 2 | | ≤ ≤ | | − | | | | 2 This appears in the proof of [ES16, Theorem 3.2] . See also Section 4 for functional equations of this kind. Then, µ satisfies (1.1) for Y = one-point set , G = h −2, Z = [0, 1], ϕ { } 0 | | F (z)= α 2z, and F (z) = (1 α 2)z + α 2. 0 | | 1 − | | | | (The notation F1 here is different from [ES16].) It is known that −2 −2 s2 h , 1 h + log 2 dim µ = | | − | | . H ϕ h −2 log α 2 (1 h −2) log(1 α 2) −| | | | − − | | − | | Hence, if a = b, then, dim µ < 1 and hence ϕ is a singular function. 6 H ϕ Further, multifractal analysis for µϕ is investigated. See [F97] for details. 3.4. De Rham’s functional equations. Before we apply our results to a class of de Rham’s functional equations, we give a short review. De Rham [dR56, dR57]3 considered a certain class of functional equations. He con- sidered the solution ϕ of the following functional equation which takes its values in a certain metric space: g (ϕ(Nx)) 0 x 1/N, 0 ≤ ≤ g (ϕ(Nx 1)) 1/N x 2/N, ϕ(x)= 1 − ≤ ≤ (3.9) · · · g (ϕ(Nx (N 1))) (N 1)/N x 1, N−1 − − − ≤ ≤ where each gi is a weak contraction (its definition is given below), g0(0) = 0, gN−1(1) = 1 and gi+1(0) = gi(1) for each i. Solutions of de Rham’s functional equations give parameterizations of several self-similar sets such as the Kˆoch curve and the P´olya curve, etc. Some singular functions such as the Cantor, Lebesgue, etc. functions are solutions of such functional equations. We give a short review of some known results. [BK00] considers self- similarity, inversion and composition of de Rham’s functions, and points out a connection with Collatz’s problem. [Kr09] shows connections between sums related to the binary sum-of-digits function and the Lebesgue’s singular function, and its partial derivatives with respect to the parameter. [Kaw11] investigates the set of points where Lebesgue’s singular function has the derivative zero. [P04] regards a de Rham curve as the limit of a polygonal arc by repeatedly cutting off the corners, obtain a formula for the local 2There is a typo in it, and it will be fixed in (3.8). 3An English translation of [dR57] is included in Edgar [E93]. 18 KAZUKIOKAMURA H¨older exponent of a de Rham curve at each point, and describe the sets of points with given local regularity. [Be08, Be12] consider multifractal and thermodynamic formalisms for the de Rham function. Recently, [BKK18] performs the multifractal analysis for the pointwise H¨older exponents of zipper fractal curves on Rd generated by affine mappings on Rd, d 2. [BKK18] and [N04] consider the Hausdorff dimension of the image measure≥ of the Lebesgue measure on an interval by the de Rham function. [PV17, Section 9.3] also consider de Rham curves in terms of matrix products. [DL91, DL92-1, DL92-2, P06] are related to wavelet theory. The length of de Rham curve is investigated in [Me98] and [DMS98]. [TGD98] uses de Rham’s functional equations to construct the Sinai-Ruelle-Bowen measures for several Baker-type maps. de Rham’s functional equations also appear in [DS76], which is concerned with gambling strategy. [G93, Z01, SB17, Ok19+] consider generalizations of de Rham’s functions. The case that some of gi are not affine is considered in [Ha85, SLK04, Ok14, Ok16, Ok19+, SB17], [SLK04] considers it from a view point of ran- dom walk. [Ha85] and [SLK04] use the technique of [La73]. [SB17] gives conditions for existence, uniqueness and continuity, and furthermore, pro- vides a general explicit formula for contractive systems. However, it seems that real-analytic properties for the solutions are not fully investigated. Let (M, d) be a metric space. Following [Ha85, Definition 2.1], we say that a function f : M M is a weak contraction if for every t> 0, → lim sup d(f(x),f(y)) < t. s→t,s>t d(x,y)≤δ By [Ha85, Corollary 6.6], if each gi is a weak contraction, g0(0) = 0, gN−1(1) = 1 and gi+1(0) = gi(1) for each i, then, (3.9) has a unique con- tinuous solution ϕ and we let µ = µϕ be the probability measure such that the solution ϕ of (3.9) is the distribution function of µϕ. If dimH µϕ < 1, then, µϕ is singular with respect to the Lebesgue measure, and hence ϕ is a singular function on [0, 1]. Therefore, it is valuable to know whether dimH µϕ < 1 or not. 3.4.1. Dimension formula. Contrary to the examples in the above subsec- tions, in this framework, µy is the invariant measure of a certain iterated function system with place-dependent probabilities, and furthermore, we have a form of dimension formula for µϕ for a large class of (gi)i, thanks to Fan-Lau [FL99]. By the Lebesgue-Stieltjes integral, for every bounded Borel measurable function F : [0, 1] R, → ′ x + i F (x)dµϕ(x)= gi(ϕ(x))F dµϕ(x). [0,1] [0,1] N Z Xi Z 2 ′ We assume that gi C ([0, 1]) and 0 < gi(z) < 1 hold for each i ΣN and every z [0, 1]. Then,∈ ϕ is H¨older continuous and ∈ ∈ 1 sup log g′(ϕ(s )) log g′(ϕ(s )) : s s t g′′(s) 1 {| i 1 − i 2 | | 1 − 2|≤ }dt sup | i | tc−1dt < + , t ≤ g′(s) ∞ Z0 s∈[0,1] i Z0 where we let c = cϕ be a positive number. 19 Let h be a positive function on [0, 1] such that x + i h(x)= g′ (g (ϕ(x))) h . i i N i∈XΣN h ϕ−1(x)= g′(g (x))h ϕ−1(g (x)). ◦ i i ◦ i Xi This is unique under the constraint that h(x)µϕ(dx) = 1. Z[0,1] See [FL99, Theorem 1.1]. Then, by [FL99, Corollary 3.5], h(x) log (1/g′(ϕ(x))) µ (dx) i∈ΣN [i/N,(i+1)/N] i ϕ dimH µϕ = . P R log N We have that H(g (y))g′(y) log (1/g′(g (y))) ℓ(dy) i∈ΣN [0,1] i i i i dimH µϕ = , P R log N where ℓ is the Lebesgue measure on [0, 1] and H is a function on [0, 1] satisfying the following conditions: ′ H(y)= gi(gi(y))H(gi(y)), and H(y)ℓ(dy) = 1. (3.10) [0,1] iX∈ΣN Z It is interesting to investigate properties for H. If each gi is linear, in other words, g′ is a constant function, then, g′ = 1, and hence, i i∈ΣN i H 1 satisfies (3.10). ≡ P We focus on the case that gi is not affine. In that case, it is difficult for knowing whether ′ ′ H(gi(y))gi(y) log 1/gi(gi(y)) ℓ(dy) < log N. [0,1] i∈ΣN Z X In the following subsection, we give a necessary and sufficient condition for dimH µϕ < 1 for a specific choice for (gi)i by using Theorem 1.1. It is interesting to investigate properties for H, but in this paper we do not analyze H directly. Furthermore, by [FL99, Theorem 1.6], log µϕ(In(x)) ′ lim − = h(z) log 1/gi(ϕ(z)) dµϕ(z), µy-a.s.x [0, 1], n→∞ n [0,1] ∈ iX∈ΣN Z (3.11) n n n n where we let In(x) := [i/2 , (i + 1)/2 ) such that x [i/2 , (i + 1)/2 ). However, we do not see how to estimate the integrand in∈ the right hand side of (3.11). Arguments in [FL99] depend on the fact µy is an invariant measure of an iterated function system, and we are not sure whether a convergence corresponding to (3.11) holds for the examples in the above subsections. 3.4.2. De Rham’s functional equations driven by N linear fractional trans- formations. [Ok14] considers de Rham’s functional equations driven by two linear fractional transformations, here we consider not only the case that 20 KAZUKIOKAMURA N = 2 and but also the case that N 3. Our outline is similar to the one in [Ok14], however several additional≥ considerations are needed. In the following, we consider the equation (3.9) for the case that all gi az + b are linear fractional transformations. Let Φ(A; z) := for a 2 2 real cz + d × a b matrix A = and z R. Let c d ∈ g (x) := Φ (A ; x) , x [0, 1], i = 0, 1,...,N 1, i i ∈ − ai bi such that 2 2 real matrices Ai = , i = 0, 1, satisfy the following × ci di conditions (A1) - (A3). (A1) Φ(A0; 0) = 0, Φ(AN−1; 1) = 1, and Φ(A ; 1) = Φ(A ; 0), i = 1,...,N 1. i−1 i − (A2) det A = a d b c > 0, i = 0, 1,...,N 1. i i i − i i − (A3) (a d b c )1/2 min d , c + d , i = 0, 1,...,N 1. i i − i i ≤ { i i i} − In several cases, we will replace (A3) with a stronger condition (sA3): (a d b c )1/2 < min d , c + d , i = 0, 1,...,N 1. i i − i i { i i i} − By (A2) and (A3), di > 0, and henceforth, we can assume di = 1 for each i, without loss of generality. By (A1), b0 = 0, and, 0 < bi < 1, 1 i N 1. By (A3), a 1. ≤ ≤ − 0 ≤ Lemma 3.15. For each i, Φ (Ai; x) is a weak contraction on [0, 1]. Proof. If ci = 0, then, Φ (Ai; x) = aix + bi. By (A2) a1 > 0. By (A1), a + b = b < 1 and b 0 for i N 2, and, a + b 1 and b > 0 for i i i+1 i ≥ ≤ − i i ≤ i i 1. Hence ai < 1. Thus Φ (Ai; x) is a contraction on [0, 1]. ≥If c = 0, then, by (A3), for every t> 0, i 6 a b c min i − i i < 1. t<|x−y| (cix + 1)(ciy + 1) Hence, Φ (Ai; x) is a weak contraction. Therefore, (A1) - (A3) guarantee that (3.9) has a unique continuous so- lution ϕ. The above (sA3) is identical with (A3) in [Ok14]. We remark that our framework contains the cases that the technique of [La73] appearing in [Ha85, Theorem 7.3] and [SLK04, Proposition 3.1] is not applicable. By (A1) - (A3), b c 1 a 1 b c . i − i ≤ − i ≤ − i i Hence, (1 a )2 + 4b c min (b + c )2, (1 + b c )2 0. − i i i ≥ { i i i i }≥ Let c a 1+ (1 a )2 + 4b c α := min 0, 0 , i − − i i i 1 i N 1 . ( 1 a0 2bi ≤ ≤ − ) − p 21 If a0 = 1, then, we replace c0/(1 a0) in the right hand side of the above definition with 1. − Let − c a 1+ (1 a )2 + 4b c β := max 0, 0 , i − − i i i 1 i N 1 . ( 1 a0 2bi ≤ ≤ − ) − p If a0 = 1, then, we replace c0/(1 a0) in the right hand side of the above definition with + . − Let Y := [α, β∞]. We consider the topology of Y defined by the Euclid metric. For k Σ and y Y , let ∈ N ∈ (ak bkck)(y + 1) aky + ck Gk(y) := − , and Hk(y) := . (bky + 1) ((ak + bk)y + ck + 1) bky + 1 If a0 = 1, then, ak G0(+ ) := 1, Gk(+ ) := 0,H0(+ ):=+ ,Hk(+ ) := , 1 k N 1. ∞ ∞ ∞ ∞ ∞ bk ≤ ≤ − Lemma 3.16. (i-1) If a = 1, then, α = 1 and β =+ . 0 − ∞ (i-2) If a0 < 1 and bN−1 + cN−1 = 0, then, 1= α β < + . (i-3) If a < 1 and b + c > 0, then, −1 < α ≤ β < +∞. 0 N−1 N−1 − ≤ ∞ (ii) Hi(y) Y for i ΣN ,y Y . (iii) (1.2) ∈holds. ∈ ∈ Proof. (i) By (A1), it holds that if 1 i N 2, then, ≤ ≤ − a 1 (1 a )2 + 4b c a 1+ (1 a )2 + 4b c i − − − i i i < 1 < i − − i i i . (3.12) p 2bi − p 2bi By (A2) and (A3), bN−1 + cN−1 0. By this and (A1), it holds that if i = N 1, then, ≥ − a 1 (1 a )2 + 4b c N−1 − − − N−1 N−1 N−1 = 1, (3.13) p 2bN−1 − and, c a 1+ (1 a )2 + 4b c 1 N−1 = N−1 − − N−1 N−1 N−1 . (3.14) − ≤ bN−1 p 2bN−1 Hence, if a0 = 1, then, α = 1. If a0 < 1, then, by (A1), a0 < c0 + 1. Hence, if a < 1 and b + c − > 0, then, α> 1. 0 N−1 N−1 − (ii) Let i = 0. First we remark that H0(z) = a0z + c0. Assume a0 < 1. Then, by α c /(1 a ) β, we see that α H (α) H (β) β. If ≤ 0 − 0 ≤ ≤ 0 ≤ 0 ≤ a0 = 1, then, β =+ . It is easy to see that α H0(α). Let 1 i N 1.∞ Then, H (z) z if and only≤ if ≤ ≤ − i ≥ a 1 (1 a )2 + 4b c a 1+ (1 a )2 + 4b c i − − − i i i z i − − i i i . (3.15) p 2bi ≤ ≤ p 2bi Hence, Hi(β) β. By (A2), for≤ every k, H (z) is monotone increasing on z 1. By this, k ≥− (3.12), (3.13), (3.14), and (3.15), Hi(α) α. (iii) It is easy to see by calculations that≥ for every i 0, ≥ (y + 1)(bi+1 bi) Gi(y)= − . (bi+1y + 1)(biy + 1) 22 KAZUKIOKAMURA The assertion follows from this and (A1). Lemma 3.17. If (sA3) holds, then, (2.6) holds for y = 0. Proof. By (sA3), 1 > a0 and bN−1 + cN−1 > 0. By this and Lemma 3.16 (i), we have that 1 < α β < + . Therefore, − ≤ ∞ 0 < G (α) G (β) < 1. 0 ≤ 0 Let i 1. Since α> 1 and 0 < b < 1, ≥ − i inf Gi(y) > 0. y≥α By the definition of G , we can show that if y 1, then, G (y) < 1. Hence, i ≥− i by continuity of Gi, sup Gi(y) < 1. y∈[α,β] Hereafter we denote the set of fixed points of a map f by Fix(f). Remark 3.18. (i) In the above proof, we have used bN−1 + cN−1 > 0. However, if i < N 1, then, b + c > 0 may fail. − i i (ii) If i 1, Gi may not be increasing. (iii) ≥ Fix(H ) Fix(H ) = 1. i ≤ | 0 | i∈ΣN \ (iv) If a0 < 1, then, Fix( H0)= c0/ (1 a0) . If a0 = 1, then, by extending the domain of H , Fix( H )= +{ . − } 0 0 { ∞} Lemma 3.19. For the above definitions of Y , Gi i and Hi i, assume that ν satisfies (1.1) for Z = (Σ )N and F ({x)=} ix. Then,{ } { y}y∈Y N i X (x) π(x)= i , N i Xi≥1 and, µ = ν π−1. ϕ 0 ◦ By this lemma, we identify ((Σ )N, F ) with ([0, 1], S ) where we let N { i}i { i}i Si(z) := (z + i)/N. ; Proof. Let pn(x) qn(x) := AX1(x) AXn(x), n 1. rn(x) sn(x) · · · ≥ We have that k k Xj(x) Xj(x) 1 µϕ (π(Ik(x))) = µϕ j , j + k " 2 2 2 !! Xi=1 Xi=1 =Φ A A ; 1 Φ A A ; 0 X1(x) · · · Xk(x) − X1(x) · · · Xk(x) p (x)s (x) q (x)r (x) = k k − k k . sk(x)(rk(x)+ sk(x)) 23 Therefore by computation, (Recall (2.7) for the definition of R0,n+1(x).) µ (π(I (x))) R (x) ν (I (x)) ϕ n+1 = 0,n+1 = 0 n+1 . e µϕ (π(In(x))) R0,n(x) ν0(In(x)) e Hence µϕ (π(In(x))) = ν0(In(x)). Since ϕ is continuous, we have that e lim µϕ (π(In(x))) = 0, n→∞ and hence, ν has no atoms. Since π−1(π(I (x))) I (x) is at most countable, 0 n \ n ν π−1(π(I (x))) I (x) = 0. 0 n \ n N Since π(In(x)) n 1,x (ΣN ) generates the Borel σ-algebra on [0, 1], { | ≥ ∈ } ν π−1 = µ . 0 ◦ ϕ The following is the main result of this subsection. Theorem 3.20 (Upper bound for Hausdorff dimension). (i) If either 1 G−1 = (3.16) i N 6 ∅ i∈\ΣN or 1 G−1 Fix(H ) (3.17) i N ⊂ i i∈\ΣN i∈\ΣN fails, then, dimH µϕ < 1. (ii) If (3.16) and (3.17) hold, then, the distribution function of µϕ is given by (N 1)x ϕ(x)= − , (3.18) N 1 c N(x 1) − − 0 − In other words, if the solution ϕ of (3.9) is not of the form of (3.18), then, dimH µϕ < 1, and hence, ϕ is a singular function. This is a special case of Theorem 3.32 below. For proof, see the proof of Theorem 3.32. Remark 3.21 (Lower bound for Hausdorff dimension). Assume a0 < 1 and bN−1 + cN−1 > 0. Let c be the constant in (2.5). Then, by Proposition 2.4 and Lemma 2.9, inf s e((p ) ) (p ) P , c p 1 c, j dim µ 1+ { N j j∈ΣN | j j∈ΣN ∈ N ≤ j ≤ − ∀ } > 0. H ϕ ≥ log N Proposition 3.22 (Regularity and singularity withe respect to self-similare measures). Let p (0, 1), 0 i N 1. Let e := c /(1 a ) and assume i ∈ ≤ ≤ − 0 0 − 0 a0 < 1. Then, (i) If there exist p (0, 1), 0 i N 1 such that i ∈ ≤ ≤ − i pi + e0 j=0 pj ai = . (3.19) i−1 1 Ppj e0 + 1 − j=0 P i−1 j=0 pj bi = . (3.20) i−1 1 P pj e0 + 1 − j=0 P 24 KAZUKIOKAMURA e 1 i p e + 1 p 0 − j=0 j 0 − i ci = . (3.21) 1 P i−1 p e + 1 − j=0 j 0 hold for every i, then, µϕ is absolutely P continuous with (p0,...,pN−1)-self- similar measure µ(p0,...,pN−1). (ii) If the assumptions of (i) fails, then, µϕ is singular with respect to every (p0,...,pN−1)-self-similar measure. We do not know about an explicit expression for the Radon-Nikodym dµ derivative ϕ . dµ(p0,...,pN−1) Proof. (i) We first remark that in this case, 0 < a0 = p0 < 1 and β < + . By computation, ∞ ′ Hi(e0)= e0, and Hi(e0)= pi, ′ where Hi denotes the derivative of Hi. Hence, each Hi is contractive on a neighborhood U of e0. Since H0 is M contractive on [α, β], there exists M such that it holds that H0 (x) U for every x [α, β]. ∈ By Azuma’s∈ inequality, M ν X = 0 = 1. 0 { i+j } i[≥1 j\=1 Hence, lim HX (x) HX1(x)(0) = e0, ν0-a.s.x. n→∞ n ◦···◦ and this convergence is exponentially fast4. Hence, 1 p G H H (0) < + , ν -a.s.x. − i i Xn(x) ◦···◦ X1(x) ∞ 0 n≥1 i∈Σ X XN q By this and [Sh80, Theorem VII.6.4], ν0 is absolutely continuous with re- spect to ν(p0,...,pN−1). Hence, µϕ is absolutely continuous with respect to µ(p0,...,pN−1). (ii) Let µ(p0,...,pN−1) be (p0,...,pN−1)-self-similar measure. By the defi- nition of π, we have that π−1(π(A)) A is at most countable for every A. \ −1 First we consider the case that p0 = a0. Then, Fix(H0) = e0 = G0 (p0). Therefore, if G (y) = p , then, G (H6 (y)) = p . Thus (3.2) holds6 for l = 1 0 0 0 0 6 0 and i1 = 0. Assume that p = a . Then, either (a) G (e ) = p , i Σ , or (b) 0 0 i 0 i ∈ N Hi(e0) = e0, i ΣN , fails, because if both (a) and (b) hold, then, (3.19), (3.20) and (3.21)∈ follow. Assume that (a) fails. Then, by using that p0 = a0 and G0 is strictly −1 increasing, iGi (pi) = . Since each Gi is continuous and Y is compact, (3.2) holds for∩ every l and∅ i ,...,i . Assume that (b) fails. Then, H (e ) = 1 l i 0 6 e0 for some i, and (3.2) holds for l = 1 and i1 = i. 4the speed of convergence depends on the choice of x. 25 Example 3.23 (Linear case). If all ci are zero, that is, all gi are affine maps, then, α = β = 0, and hence Y = 0 and G (0) = a . Then, µ is { } i i ϕ (a0, . . . , aN−1)-self-similar measure. We have that s ((a , . . . , a )) dim µ =1+ N 0 N−1 . H ϕ log N Example 3.24. Let N = 2. Consider (3.9) for 1 0 0 1 A = , A = , 0 1 1 1 1 2 − then, g0(x) = x/(x + 1) and g1(x) = 1/(2 x). Then, (A1)-(A3) holds, and hence, a unique solution ϕ of (3.9) exists.− ϕ is the inverse function of Minkowski’s question-mark function [Mi1904]. But (sA3) fails. Then, Y = [ 1, + ], − ∞ x + 1 1 G (x)= ,H (x)= x + 1,H (x)= , 0 x + 2 0 1 −x + 2 and, µ = µ . In this case, (2.6) may fail, but (1.5) holds for every i , i ϕ 0 1 2 ∈ 0, 1 . By Lemma 2.6, there is a constant ǫ1 > 0 such that for every i N and{ }every x 0, 1 N, ∈ ∈ { } s (p (y; x)) + s (p (y; x)) < ǫ . (3.22) 2 i 2 i+1 − 1 Hence, by Theorem 1.1 (i), we have that dimH µϕ < 1. In this case, for y = 0, (1.7) fails. We are not sure whether dimH µϕ > 0. By Proposition 3.3, µϕ is singular with respect to every (p0,p1)-self-similar measure. (3.22) is stable with respect to small perturbations for g0 and g1. See [Ok16, Example 2.8 (ii)] for a way of small perturbations for g0 and g1. It is known that µϕ is regular ([MT19+]). Remark 3.25. If µ is the measure on [0, 1] such that its distribution func- tion is Minkowski’s question-mark function, then, by [Kin60] it is known that e log 2 1 dimH µ = > . (3.23) 2 [0,1] log(1 + x)µ(dx) 2 Remark 3.26. e R (i) In the case that each gi ise a linear fractional transfor- mation, we can take Y as a subset of the set of real numbers. However, if some gi are not linear fractional transformations, then, we may not be able to take Y as a subset of R. Therefore, our approach may not work well, at least in a direct manner. One difficulty is that a set of functions can be very large, informally speaking. (ii) The approaches in [Ha85, SLK04, Ok16] are different from the one used here. As an outline level, they are somewhat similar to each other. 3.5. Other homogeneous examples. Example 3.27. We give an example that (1.5) fails for every i1,...,il, l = 1, 2 both, but (1.5) holds for every i1,...,il, l = 3. Let N = 2. Let Y = R. Let 1 1 7 G (x) := 1 + x + 1 + x 1 . 0 6 {x<0,x>1} 6 {0≤x≤1/2} 6 − {1/2≤x≤1} 26 KAZUKIOKAMURA Let 5 3y H (y)= H (y)= − . 0 1 6 Then, 1 1 2 1 G = G H = G = . 0 3 0 0 3 0 3 2 1 1 2 G H2 = G = . 0 0 3 0 2 3 2 1 7 7 G H2 = G H = G = . 0 0 3 0 0 2 0 12 12 Example 3.28. Let N = 2 and Y = Z = [0, 1]. Let 1 3 G (x)= H (x)= 1 + x1 + 1 . 0 0 4 {x<1/4} {1/4≤x≤3/4} 4 {x>3/4} Let H (x) = 1 G (x). Then, 1 − 0 1 1 1 1 1 G = G = H = H = , 0 2 1 2 0 2 1 2 2 and for every y = 1/2, dimH µy < 1. Therefore, if we did not introduce Y (y) in the condition6 that (1.5) holds for some ǫ (0, 1/N), l 1 and 0 ∈ ≥ i1,...,il ΣN , and furthermore simply assumed that the intersection of −∈1 1 1 −1 1 1 G ǫ0, + ǫ0 and (Gj Hi Hi1 ) ǫ0, + ǫ0 i∈ΣN i N − N j∈ΣN ◦ l ◦···◦ N − N is empty, then, the converse of Theorem 1.1 (i) would not hold. T T Example 3.29. Fix p (0, 1). Let N =2 and ∈ Y = min p, 1 p 2, min p, 1 p 2 − { − } { − } and Z = [0, 1]. Let y G (y)= p + y , and H (y)= H (y)= | | . 0 | | 0 1 y + 1 | | Np Then, for every x (Σ ) and y Y , ∈ N ∈ lim G0 HX (x) HX1(x)(y)= p. n→∞ ◦ n ◦···◦ By this and Proposition 2.4, log Ry,n(x) lim − = lim s2 (Gj HX (x) HX1(x)(y))j = s2(p, 1 p), νy-a.s.x. n→∞ n n→∞ ◦ n ◦···◦ − By using Example 3.2 (i) and Lemmas 2.5 and 2.9, we can show that for every y Y , ∈ s (p, 1 p) dim µ =1+ 2 − . H y log 2 Furthermore, µ is the product measure on 0, 1 N such that y { } n−1 y µy(Xn =0)= p + H0 (y)= p + | | . s(n 1) y + 1 q − | | If y = 0, then, µ is the (p, 1 p)-self-similar measure. If y = 0, then, by y − 6 Kakutani’s dichotomy [Kak48], µy is singular with respect to the (p, 1 p)- self-similar measure. − 27 Example 3.30. Let Z = [0, 1] and N = 2. Let Y = R. Let 1 y G (y) := max y , 0 ,H (y) := (1 ǫ)y, and, H (y) := . 0 2 − | | 0 − 1 ǫ Let y = 0. Then, (2.6) fails. By using that H is contractive and G (0) = 6 0 0 G1(0) = 1/2 and 0 is the fixed points of H0 and H1 both, we see that it does not hold that (1.5) holds for some ǫ (0, 1/N), l 1 and i ,...,i Σ . 0 ∈ ≥ 1 l ∈ N We will show that dimH µy < 1. For simplicity we assume y = 1/4. By Azuma’s inequality, if we take sufficiently small ǫ > 0, then, there exists D> 1 such that for every n 1, ≥ N 1 1 ℓ x 0, 1 H H < D−n, ∈ { } Xn(x) ◦···◦ X1(x) 4 2 ≤ N where ℓ is the (1/2, 1/2)-Bernoulli measure on 0, 1 . Therefore, by the { } definition of νy, for each n, (i ,...,i ) 0, 1 n ν (I(i ,...,i , 0)) > 0 |{ 1 n ∈ { } | y 1 n }| 1 1 2 n = (i ,...,i ) 0, 1 n H H < . 1 n ∈ { } in ◦···◦ i1 4 2 ≤ D Hence, by taking sum with respect to n, (i ,...,i ) 0 , 1 m ν (I(i ,...,i )) > 0 |{ 1 m ∈ { } | y 1 m }| = (i ,...,i ) 0, 1 n ν (I(i ,...,i , 0, 1,..., 1)) > 0 |{ 1 n ∈ { } | y 1 n }| n Example 3.31. Let Z = [0, 1] and N = 2and Si(z) = (z+i)/2. It is easy to construct an example such that µy is not absolutely continuous or singular with respect to (p, 1 p)-self-similar measure µ(p,1−p). Let Y = 0, 1, 2 . Let − { } H0(0) = 1,H1(0) = 2, and Hi(j)= j, j = 1, 2, i = 0, 1. Let 1 G (j)= , j = 1, 2. 0 2 Let µ1 = µ(p,1−p) and µ2 be a probability measure which is singular with respect to µ(p,1−p). Then, by the uniqueness of the Lebesgue decomposition, µ0 is not absolutely continuous or singular with respect to µ(p,1−p). 3.6. Inhomogeneous cases. We finally deal with the case that Z is an inhomogeneous self-similar set. This has an application of a further gener- alization of (3.9). Let g0(x) := p0x and g (x) := p x + (p + + p ), i 1. i i 0 · · · i−1 ≥ 28 KAZUKIOKAMURA Consider g ϕ(S−1(x)) 0 x q , 0 0 ≤ ≤ 0 g ϕ(S−1(x)) q x q + q , ϕ(x)= 1 1 0 ≤ ≤ 0 1 (3.24) · · · g ϕ(S−1 (x)) q + + q x 1, N−1 N−1 0 · · · N−2 ≤ ≤ Then there exists a unique solution ϕ of (3.24) and let µ = µϕ be the probability measure whose density function is ϕ. We now consider the case that gi i are linear fractional transformations on [0, 1] satisfying (A1) -(A3) in Subsubsection{ } 3.4.2. Define a space Y and functions Gi,Hi, i ΣN on Y in the same manner as in Subsubsection 3.4.2. Let q , i Σ be positive∈ numbers satisfying that i ∈ N q + + q = 1. 0 · · · N−1 Let S0(x) := q0x and S (x) := q x + (q + + q ), i 1. i i 0 · · · i−1 ≥ Then we have the following generalization of Theorem 3.20. Theorem 3.32. (i) If either G−1 (q ) = (3.25) i i 6 ∅ i∈\ΣN or G−1 (q ) Fix(H ) (3.26) i i ⊂ i i∈\ΣN i∈\ΣN fails, then, dimH µϕ < 1. (ii) If (3.25) and (3.26) hold, then, the distribution function ϕ of µϕ is given by (1 q )x c ϕ(x)= − 0 , where we let C := 0 . (3.27) 1 q + c c x 0 1 q − 0 0 − 0 − 0 Proof. (i) If (3.25) fails and β < + , then, by the continuity of Gi and the compactness of Y , we have that ∞ inf Gi(y) qi > 0. y∈Y | − | iX∈ΣN This holds even when β = + , by recalling the definition of Gi(+ ). Hence, ∞ ∞ sup sN (pi(y; x)) < 0, N x∈(ΣN ) ,y∈Y,i∈N where sN denotes the relative entropy with respect to (qi)i. Now the asser- tion follows from Proposition 2.4 and Lemma 2.5. Assume that (3.25) holds and (3.26) fails. Let −1 y0 := G0 (q0) . Then, by (3.25), we have that G−1 (q )= y . (3.28) i i { 0} i∈\ΣN 29 Since (3.26) fails, we have that for some i, H (y ) = y . i 0 6 0 Since G0 is monotone increasing, (1.5) holds for l = 1 and i1 = i. (ii) By the assumption, (3.28) holds, and hence it holds that for each i Σ , ∈ N Gi (y0)= qi, and Hi (y0)= y0. By using them, we see that for every i Σ , ∈ N a = (1+ q )y b + q , and c = y (1 q q y b ). (3.29) i i 0 i i i 0 − i − i 0 i By induction in i, we can show that q + + q b = 0 · · · i−1 , i 1. (3.30) i 1+ y (1 (q + + q )) ≥ 0 − 0 · · · i−1 If i = 0, then, by (A1), b0 = 0. We see that c y = 0 . 0 1 q − 0 By (3.29) and (3.30), it is easy to see that ϕ given by (3.27) satisfies (3.24). In the special case that each gi is linear, we have the following: Corollary 3.33. Let N 2. Let pi,qi, i ΣN be positive numbers satisfying that ≥ ∈ p + + p = q + + q = 1. 0 · · · N−1 0 · · · N−1 Let S0(x) := q0x and S (x) := q x + (q + + q ), i 1. i i 0 · · · i−1 ≥ Let g0(x) := p0x and g (x) := p x + (p + + p ), i 1. i i 0 · · · i−1 ≥ Consider (3.24). If (p , ,p ) = (q , ,q ), then, 0 · · · N−1 6 0 · · · N−1 dimH µϕ < 1. Otherwise, µ is the Lebesgue measure on [0, 1]. In this case, [Ok19+, Theorem 3.9] could also give singularity of µϕ. However, it is not sufficient to give singularity if gi is not linear for some i, for example, it is a linear fractional transformation as above. 4. Open problems (1) Give estimates for the upper and lower local dimensions of µy and consider the Hausdorff dimensions for the level sets. Consider other notions p of dimensions for µy, such as L -dimensions. See [St93]. (2) If Z = [0, 1], then, under what conditions is µy a Rajchman measure [R28], that is, a measure whose Fourier transform vanishes at infinity? See [Ly95] for more information of Rajchman measures. Recently, [JS16] shows that the Fourier coefficients of µ decay to zero, by considering conditions which assure a given measure invariant with respect to the Gauss map is a Rajchman measure, and using (3.23).e 30 KAZUKIOKAMURA 2 (3) Let Z = [0, 1]. It is natural to consider structures of L ([0, 1],µy ). For example, find a subset P of N such that exp(2πik) : k P forms an 2 { ∈ } orthonormal basis of L ([0, 1],µy). [JP98] considers this question for a class of self-similar measures. Acknowledgment. The author appreciates a referee for suggesting an ex- tension of our main results and many comments. The author was supported by JSPS KAKENHI 16J04213, 18H05830, 19K14549 and also by the Re- search Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University. 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