arXiv:1902.01165v3 [math.MG] 1 May 2020 1 ,5 ,1,2,2]frexamples. for F 26] recurrent 25, and 17, FIFs recu o 9, of certain notion 5, applications of 3, the and [1, sets naturally generalized works invariant theoretic of Hardin are many approximation and graphs are Elton the whose Barnsley, for FIFs, [4], recurrent suited In ideally iterat are certain functions. of they sets invariant and are (IFSs) functions these of graphs The iia FsadrcretIS si n iesoa ae ti adt hard functions. is continuous it of case, graphs dimensional are one eas sets in invariant is as their it IFSs that while recurrent However, and IFSs case. similar two-dimensional the especially, cases, Iso etnua rd,wieteeaersrciecniin on conditions restrictive general are more there a interp while presented factors. the grids, [12] where Feng rectangular on grids, collinear. t FISs are rectangular required boundary are on the boundary FISs on the constructed on [10] points Dalla interpolation the where gles, nrcaglrgis ae nti ok em n iwnta [2 Viswanathan and so-called Verma the co to work, to analogous this framework operator on general linear Based a bounded presented a [22] grids. Xu rectangular and on Ruan [18], Yun and insses etclsaigfactors. scaling vertical systems, tion datg fblna Isi htteei orsrcino interpola on restriction F no bilinear is introduced there also that [22] is Xu FISs and bilinear Ruan of advantage [19]. FIFs univariate the with LR14A010001. rca neplto ucin FF)wr nrdcdb Barns by introduced were (FIFs) functions interpolation Fractal 2010 aual,w att eeaieFF n eurn Ist higher to FIFs recurrent and FIFs generalize to want we Naturally, n[6,Msoutitoue rca neplto ufcs(FI surfaces interpolation fractal introduced Massopust [16], In sn fl-u’tcnqeitoue yM yz[5 n develop and [15] Ma by lysz introduced technique ‘fold-out’ a Using h eerhwsspotdi atb h SCgat11771391 grant NSFC the by part in supported was research The orsodn uhr u-u Ruan. Huo-Jun author: Corresponding e od n phrases. and words Key EURN RCA NEPLTO SURFACES INTERPOLATION FRACTAL RECURRENT ahmtc ujc Classification. Subject Mathematics anasmto sta etclsaigfcoshv unifo have factors scaling const vertical partition. certain that compatible under is RFISs bilinear assumption of whi scaling main dimension vertical generated box and be the points to tain interpolation easy on are which restrictions rectang RFISs, on bilinear (RFISs) introduce surfaces interpolation fractal current Abstract. OSRCINADBXDMNINOF DIMENSION BOX AND CONSTRUCTION nti ae,w rsn eea rmwr ocntutre construct to framework general a present we paper, this In eurn rca neplto ufcs o dimension, box surfaces, interpolation fractal Recurrent HNLAGADHOJNRUAN HUO-JUN AND LIANG ZHEN 1. Introduction rmr 88;Scnay41A30. Secondary 28A80; Primary 1 α fatloeao associated operator -fractal atr.W loob- also We factors. lrgis hnwe Then grids. ular msm ne a under sums rm ans hr the where raints, n yZNF grant ZJNSFC by and , eteeaeno are there le dfnto systems function ed rn Fs There IFSs. rrent inpit and points tion osrcinof construction F.Pes see Please IFs. etclscaling vertical e 2 n1986. in [2] ley ecoplanar. be o oconstruct to y lto points olation db Metzler by ed s ntrian- on Ss) src FISs nstruct dimensional guarantee o trtdfunc- iterated ]studied 4] Isto FIFs f occurring - S.One ISs. 2 ZHENLIANGANDHUO-JUNRUAN vertical scaling factors. We remark that some ideas of [22] are similar to the paper on the one-dimensional bilinear FIFs by Barnley and Massopust [6]. As pointed in [22], we expect that bilinear FISs will be used to generate some natural scenes. However, in order to fit given data more effectively, we need re- current FISs (RFISs). Bouboulis, Dalla and Drakopoulos [7, 8], and Yun, Choi and O [27] presented some nice methods to generate RFISs with function scaling factors, while the restrictive conditions for continuity are hard to check. In this paper, by extending the methods in [4] and [22], we present a general framework to construct RFISs on rectangular grids and introduce bilinear RFISs. Similarly to bilinear FISs constructed in [22], bilinear RFISs can be defined on rect- angular grids without any restrictions on interpolation points and vertical scaling factors. It is one of the basic problems in fractal geometry to obtain the box dimension of fractal sets. In [4], Barnely, Elton and Hardin obtained the box dimension of affine recurrent FIFs in one-dimensional case, where they assumed that the correlation matrix is irreducible. The result was generalized by Ruan, Sha and Ye [20], where there is no restrict condition on the correlation matrix. We remark that in [4, 20] and in the later works such as [7, 8], the vertical scaling factors are constants. In the case of bilinear FIFs (in one dimensional case) and bilinear FISs, it is very difficult to obtain their box dimension without additional assumptions since vertical scaling factors are functions. The main assumption in [6, 13] is that vertical scaling factors have a uniform sum. In this paper, stronger restrict conditions are needed to obtain the box dimension since we deal with bilinear RFISs. We assume that vertical scaling factors have uniform sums under a compatible partition. From Lemmas 4.3 and 4.4, we can see that the assumption is reasonable. We remark that by using the method of this paper and other previous papers, the authors also studied the construction and the box dimension of RFISs on triangular domains in [14]. The paper is organized as follows. In section 2, we present a general framework to construct RFISs. Bilinear RFISs are introduced in Section 3. In Section 4 we obtain the box dimension of bilinear RFISs under certain constraints.
2. Construction of Recurrent Fractal Interpolation Surfaces
For every positive integer N, we denote ΣN = 1, 2,...,N and ΣN,0 = 0, 1,...,N . 3 { } { } Given a data set (xi,yj ,zij ) R : i ΣN, , j ΣM, with { ∈ ∈ 0 ∈ 0} x < x < < xN , y
′ ′ Clearly, ui(xi)= ui+1(xi)= xi for all i ΣN−1, which is an important property in our construction. ∈ ′ ′ ′ Similarly, we choose y0,y1,...,yM to be a finite sequence in yj : j ΣM,0 such that { ∈ } ′ ′ yj yj−1 >yj yj−1, j ΣM . | − |′ − ∀ ∈ ′ ′ For each j ΣM , we denote by Jj the closed interval with endpoints yj−1 and yj, ∈ ′ and let vj : J Jj be a contractive homeomorphism with j → ′ ′ (2.2) vj (yj−1)= yj−1, vj (yj )= yj. ′ ′ Clearly, vj (yj )= vj+1(yj )= yj for all j ΣM−1. ∈ ′ ′ For all i ΣN,0 and j ΣM,0, we denote by zij to be the zpq satisfying xp = xi ′∈ ∈ and yq = yj . ′ ′ ′ For i ΣN and j ΣM , we denote Dij = Ii Jj , and define a continuous ∈ ′ ∈ × function Fij : D R R satisfying ij × → ′ ′ ′ (2.3) Fij (xk,yℓ,zkℓ)= zkℓ for all (k,ℓ) i 1,i j 1, j , and ∈{ − }×{ − } ′ ′′ ′ ′′ (2.4) Fij (x,y,z ) Fij (x,y,z ) αij z z | − |≤ | − | ′ ′ ′′ for all (x, y) Dij and all z ,z R, where αij is a given constant with 0 < αij < 1. ∈ ∈′ Now we define a map Wij : D R Dij R by ij × → × (2.5) Wij (x,y,z) = (ui(x), vj (y), Fij (x,y,z)). Then ′ ′ ′ Wij (xk,yℓ,zkℓ) = (xk,yℓ,zkℓ), (k,ℓ) i 1,i j 1, j , ′ ∈{ − }×{ − } and D R, Wij ; (i, j) ΣN ΣM is a recurrent iterated function system (re- { ij × ∈ × } current IFS). For i ΣN and j ΣM , we denote by (Dij R) the family of all non-empty ∈ ∈ H × compact subsets of Dij R. Let be the product of all (Dij R), i.e., × H H × = (Aij ) i N, jeN : Aij (Dij R) for all i, j . H { 1≤ ≤ 1≤ ≤ ∈ H × } For any A =e (Aij ) , we define W (A) by ∈ H ∈ H ′ W (A) = eWij (Akℓ) : (k,ℓ) ΣeN ΣM and Dkℓ Dij ij ∪{ ∈ × ⊂ } for all 1 i N and 1 j M. ≤ ≤ ≤ ≤ For any continuous function f on I J and (i, j) ΣN ΣM , we denote × ∈ × Γf D = (x,y,f(x, y)) : (x, y) Dij , | ij { ∈ } which is the graph of f restricted on Dij . Similarly as in [2, 4, 22], we have the following theorem. ′ Theorem 2.1. Let D R, Wij ; (i, j) ΣN ΣM be the recurrent IFS defined { ij × ∈ × } by (2.5). Assume that Fij : (i, j) ΣN ΣM satisfies the following matchable conditions: { ∈ × } ∗ −1 −1 (1). for all i ΣN , j ΣM , and x = u (xi)= u (xi), ∈ −1 ∈ i i+1 ∗ ∗ ′ (2.6) Fij (x ,y,z)= Fi ,j (x ,y,z), y J , z R, and +1 ∀ ∈ j ∈ ∗ −1 −1 (2). for all i ΣN , j ΣM , and y = v (yj )= v (yj ), ∈ ∈ −1 j j+1 ∗ ∗ ′ (2.7) Fij (x, y ,z)= Fi,j (x, y ,z), x I , z R. +1 ∀ ∈ i ∈ 4 ZHENLIANGANDHUO-JUNRUAN
Then there exists a unique continuous function f on I J, such that f(xi,yj)= zij × for all i ΣN, and j ΣM, , and (Γf D ) i N, j M is the invariant set of W , ∈ 0 ∈ 0 | ij 1≤ ≤ 1≤ ≤ i.e., Γf D = Wij (Γf D′ ) for all (i, j) ΣN ΣM . | ij | ij ∈ × Proof. Let C∗(I J) be the collection of all continuous functions ϕ on I J × ∗ × satisfying ϕ(xi,yj ) = zij for all (i, j) ΣN,0 ΣM,0. Define T : C (I J) C∗(I J) as follows: given ϕ C∗(I ∈J), × × → × ∈ × −1 −1 −1 −1 (2.8) T ϕ(x, y)= Fij u (x), v (y), ϕ(u (x), v (y)) , (x, y) Dij , i j i j ∈ for all (i, j) ΣN ΣM . ∈ ∗ × Given ϕ C (I J), it is clear that for all (i, j) ΣN ΣM , T ϕ is ∈ × ∈ × |int(Dij ) continuous, where we use int(E) to denote the interior of a subset E of R2. For all (xp,yq) Dij , ∈ −1 −1 −1 −1 Fij (ui (xp), vj (yq), ϕ(ui (xp), vj (yq))) ′ ′ ′ ′ ′ ′ ′ = Fij (xp,yq, ϕ(xp,yq)) = Fij (xp,yq,zpq)= zpq so that T ϕ(xp,yq) = zpq. Furthermore, from matchable conditions, we know that T ϕ is well defined on the boundary of Ii Jj for all (i, j) ΣN ΣM . Thus T : C∗(I J) C∗(I J) is well defined.× ∈ × × → ∗ × For any ϕ C (I J), we define ϕ ∞ = max ϕ(x, y) : (x, y) I J . From (2.4), we∈ can easily× see that T is| contractive| on{ the complete metric∈ × space} ∗ ∗ (C (I J), ∞). Thus there exists a unique function f C (I J) such that Tf = f×, that| · is, | ∈ × −1 −1 −1 −1 (2.9) f(x, y)= Fij u (x), v (y),f(u (x), v (y)) , (x, y) Dij , i j i j ∈ for all (i, j) ΣN ΣM . Combining this with (2.5), we know that for all (i, j) ∈ × ∈ ΣN ΣM , × Γf D = (x,y,f(x, y)) : (x, y) Dij | ij { ∈ } −1 −1 −1 −1 = (x, y, Fij u (x), v (y),f(u (x), v (y)) ) : (x, y) Dij { i j i j ∈ } ′ = (ui(x), vj (y), Fij (x,y,f(x, y))) : (x, y) D { ∈ ij } ′ = Wij (x,y,f(x, y)) : (x, y) D . { ∈ ij }
Thus (Γf Dij )1≤i≤N,1≤j≤M is the invariant set of W . | ∗ Assume that there exists f C (I J), such that (Γf Dij )1≤i≤N,1≤j≤M is the invariant set of W . Then we must∈ have× | e e ′ Fij (x, y, f(x, y)) = f(ui(x), vj (y)), (x, y) D , ∀ ∈ ij so that T f = f. Since T eis contractivee on (C∗(I J), ), we know that f = f. × |·|∞ We callef thee recurrent fractal interpolation function (RFIF) defined bye D′ { ij × R, Wij ; (i, j) ΣN ΣM , and call the graph of f a recurrent fractal interpolation surface (RFIS).∈ From× (2.9),} we have the following useful property: for all (i, j) ∈ ΣN ΣM , × ′ (2.10) f(ui(x), vj (y)) = Fij (x,y,f(x, y)) for all (x, y) D . ∈ ij ′ Generally, the recurrent IFS D R, Wij ; (i, j) ΣN ΣM is not hyperbolic. { ij × ∈ × } However, we can still show that Γf = (x,y,f(x, y)) : (x, y) I J is the attractor { ∈ × } RECURRENT FRACTAL INTERPOLATION SURFACES 5 of the recurrent IFS. The spirit of the proof follows from [2, 4, 22]. For any two nonempty compact subsets A and B of R3, we define their Hausdorff metric by
dH (A, B) = max max min d(x, y), max min d(x, y) , { x∈A y∈B y∈B x∈A } where we use d( , ) to denote the Euclidean metric. For any A, B with A = (A ) · · and B = (B ) , we define ∈ H ij 1≤i≤N,1≤j≤M ij 1≤i≤N,1≤j≤M e dH (A, B) = max dH (Aij ,Bij ) : (i, j) ΣN ΣM . { ∈ × } It is well knowne that ( , dH ) is a complete metric space. For details about the H Hausdorff metric and the metric d , please see [3, 4, 11]. e e H ′ Theorem 2.2. Let f be the RFIFe defined by D R, Wij ; (i, j) ΣN ΣM . { ij × ∈ × } Then for any A = (Aij ) i N, j M with Aij = for all (i, j) ΣN ΣM , 1≤ ≤ 1≤ ≤ ∈ H 6 ∅ ∈ × n (2.11) lim dH (W (A), (Γf eD )1≤i≤N,1≤j≤M )=0, n→∞ | ij where W 0(A)= A and Wen+1(A)= W (W n(A)) for any n 0. ≥ Proof. For any A = (Aij )1≤i≤N,1≤j≤M , we define AXY to be the projection of A onto I J, i.e., ∈ H × e AXY = (x, y) : there exists (i, j) ΣN ΣM and z R with (x,y,z) Aij . { ∈ × ∈ ∈ } Let (I J) be the family of all nonempty compact subsets of I J. It is clear H × × that AXY (I J). Define L∈: H (I× J) (I J) by H × → H × ′ L(U)= (ui(x), vj (y)) : (x, y) D U ,U (I J). { ∈ ij ∩ } ∈ H × 1≤[i≤N 1≤j≤M
It is clear that L is contractive on (I J). Since Aij = for all (i, j) ΣN ΣM , we have H × 6 ∅ ∈ × n lim dH (L (AXY ), I J)=0. n→∞ × n n Let Ωij = (x,y,f(x, y)) : (x, y) L (AXY ) Dij . Noticing that f is uniformly continuous{ on I J, we have ∈ ∩ } × n (2.12) lim dH (Ωij , Γf Dij )=0 n→∞ | for all (i, j) ΣN ΣM . For n 1,∈ we define× ≥ n ∆n = sup f(x, y) z : (x,y,z) (W (A))ij for some (i, j) ΣN ΣM . {| − | ∈ ∈ × } n n By the definition of Hausdorff metric, we have dH (Ω , (W (A))ij ) ∆n for all ij ≤ (i, j) ΣN ΣM and n 1. ∈ × ≥ n Given (i, j) ΣN ΣM and n 1, for any (x,y,z) (W (A))ij , there exist ∈ × ∗ ∗ ∗ ≥ n−1 ∈ (k,ℓ) ΣN ΣM and (x ,y ,z ) (W (A))kℓ such that ∈ × ∈ ∗ ∗ ∗ ∗ ∗ (x,y,z) = (ui(x ), vj (y ), Fij (x ,y ,z )).
Let α = max αij : (i, j) ΣN ΣM . By (2.4) and (2.10), { ∈ × } ∗ ∗ ∗ ∗ ∗ ∗ ∗ f(x, y) z = Fij (x ,y ,f(x ,y )) Fij (x ,y ,z )) | − | | − | ∗ ∗ ∗ α f(x ,y ) z α∆n ≤ | − |≤ −1 6 ZHENLIANGANDHUO-JUNRUAN so that ∆n α∆n−1. Since 0 <α< 1, we have lim ∆n = 0. Thus ≤ n→∞ n n lim dH (Ωij , (W (A))ij )=0 n→∞ for all (i, j) ΣN ΣM . Combining this with (2.12), we have ∈ × n lim dH ((W (A))ij , Γf D )=0 n→∞ | ij for all (i, j) ΣN ΣM . This completes the proof of the theorem. ∈ × Remark 2.3. The assumption “Aij = for all (i, j) ΣN ΣM ” in the above theorem can be weaken. In fact, from6 the∅ proof, we can∈ see that× the theorem still n holds if we only require that limn dH (L (AXY ), I J) = 0. →∞ × 3. Construction of Bilinear RFISs Let h(x, y),S(x, y) be two continuous functions on I J satisfying × (3.1) h(xi,yj )= zij , (i, j) ΣN, ΣM, , and ∈ 0 × 0 max S(x, y) : (x, y) I J < 1. {| | ∈ × } ′ Let gij (x, y), (i, j) ΣN ΣM be continuous functions on D satisfying ∈ × ij ′ ′ ′ (3.2) gij (x ,y )= z , (k,ℓ) i 1,i j 1, j . k ℓ kℓ ∈{ − }×{ − } ′ For (i, j) ΣN ΣM , we define functions Fij (x,y,z): D R R by ∈ × ij × → (3.3) Fij (x,y,z)= S(ui(x), vj (y))(z gij (x, y)) + h(ui(x), vj (y)), − ′ ′ where ui C(I ) and vj C(J ) are functions defined in Section 2. Then for all ∈ i ∈ j (i, j) ΣN ΣM and (k,ℓ) i 1,i j 1, j , ∈ × ∈{ − }×{ − } ′ ′ ′ ′ ′ Fij (xk,yℓ,zkℓ)= h(ui(xk), vj (yℓ)) = h(xk,yℓ)= zkℓ so that (2.3) holds. Furthermore, it follows from max S(x, y) : (x, y) I J < 1 that (2.4) also holds. {| | ∈ × } ∗ −1 −1 ′ Given i ΣN−1, let x = ui (xi)= ui+1(xi). Given j ΣM , for all y Jj and z R, we have∈ ∈ ∈ ∈ ∗ ∗ Fij (x ,y,z)= S(xi, vj (y))(z gij (x ,y)) + h(xi, vj (y)), − ∗ ∗ Fi ,j (x ,y,z)= S(xi, vj (y))(z gi ,j (x ,y)) + h(xi, vj (y)). +1 − +1 ∗ ∗ ′ Thus, the matchable condition (2.6) holds if gij (x ,y)= gi+1,j (x ,y) for all y Jj . ∗ ∗ ∈ Similarly, the matchable condition (2.7) holds if gij (x, y ) = gi,j+1(x, y ) for all ′ x Ii. Thus, from Theorem 2.1, we have the following result. We remark that Metzler∈ and Yun [18], and Yun, Choi and O [27] obtained similar results.
Theorem 3.1. Let Fij , (i, j) ΣN ΣM be defined as in (3.3). Assume that the following matchable conditions∈ hold:× ∗ −1 −1 (1). for all i ΣN , j ΣM , and x = u (xi)= u (xi), ∈ −1 ∈ i i+1 ∗ ∗ ′ (3.4) gij (x ,y)= gi ,j (x ,y), y J , and +1 ∀ ∈ j ∗ −1 −1 (2). for all i ΣN , j ΣM , and y = v (yj )= v (yj ), ∈ ∈ −1 j j+1 ∗ ∗ ′ (3.5) gij (x, y )= gi,j (x, y ), x I . +1 ∀ ∈ i Then there exists a unique continuous function f on I J, such that (Γf D ) i N, j M × | ij 1≤ ≤ 1≤ ≤ is the invariant set of W with f(xi,yj )= zij for all i ΣN, and j ΣM, . ∈ 0 ∈ 0 RECURRENT FRACTAL INTERPOLATION SURFACES 7
We remark that from (3.2), it is easy to check that for all i ΣN−1, j ΣM , ∗ −1 −1 ∈ ∈ and x = ui (xi)= ui+1(xi), ∗ ′ ∗ ′ ∗ ′ ∗ ′ (3.6) gij (x ,yj−1)= gi+1,j (x ,yj−1), gij (x ,yj )= gi+1,j (x ,yj). ∗ −1 −1 Similarly, for all i ΣN , j ΣM , and y = v (yj )= v (yj ), ∈ ∈ −1 j j+1 ′ ∗ ′ ∗ ′ ∗ ′ ∗ (3.7) gij (xi−1,y )= gi,j+1(xi−1,y ), gij (xi,y )= gi,j+1(xi,y ). However, the match conditions in Theorem 3.1 may not be satisfied if we only require that (3.6) and (3.7) hold. Now we want to construct bilinear RFISs which are special RFISs in the above theorem. The basic idea is similar to that in [6, 22]. For all i ΣN and j ΣM , ∈ ∈ we define ui and vj to be linear functions satisfying (2.1) and (2.2). We also define gij to be the bilinear function satisfying (3.2). That is, if we denote ′ ′ x x yj y λ (x)= i − , µ (y)= − , i x′ x′ j y′ y′ i − i−1 j − j−1 then for all (x, y) D′ , ∈ ij ′ ′ gij (x, y)=λi(x)µj (y)z + (1 λi(x))µj (y)z i−1,j−1 − i,j−1 ′ ′ + λi(x)(1 µj (y))z + (1 λi(x))(1 µj (y))z . − i−1,j − − i,j Notice that once x is fixed, gij (x, y) is a linear function of y. Combining this with (3.6), the matchable condition (3.4) in Theorem 3.1 is satisfied. Similarly, (3.5) is also satisfied. Thus, in the case that gij are bilinear for all i, j, we do generate RFIF and RFIS. In order to construct bilinear RFISs, we define h : I J R to be the function × → satisfying (3.1) and h I J is bilinear for all (i, j) ΣN ΣM . | i× j ∈ × Let sij : (i, j) ΣN, ΣM, be a given subset of R with sij < 1 for all i, j. { ∈ 0 × 0} | | We define S : I J R to be the function such that S I J is bilinear for all × → | i× i (i, j) ΣN ΣM and ∈ × S(xi,yj )= sij , (i, j) ΣN, ΣM, . ∀ ∈ 0 × 0 ′ For all (i, j) ΣN ΣM , we define Fij : D R R by (3.3). Then the RFIF ∈ × ij × → f is called a bilinear RFIF. We call Γf a bilinear RFIS. The function S(x, y) is called the vertical scale factor function of f. We also call sij , (i, j) ΣN,0 ΣM,0 vertical scaling factors of f if there is no confusion. ∈ × From the construction, we know that a bilinear RFIS is determined by inter- polation points (xi,yj ,zi,j ) (i,j)∈ΣN,0×ΣM,0 , vertical scaling factors sij : (i, j) { } ′ ′ { ∈ ΣN, ΣM, , and domain points (x ,y ) . This property is similar 0 × 0} { i j }(i,j)∈ΣN,0×ΣM,0 to the linear recurrent FIF in the one-dimensional case [4]. In particular, a bilinear RFIS is easy to be generated, while there are no restrictions on interpolation points and vertical scaling factors.
4. Box Dimension of Bilinear RFISs 3 For any k1, k2, k3 Z and ε> 0, we call Πi=1[kiε, (ki +1)ε] an ε-coordinate cube 3 ∈ 3 in R . Let E be a bounded set in R and E(ε) the number of ε-coordinate cubes intersecting E. We define N
log E(ε) log E(ε) (4.1) dimBE = lim N and dimBE = lim N , ε→0+ log 1/ε ε→0+ log 1/ε 8 ZHENLIANGANDHUO-JUNRUAN and call them the upper box dimension and the lower box dimension of E, respec- tively. If dimBE = dimBE, then we use dimB E to denote the common value and call it the box dimension of E. It is easy to see that in the definition of the upper 1 + and lower box dimensions, we can only consider εn = KnN , where n Z . That is, ∈
log E(εn) log E(εn) (4.2) dimBE = lim N and dimBE = lim N . n→∞ n log K n→∞ n log K
It is also well known that dimBE 2 when E is the graph of a continuous function on a domain of R2. Please see [11]≥ for details. In this section, we will estimate the box dimension of Γf, where f is the bilinear RFIF defined in the section 3. Without loss of generality, we can assume that I = J = [0, 1].
4.1. A method to calculate the box dimension. It is difficult to obtain the box dimension of general bilinear RFIS. In this paper, we assume that i j (4.3) M = N, and xi = ,yj = , i, j ΣN, . N N ∀ ∈ 0 Furthermore, we assume that there exists a positive integer K 2 such that ≥ ′ ′ Ii Jj (4.4) | | = | | = K, i, j ΣN , Ii Jj ∀ ∈ | | | | and for all (i , j ), (i , j ) ΣN ΣN , we have 1 1 2 2 ∈ × (4.5) D′ = D′ or int(D′ ) int(D′ )= . i1j1 i2j2 i1j1 ∩ i2j2 ∅ For each n Z+ and 1 k,ℓ KnN, we denote ∈ ≤ ≤ k 1 k ℓ 1 ℓ Dn = − , − , . kℓ KnN KnN × KnN KnN Given n Z+ and U [0, 1]2, we define ∈ ⊂ n O(f,n,U)= O(f,Dkℓ), 1≤k,ℓX≤KnN n Dkℓ⊂U where we use O(f, E) to denote the oscillation of f on E [0, 1]2, that is, ⊂ O(f, E) = sup f(x′) f(x′′): x′, x′′ E . { − ∈ } We also denote O(f,n)= O(f, n, [0, 1]2) for simplicity. We will use the following simple lemma, which presents a method to estimate the upper and lower box dimensions of the graph of a function from its oscillation. Similar results can be found in [11, 13, 21]. Lemma 4.1. Let f be the bilinear RFIF defined in Section 3. Then log O(f,n) (4.6) dimB(Γf) max 2, 1 + lim , and ≥ n n→∞ n log K o log(O(f,n)+2KnN) (4.7) dimB(Γf) 1+ lim , ≤ n→∞ n log K where we define log 0 = according to the usual convention. −∞ RECURRENT FRACTAL INTERPOLATION SURFACES 9
Proof. It is clear that
−1 n −1 (4.8) f (εn) ε O(f,D )= ε O(f,n) NΓ ≥ n ij n 1≤i,jX≤KnN so that (4.6) holds. On the other hand, we note that E(ε) and E(εn) can be N N replaced by E (ε) and E(εn) in (4.1) and (4.2) respectively, where NE(ε) is the smallest numberN of cubesN of side ε that cover E (see [11] for details). In our case, e e e −1 n n n N f (εn) (ε O(f,D )+2)= K N O(f,n)+2K N Γ ≤ n ij 1≤i,jX≤KnN e so that (4.7) holds.
log O(f,n) Remark 4.2. From Lemma 4.1, dimB (Γf) = 2 if lim 1, and dimB(Γf)= n→∞ n log K ≤ 1 + lim log O(f,n) if the limit exists and is larger than 1. n→∞ n log K 4.2. Compatible partitions and uniform sums. The vertical scaling factors sij : i, j ΣN, are called steady if for each (i, j) ΣN ΣN , either all of { ∈ 0} ∈ × si−1,j−1,si−1,j ,si,j−1 and sij are nonnegative or all of them are nonpositive. m 2 Given = Br r=1, where B1,...,Bm are nonempty subsets of [0, 1] , is B { } 2 m 2 B called a partition of [0, 1] if r=1 Br = [0, 1] and int(Br) int(Bt) = for all m ∩ ∅ r = t. A partition = BSr r=1 of [0, 1] is called compatible with respect to 6 ′ B { } Dij,D ; (i, j) ΣN ΣM if the following two conditions hold: { ij ∈ × } (1) for each (i, j) ΣN ΣN , there exist r, t 1, 2,...,m , such that Dij ′ ∈ × ∈{ } ⊂ Br and D Bt, and ij ⊂ (2) Assume that r, t 1, 2,...,m . If there exists (i, j) ΣN ΣN , such that ′∈{ } ∈ × Dij Br and D Bt, then ⊂ ij ⊂ ′ ′ Bt = Dkℓ : Dkℓ Br,Dkℓ Bt . [{ ⊂ ⊂ } For 1 r, t m, we denote ≤ ≤ Λr = (i, j) ΣN ΣN : Dij Br , { ∈ × ⊂ } ′ ′ Λ = (i, j) ΣN ΣN : D Bt , t { ∈ × ij ⊂ } We remark that the second condition is equivalent to
′ ′ Bt = Dij , if Λr Λt = . ′ ∩ 6 ∅ (i,j)∈[Λr ∩Λt Given 1 r m and α, β 0, 1,...,N K , we denote ≤ ≤ ∈{ − } ′ Λr(α, β)= (i, j) Λr : D = [xα, xα K ] [yβ,yβ K] . { ∈ ij + × + } We say that the vertical scaling factors sij : i, j ΣN,0 have uniform sums under m { ∈ } ′ a compatible partition = Br if for all r, t 1, 2,...,m with Λr Λ = , B { }r=1 ∈{ } ∩ t 6 ∅ there exists a constant γrt, such that
γrt = S(ui(xα), vj (yβ)) = S(ui(xα), vj (yβ K )) | | | + | (i,j)∈XΛr (α,β) (i,j)∈XΛr (α,β)
= S(ui(xα K ), vj (yβ)) = S(ui(xα K ), vj (yβ K )) | + | | + + | (i,j)∈XΛr (α,β) (i,j)∈XΛr (α,β) 10 ZHENLIANGANDHUO-JUNRUAN
′ for all α, β 0, 1,...,N K with [xα, xα+K ] [yβ,yβ+K] Dij : (i, j) ′ ∈ { − } ′ × ∈ { ∈ Λr Λt . In this case, we also call γrt :Λr Λt = the uniform sums of vertical scaling∩ } factors. { ∩ 6 ∅}
Lemma 4.3. Assume that the vertical scaling factors sij : i, j ΣN,0 are ′ { ∈ } steady and have uniform sums γrt :Λr Λt = under a compatible parti- m { ∩ 6 ∅} ′ tion = Br r=1. Then for all r, t 1, 2,...,m with Λr Λt = , and all B { } ∈ { } ′ ∩ 6 ∅ ′ α, β 0, 1,...,N K with [xα, xα+K ] [yβ,yβ+K] Dij : (i, j) Λr Λt , we have∈{ − } × ∈{ ∈ ∩ }
S(ui(x), vj (y)) = γrt, (x, y) [xα, xα K ] [yβ,yβ K]. | | ∀ ∈ + × + (i,j)∈XΛr (α,β) ′ Proof. Given r, t 1, 2,...,m with Λr Λt = , and given α, β 0, 1,...,N ∈{ } ′ ∩ 6 ∅ ′ ∈{ − K with [xα, xα K ] [yβ,yβ K] D : (i, j) Λ , we define } + × + ∈{ ij ∈ t} ∗ S (x, y)= S(ui(x), vj (y)) γrt, | |− (i,j)∈XΛr (α,β) where (x, y) [xα, xα K ] [yβ,yβ K]. Since the vertical scaling factors are steady, ∈ + × + we know that for each (i, j) Λr(α, β), the function S is nonnegative or nonpositive ∈ on Dij . It follows that S(ui(x), vj (y)) is nonnegative or nonpositive on [xα, xα+K ] ∗ × [yβ,yβ+K]. As a result, S is a bilinear function on [xα, xα+K ] [yβ,yβ+K]. Thus, ∗ × from S =0on(xα,yβ), (xα,yβ+K), (xα+K ,yβ) and (xα+K ,yβ+K), we know that ∗ S = 0 on [xα, xα K ] [yβ,yβ K] so that the lemma holds. + × + 4.3. Calculation of box dimension. In this subsection, we always assume that f is the bilinear RFIF determined in the section 3 with conditions (4.3), (4.4) and (4.5). Furthermore, we assume that the vertical scaling factors sij : i, j ΣN,0 ′ { ∈ } are steady and have uniform sums γrt :Λr Λt = under a compatible partition m { ∩ 6 ∅} = Br r=1. B Using{ } our assumptions, we can obtain the following basic result. Lemma 4.4. There exists a positive constant C > 0 such that m n (4.9) O(f,n +1,Br) γrtO(f,n,Bt) CK − ≤ Xt=1 + ′ for all 1 r m and n Z , where we define γrt =0 if Λr Λ = . ≤ ≤ ∈ ∩ t ∅ Proof. From the first condition of compatible partition, ′ Br = Dij : (i, j) Λr Λt ′ ∪{ ∈ ∩ } t: Λr[∩Λt6=∅ for all 1 r m. Thus, in order to prove the lemma, it suffices to show that for ≤ ≤ ′ all 1 r, t m with Λr Λ = , there exists a constant Crt > 0, such that ≤ ≤ ∩ t 6 ∅ n O(f,n +1,Dij ) γrtO(f,n,Bt) CrtK . − ≤ i,j X ′ ( )∈Λr ∩Λt
For all α, β 0, 1,...,N K , we denote Dαβ = [xα, xα+K ] [yβ,yβ+K]. From the second∈ condition { of− compatible} partition, for all r, t = 1, 2×,...,m with ′ e Λr Λ = , ∩ t 6 ∅ ′ ′ Bt = Dαβ : (i, j) Λr Λ , such that Dαβ = D . ∪{ ∃ ∈ ∩ t ij } e e RECURRENT FRACTAL INTERPOLATION SURFACES 11
′ On the other hand, it is clear that for each (i, j) Λr Λt, there exists a unique 2 ′ ∈ ∩ (α, β) 0, 1,...,N K , such that Dij = Dαβ Bt. Thus, in order to prove ∈ { − } ⊂ ′ the lemma, it suffices to show that for all r, t = 1, 2,...,m with Λr Λt = , and ′ e ′ ∩ 6 ∅ all α, β =0, 1,...,N K with Dαβ = Dij for some (i, j) Λr Λt, there exists a constant C > 0,− such that ∈ ∩ r,t,α,β e n (4.10) O(f,n +1,Dij) γrtO(f, n, Dα,β) Cr,t,α,βK − ≤ (i,j)∈XΛr (α,β) e for all positive integers n. ∗ 2 Denote M = max f(x, y) : (x, y) [0, 1] . For (i, j) ΣN ΣN and z∗ [ M ∗,M ∗], we define{| | ∈ } ∈ × ∈ − ∗ ′ Fˆi,j,z∗ (x, y)= Fij (x,y,z ), (x, y) D . ∈ ij ′ Since S(ui(x), vj (y)),gij (x, y),h(ui(x), vj (y)) are all bilinear functions on Dij , we know that Cij = sup Fˆi,j,z∗ (x, y) < , ′ k∇ k ∞ (x,y)∈int(Dij ) ∗ ∗ ∗ z ∈[−M ,M ] where is the standard Euclidean norm. k·k n n ′ n Given 1 k,ℓ K N with D D , we fix a point (xk,n,yℓ,n) D . It is ≤ ≤ kℓ ⊂ ij ∈ kℓ clear that for all (x, y) Dn , ∈ kℓ (4.11) Fij (xk,n,yℓ,n,f(x, y)) Fij (x,y,f(x, y)) √2Cij εn. | − |≤ On the other hand, for all (x′,y′), (x′′,y′′) Dn , we have ∈ kℓ ′ ′ ′′ ′′ Fij (xk,n,yℓ,n,f(x ,y ) Fij (xk,n,yℓ,n,f(x ,y )) − ′ ′ ′′ ′′ =S(ui(xk,n), vj (yℓ,n)) f(x ,y ) f(x ,y ) . − Combining this with (2.10) and (4.11), we have ′ ′ ′′ ′′ f(ui(x ), vj (y )) f(ui(x ), vj (y )) | − | ′ ′ ′′ ′′ S(ui(xk,n), vj (yℓ,n)) f(x ,y ) f(x ,y ) +2√2Cij εn ≤| | · | − | so that n n O(f, (ui vj )(D )) S(ui(xk,n), vj (yℓ,n)) O(f,D )+2√2Cij εn. × kℓ ≤ | | · kℓ Thus, from Lemma 4.3,
O(f,n +1,Dij)
(i,j)∈XΛr (α,β) n n+2 S(ui(xk,n), vj (yℓ,n)) O(f,Dkℓ)+2√2Cij NK ≤ n | | · 1≤k,ℓX≤K N (i,j)∈XΛr (α,β) n e Dkℓ⊂Dαβ n+2 =γrt O(f, n, Dαβ)+2√2Cij NK . · Similarly, we havee n+2 O(f,n +1,Dij ) γrtO(f, n, Dαβ ) 2√2Cij NK ≥ − (i,j)∈XΛ (α,β) r e so that (4.10) holds. 12 ZHENLIANGANDHUO-JUNRUAN
′ Define G = (γrt)m×m, where γrt = 0 if Λr Λt = . Given 1 r, t m and + n ∩ ∅ ≤ ≤ n Z , a finite sequence ik in 1, 2,...,m is called an n-path (a path for ∈ { }k=0 { } short) from t to r if i0 = t, in = r, and
γi ,i − > 0, k =1, 2, . . . , n. k k 1 ∀ r and t are called connected, denoted by r t, if there exist both a path from r to t, and a path from t to r. We remark that∼ in general, it is possible that there is no path from r to itself. A subset V of 1, 2,...,m is called connected if r t for all r, t V . Furthermore, V is called{ a connected} component of 1, 2,...,m∼ ∈ { } if V is connected and there is no connected subset V of 1, 2,...,m such that { } V ( V . It is well known that the matrix G is irreducible if and only if 1, 2,...,m e is connected. Please see [23] for details. { } e Given a connected component V = r ,...,rt of 1, 2,...,m , where r < r < { 1 } { } 1 2 < rt, we define a submatrix G V of G by ··· | (G V )kℓ = γr ,r , 1 k,ℓ t. | k ℓ ≤ ≤ Definition 4.5. Given (i, j), (k,ℓ) ΣN ΣN , we call (k,ℓ) an ancestor of (i, j) ∈ n × if there exists a finite sequence (iτ , jτ ) in ΣN ΣN such that (i , j ) = (i, j), { }τ=0 × 0 0 (in, jn) = (k,ℓ), and ′ Di ,j D , τ =1, 2, . . . , n. τ τ ⊂ iτ−1,jτ−1 ∀ Let A (i, j) be the collection of all ancestors of (i, j) and A(i, j)= (i, j) A (i, j). 0 { } 0 We call (i, j) degenerate if for all (k,ℓ) A(i, j), we have either S ∈ (4.12) sk−1,ℓ−1 = sk−1,ℓ = sk,ℓ−1 = skℓ =0, or
′ (4.13) zpq = gkℓ(xp,yq), p, q ΣN, with (xp,yq) D . ∀ ∈ 0 ∈ kℓ Given 1 r m, we call r is degenerate if (i, j) is degenerate for all (i, j) Λr. A connected≤ component≤ V of 1, 2,...,m is called degenerate if r is degenerate∈ for all r V . Otherwise, V is called{ non-degenerate.} ∈ By definition, a connected component V of 1, 2,...,m is non-degenerate if { } there exists (i, j) Λr : r V such that (i, j) is non-degenerate. ∈ ∪{ ∈ }
Remark 4.6. It is clear that (4.12) holds if and only if S Dkℓ = 0. Since gkℓ is bilinear, we can see that (4.13) holds if and only if the following| two conditions hold: ′ ′ (1) for each p ΣN,0 with xp Ik, points in (xp,yq,zpq): q ΣN,0,yq Jℓ are collinear;∈ and ∈ { ∈ ∈ } ′ ′ (2) for each q ΣN,0 with yq Jℓ, points in (xp,yq,zpq): p ΣN,0, xp Ik are collinear.∈ ∈ { ∈ ∈ }
Lemma 4.7. Given (i, j) ΣN ΣN , if (i, j) is degenerate, then ∈ × (1) for all (k,ℓ) A(i, j), we have f(x, y)= h(x, y), (x, y) Dkℓ, ∈ ∈ n + (2) there exists a constant C > 0, such that O(f,n,Dij ) CK for all n Z , ≤ ∈ (3) dimB Γf D =2. | ij Proof. (1). Let T and C∗([0, 1]2) be the same as defined in the proof of Theorem 2.1. Let C∗ ([0, 1]2) be the collection of all continuous functions ϕ C∗([0, 1]2) satisfying ij ∈ ϕ(x, y)= h(x, y), (x, y) Dkℓ, ∈ for all (k,ℓ) A(i, j). ∈ RECURRENT FRACTAL INTERPOLATION SURFACES 13
∗ 2 Given ϕ Cij ([0, 1] ) and (k,ℓ) A(i, j), if S Dkℓ = 0, then (4.13) holds. In this ∈ ∈ ′ | 6 case, for all (α, β) ΣN ΣN with Dαβ D , we have (α, β) A(i, j) so that ∈ × ⊂ kℓ ∈ ϕ(x, y)= h(x, y), (x, y) Dαβ. ∈ ′ Combining this with (4.13), we know that ϕ is bilinear on Dkℓ. Thus, ′ ϕ(x, y)= gkℓ(x, y), (x, y) D . ∈ kℓ Hence, using (2.8) and (3.3), we have −1 −1 −1 −1 T ϕ(x, y)= Fkℓ uk (x), vℓ (y), ϕ(uk (x), vℓ (y)) −1 −1 −1 −1 = S(x, y) ϕ u (x), v (y) gkℓ u (x), v (y) + h(x, y) k ℓ − k ℓ = h(x, y) for all (x, y) Dkℓ. In the case that S Dkℓ = 0, it is clear that we still have ∈ | ∗ 2 T ϕ(x, y)= h(x, y) on Dkℓ by using (2.8) and (3.3). Thus T is a map from Cij ([0, 1] ) ∗ 2 ∗ 2 to itself. Notice that Cij ([0, 1] ) is complete since it is closed in C ([0, 1] ). Hence f C∗ ([0, 1]2). ∈ ij (2) and (3) directly follow from (1).
Given a matrix X = (Xij )n n, we say X is non-negative if Xij 0 for all i and × ≥ j. X is called strictly positive if Xij > 0 for all i and j. The following lemma is well known. Please see [23] for details.
Lemma 4.8 (Perron-Frobenius Theorem). Let X = (Xij )n×n be an irreducible non-negative matrix. Then (1) ρ(X), the spectral radius of X, is an eigenvalue of X and has strictly posi- tive eigenvector. (2) ρ(X) increases if any element of X increases. Given three points (x(k),y), k =1, 2, 3 in [0, 1]2 with (x(2) x(1))(x(3) x(2)) > 0, we denote − − (1) (2) (3) (2) (1) (3) df (x , x , x ; y)= f(x ,y) λf(x ,y)+(1 λ)f(x ,y) , | − − | where λ = (x(3) x(2))/(x(3) x(1)). By definition, (x(k),y,f(x(k),y)), k = 1, 2, 3 − −(1) (2) (3) (k) are collinear if and only if df (x , x , x ; y) = 0. Furthermore, if all of (x ,y), 2 (1) (2) (3) k =1, 2, 3 lie in a subset E of [0, 1] , then O(f, E) df (x , x , x ; y). The following result is the key lemma to obtain≥ the exact box dimension of bilinear RFISs. Lemma 4.9. Let V be a non-degenerate connected component of 1, 2,...,m . If { } ρ(G V ) >K, then for all r V , | ∈ O(f,n,B ) (4.14) lim r = . n→∞ Kn ∞
Proof. Denote by nV the cardinality of V , and assume that V = r ,...,rn . { 1 V } Firstly, we show that following claim holds: for all 1 k nV , there exist (k) (k) (k) ≤ ≤ (i , j ) Λr and three points (xk,τ ,y ), τ =1, 2, 3 in D (k) (k) , such that ∈ k i ,j (k) δk := df (xk,1, xk,2, xk,3; y ) > 0. 14 ZHENLIANGANDHUO-JUNRUAN
∗ ∗ ∗ Since V is non-degenerate, there exist t V and (i , j ) Λt∗ such that ∗ ∗ ∈ ∗ ∗ ∈ (i , j ) is non-degenerate. Thus there exists (k0,ℓ0) A(i , j ) such that both (4.12) and (4.13) do not hold. By Remark 4.6, we can∈ assume without loss of ′ generality that there exist three interpolation points (xp ,yq,zp ,q) 1≤i≤3 in D { i i } k0ℓ0 with p1 < p2 < p3, such that they are not collinear. That is, if we denote λ0 = (xp xp )/(xp xp ), then 3 − 2 3 − 1 δ := df (xp , xp , xp ; yq)= zp ,q (λ zp ,q + (1 λ )zp ,q) > 0. 0 1 2 3 | 2 − 0 1 − 0 3 | ′ Let r0 and t0 be elements in 1, 2,...,m satisfying (k0,ℓ0) Λr0 Λt0 . Let α { } ∈ ∩ ′ and β be elements in 0, 1,...,N K satisfying [xα, xα+K ] [yβ,yβ+K]= D . { − } × k0ℓ0 Notice that gij are the same for all (i, j) Λr (α, β), since they are the same ∈ 0 bilinear function passing through the four points (xα,yβ,zα,β), (xα,yβ+K,zα,β+K), (xα+K ,yβ,zα+K,β) and (xα+K ,yβ+K,zα+K,β+K). We denote it by gαβ. For (i, j) Λr (α, β), we define θij = 1 if S is nonnegative on Dij , and define ∈ 0 θij = 1 otherwise. Then S(x, y) = θij S(x, y) for all (x, y) Deij . Combining this with− (2.10) and (3.3), | | ∈
θij f(ui(x), vj (y)) = S(ui(x), vj (y)) f(x, y) gij (x, y) + θij h(ui(x), vj (y)) | | − ′ for (x, y) D = [xα, xα K ] [yβ,yβ K]. By Lemma 4.3, ∈ ij + × + θij f(ui(x), vj (y))
(i,j)∈XΛr0 (α,β)
=γr t f(x, y) gαβ(x, y) + θij h(ui(x), vj (y)) 0 0 − (i,j)∈XΛr0 (α,β) e for all (x, y) [xα, xα K ] [yβ,yβ K]. On the other hand, since both gαβ and ∈ + × + h Dij , (i, j) Λr0 (α, β) are bilinear, we have | ∈ e gαβ(xp ,yq)= λ gαβ(xp ,yq)+(1 λ )gαβ(xp ,yq), and 2 0 1 − 0 3 h(ui(xp ), vj (yq)) = λ h(ui(xp ), vj (yq)) + (1 λ )h(ui(xp ), vj (yq)) e 2 e 0 1 e − 0 3 for all (i, j) Λr (α, β). Thus ∈ 0
df (ui(xp1 ),ui(xp2 ),ui(xp3 ); vj (yq))
(i,j)∈XΛr0 (α,β)
θij f(ui(xp ), vj (yq)) λ θij f(ui(xp ), vj (yq)) ≥ 2 − 0 1 (i,j)∈XΛ (α,β) (i,j)∈XΛ (α,β) r0 r0
(1 λ ) θij f(ui(xp ), vj (yq)) − − 0 3 (i,j)∈XΛ (α,β) r0
= γr t f(xp ,yq) λ f(xp ,yq)+(1 λ )f(xp ,yq) = γr t δ . 0 0 2 − 0 1 − 0 3 0 0 0 As a result, there exists (i, j) Λr0 (α, β) such that ∈
df (ui(xp1 ),ui(xp2 ),ui(xp3 ); vj (yq)) > 0.
We remark that (ui(xpτ ), vj (yq)), τ =1, 2, 3 are three points in Dij . ∗ ∗ ∗ ∗ ∗ ′ From (i , j ) Λt and (k0,ℓ0) A(i , j ) Λr0 Λt0 , there is a path from t0 ∗ ∈ ∈ ∩ ∩ to t . Since V is connected, we know that for all 1 k nV , there is a path from ∗ ≤ ≤ t to rk so that there is a path from t0 to rk. Similarly as above, we can prove by induction that the claim holds. Now we will show (4.14) holds if ρ(G V ) >K. | RECURRENT FRACTAL INTERPOLATION SURFACES 15
T Let ξ = (ξ , ξ ,...,ξn ) be a strictly positive eigenvector of G V with eigenvalue 1 2 V | ρ(G V ) such that ξk δk for all 1 k nV . | ≤ +≤ ≤ (n) Given 1 k,ℓ nV and n Z , we denote by V (rk, rℓ) the set of all n-paths ≤ ≤ ∈ P (n) in V from rk to rℓ. Given −→r = r(0), r(1),...,r(n) V (rk, rℓ), we denote { }∈Pn+1 n+1 by (−→r ) the set of all elements (i0i1 in, j0j1 jn) in ΣN ΣN satisfying Q (k) (k) ′ ··· ··· × (i , j ) = (i , j ), Di − ,j − D and (iτ , jτ ) Λ for all 1 τ n. 0 0 τ 1 τ 1 ⊂ iτ ,jτ ∈ r(τ) ≤ ≤ Given (i, j) = (i i in, j j jn) ( r ) and (x, y) Di j , we define 0 1 ··· 0 1 ··· ∈Q −→ ∈ 0 0 ui(x)= ui ui − ui (x), vj(y)= vj vj − vj (y). n ◦ n 1 ◦···◦ 1 n ◦ n 1 ◦···◦ 1 Given 1 k,ℓ nV and n 1, for each −→r = r(0), r(1), , r(n) (n) ≤ ≤ ≥ { ··· } ∈ (rk, rℓ), similarly as above, we have PV n (k) df (ui(xk,1),ui(xk,2),ui(xk,3); vj(y )) = δk γr(t),r(t−1) (i,j)X∈Q(−→r ) tY=1 so that
O f, (ui vj)(D (k) (k) ) × i ,j −→ (Xn) (i,j)X∈Q(−→r ) r ∈PV (rk,rℓ) n δk γ ≥ r(t),r(t−1) X (n) tY=1 {r(0),...,r(n)}∈PV (rk,rℓ) n n = δk γ = δk (G V ) . r(t),r(t−1) | ℓk r(1),...,rX(n−1)∈V tY=1 r(0)=rk,r(n)=rℓ As a result,
O(f,n,Brℓ ) O f, (ui vj)(Di(k) ,j(k) ) ≥ −→ × 1≤Xk≤nV −→ (Xn) (i,j)X∈Q( r ) r ∈PV (rk,rℓ) n (G V ) δk. ≥ | ℓk 1≤Xk≤nV Thus T n T O(f,n,Br ),...,O(f,n,Br ) (G V ) (δ1,...,δn ) 1 nV ≥ | V n T (G V ) (ξ ,...,ξn ) ≥ | 1 V n T (ρ(G V )) (ξ ,...,ξn ) . ≥ | 1 V Hence for all 1 k nV , we have ≤ ≤ n O(f,n,Br ) (ρ(G V )) ξk. k ≥ | The lemma follows from ρ(G V ) >K. | For each 1 r m, we define ≤ ≤ A∗(r)= t 1,...,m : t r and there exists a path from t to r . { ∈{ } 6∼ } Then we recurrently define P (r), r =1, 2,...,m as follows: P (r) = 1 if A∗(r)= , and P (r) = 1+max P (t): t A∗(r) if A∗(r) = . We call P (r) the position of ∅r. Now we can use Lemmas{ 4.4∈ and 4.9} to obtain6 the∅ exact box dimension of bilinear RFISs under certain constraints. The spirit of the proof follows from [4, 13, 20]. 16 ZHENLIANGANDHUO-JUNRUAN
Theorem 4.10. Let f be the bilinear RFIF determined in the section 3 with con- ditions (4.3), (4.4) and (4.5). Assume that the vertical scaling factors sij : i, j ′ { ∈ ΣN,0 are steady and have uniform sums γrt :Λr Λt = under a compatible } m {∗ ∩ 6 ∅} partition = Br r=1. Let V1, V2,...,Vnc be the set of all non-degenerate con- B { } { } ∗ nected components of 1, 2,...,m . Define di = log ρ(G V )/ log K for 1 i n { } | i ≤ ≤ c and let d = max d ,...,dn∗ , 1 . Then dimB(Γf)=1+ d . ∗ { 1 c } ∗ Proof. Denote ρ = Kd∗ . Firstly, we will prove dim (Γf) 1+ d . It is clear 0 B ≥ ∗ that dimB(Γf) 2 since f is continuous. Thus we only need to consider the case ≥ ∗ d > 1. Let i be an element in 1,...,n satisfying di = d . Assume that ∗ 0 { c } 0 ∗ Vi = r1,...,rq for some 1 q m, where r1 < < rq. Then ρ0 = ρ(G V ). 0 { } ≤ ≤ ··· | i0 Let ξ = (ξ ,...,ξ )T be a strictly positive eigenvector of G with eigenvalue ρ . 1 q Vi0 0 Then | q
γrk,rℓ ξℓ = ρ0ξk Xℓ=1 for all 1 k,ℓ q. From Lemma 4.4, there exists a constant C > 0 such that ≤ ≤ m n O(f,n +1,Br ) γr ,tO(f,n,Bt) CK k ≥ k − Xt=1 + for all 1 k q and n Z . Choose a constant C1 > 0 such that C1ξk C for all k. Then≤ for≤ 1 k q∈and n Z+, we have ≥ ≤ ≤ ∈ q n (4.15) O(f,n +1,Br ) γr ,r O(f,n,Br ) C ξkK . k ≥ k ℓ ℓ − 1 Xℓ=1
d∗ It follows from d∗ > 1 that ρ0 = K >K. Thus, from Lemma 4.9, we can pick n Z+ large enough and C > 0 small enough such that 0 ∈ 2 n0 n0 C1K O(f,n ,Br ) C ρ ξk + ξk 0 k ≥ 2 0 ρ K 0 − for 1 k q. Combining this with (4.15), we know that for 1 k q, ≤ ≤ ≤ ≤ q q n0 n0 C1K n0 O(f,n0 +1,Brk ) C2ρ0 γrk,rℓ ξℓ + γrk,rℓ ξℓ C1ξkK ≥ ρ0 K − Xℓ=1 − Xℓ=1 C Kn0+1 = C ρn0+1ξ + 1 ξ . 2 0 k ρ K k 0 − By induction, we can obtain that for all 1 k q and n n0, ≤ ≤n ≥ n C1K nd∗ O(f,n,Br ) C ρ ξk + ξk C K ξk. k ≥ 2 0 ρ K ≥ 2 0 − + Notice that O(f,n) O(f,n,Br ) for all n Z . Thus using Lemma 4.1, we have ≥ 1 ∈ dimB(Γf) 1+ d∗. Now we≥ will show that the following claim holds: for all 1 r m and all ≤ ≤ δ > 0, there exists C > 0 such that n (4.16) e O(f,n,Br) C(ρ0 + δ) ≤ for all n Z+. e ∈ We will prove this by using induction on P (r). Denote Pmax = max P (r): 1 r m . { ≤ ≤ } RECURRENT FRACTAL INTERPOLATION SURFACES 17
In the case that P (r) = 1, we have A∗(r) = . If r is degenerate, then from d∗ ∅ Lemma 4.7 (2) and ρ0 = K K, we know that the claim holds. Thus we can assume that r is non-degenerate.≥ Combining this with P (r) = 1, there exists a non-degenerate connected component Vi such that r Vi. Assume that Vi = ∈ r1,...,rq for some 1 q m, where r1 < < rq. Denote ρi = ρ(G Vi ). Let { } T ≤ ≤ ··· | η = (η1,...,ηq) be a strictly positive eigenvector of G Vi with eigenvalue ρi. It ∗ | follows from A (r) = that γrk,t = 0 for all 1 k q and t Vi. Thus, from Lemma 4.4, there exists∅ a constant C > 0 such that≤ ≤ 6∈ q n O(f,n +1,Br ) γr ,r O(f,n,Br )+ CK k ≤ k ℓ ℓ Xℓ=1 + for all 1 k q and n Z . Choose a constant C1 > 0 such that C1ηk C for all 1 k≤ q.≤ Then for all∈ 1 k q and n Z+, ≥ ≤ ≤ ≤ ≤ ∈ q n (4.17) O(f,n +1,Br ) γr ,r O(f,n,Br )+ C ρ ηk. k ≤ k ℓ ℓ 1 0 Xℓ=1
Arbitrarily pick δ > 0. Let C2 be a positive constant such that −1 O(f, 1,Br ) C ρiηk + C δ (ρ + δ)ηk k ≤ 2 1 0 for all 1 k q. By induction and using (4.17), we have ≤ ≤ n −1 n (4.18) O(f,n,Br ) C ρ ηk + C δ (ρ + δ) ηk k ≤ 2 i 1 0 for all 1 k q and n Z+, where we use ≤ ≤ ∈ q n −1 n n γrk,rℓ C2ρi ηℓ + C1δ (ρ0 + δ) ηℓ + C1ρ0 ηk Xℓ=1 n+1 −1 n n C ρ ηk + C δ (ρ + δ) ρiηk + C (ρ + δ) ηk ≤ 2 i 1 0 1 0 n+1 −1 n+1 C ρ ηk + C δ (ρ + δ) ηk. ≤ 2 i 1 0 It follows that the claim holds if P (r) = 1. Assume that the claim holds for all 1 r m satisfying P (r) P , where ≤ ≤ ≤ 1 P < Pmax. Let r be an element in 1, 2,...,m satisfying P (r) = P +1. It is≤ clear that (4.16) holds if r is degenerate.{ Thus we} can assume that r is non- degenerate. In the case that r does not belong to any connected component, we have P (t) P for any 1 t m with γrt > 0. Combining this with Lemma 4.4, there exists≤ a constant C≤ > 0,≤ such that m n O(f,n +1,Br) γrt O(f,n,Bt)+ CK , n 1. ≤ ∀ ≥ Xt=1 t: PX(t)≤P By inductive assumption, we can see that the claim holds in this case. Now we consider the case that r belongs to a connected component Vi = r1,...,rq , T { } where r < < rq. Similarly as above, let η = (η ,...,ηq) be a strictly positive 1 ··· 1 eigenvector of G Vi with eigenvalue ρi = ρ(G Vi ). From Lemma 4.4, there exists a constant C > 0 such| that | q n O(f,n +1,Br ) γr ,r O(f,n,Br )+ γr ,tO(f,n,Bt)+ CK k ≤ k ℓ ℓ k Xℓ=1 t:PX(t)≤P 18 ZHENLIANGANDHUO-JUNRUAN for all 1 k q and n Z+. Thus, given δ > 0, by using inductive assumption, ≤ ≤ ∈ there exists a constant C1 > 0 such that q n (4.19) O(f,n +1,Br ) γr ,r O(f,n,Br )+ C ηk(ρ + δ) k ≤ k ℓ ℓ 1 0 Xℓ=1 + for all 1 k q and n Z . Similarly as above, there exists C2 > 0 such that for all 1 k≤ ℓ≤and n Z∈+, ≤ ≤ ∈ n −1 n O(f,n,Br ) C ρ ηk + C δ (ρ + δ) ηk. k ≤ 2 i 1 0 Hence the claim holds for P (r)= P + 1. By induction, the claim holds for all 1 r m. Combining this with Lemma 4.1, ≤ ≤ we can see from the arbitrariness of δ that dimBΓf B 1+ d for all 1 r m. | r ≤ ∗ ≤ ≤ Thus dimB(Γf) 1+ d . As a result, dimB(Γf)=1+ d . ≤ ∗ ∗ We can obtain the following corollary immediately. Corollary 4.1. Let f be the bilinear RFIF determined in the section 3 with con- ditions (4.3), (4.4) and (4.5). Assume that the vertical scaling factors sij : i, j ′ { ∈ ΣN,0 are steady and have uniform sums γrt :Λr Λt = under a compatible } m { ∩ 6 ∅} partition = Bi . Then, in the case that G is irreducible, we have B { }i=1 (1) dimB(Γf) = 1 + log ρ(G)/ log K if 1, 2,...,m is non-degenerate and ρ(G) >K, or { } (2) dimB(Γf)=2 otherwise.
Remark 4.11. In the case that K = N, it is clear that = B = [x , xN ] B { 1} { 0 × [y0,yN ] is the compatible partition. From Theorem 4.10, we can obtain the box dimension} of bilinear FISs under certain constraints, which is the main result in [13].
0.0 x 0.5 1.0 1.0 y 0.5 0.0 4
3
z 2
1
0
Figure 1. Bilinear RFIS in Example 4.1
4.4. An example. ′ ′ ′ ′ ′ 1 ′ ′ Example 4.1. Let N = 4, K = 2, and x0 = x2 = x4 = 0, x1 = x3 = 2 , y0 = y2 = ′ 1 ′ ′ y4 = 2 , y1 = 1, y3 = 0. By definition, we have
u1(x0)= x0, u1(x2)= u2(x2)= x1, u2(x0)= u3(x0)= x2,...,
v1(y2)= y0, v1(y4)= v2(y4)= y1, v2(y2)= v3(y2)= y2,.... RECURRENT FRACTAL INTERPOLATION SURFACES 19
1 1 1 1 1 Let B1 = [0, 2 ] [0, 2 ], B2 = [0, 2 ] [ 2 , 1] and B3 = [ 2 , 1] [0, 1]. It is easy to check × × × ′ that = B1,B2,B3 is a compatible partition with respect to Dij ,Dij ;1 i, j B { } ′ { ≤ ≤ 4 . Furthermore, Λr Λt = if and only if (r, t) (1, 2), (2, 1), (3, 1), (3, 2) . } Let (r, t)=(1, 2).∩ Then6 Λ∅(0, 2) = (1, 1), (1, 2)∈{, (2, 1), (2, 2) so that } 1 { } S(ui(x ), vj (y )) = S(x ,y ) + S(x ,y ) + S(x ,y ) + S(x ,y ) | 0 2 | | 0 0 | | 0 2 | | 2 0 | | 2 2 | (i,j)∈XΛ1(0,2) = s + s + s + s . | 00| | 02| | 20| | 22| Similarly, we have
S(ui(x ), vj (y )) = 2( s + s ), | 0 4 | | 01| | 21| (i,j)∈XΛ1 (0,2)
S(ui(x ), vj (y )) = 2( s + s ), | 2 2 | | 10| | 12| (i,j)∈XΛ1 (0,2)
S(ui(x ), vj (y )) =4 s . | 2 4 | | 11| (i,j)∈XΛ1 (0,2)
Assume that vertical scaling factors sij : 0 i, j 4 have uniform sums under the compatible partition B ,B ,B .{ Then we≤ must≤ have} { 1 2 3} γ = s + s + s + s = 2( s + s )=2( s + s )=4 s . 12 | 00| | 02| | 20| | 22| | 01| | 21| | 10| | 12| | 11| Similarly, γ = s + s + s + s = 2( s + s )=2( s + s )=4 s , 21 | 02| | 04| | 22| | 24| | 03| | 23| | 12| | 14| | 13| γ = s + s + s + s = 2( s + s )=2( s + s )=4 s , 31 | 22| | 24| | 42| | 44| | 23| | 43| | 32| | 34| | 33| γ = s + s + s + s = 2( s + s )=2( s + s )=4 s . 32 | 20| | 22| | 40| | 42| | 21| | 41| | 30| | 32| | 31| Let 0.85 0.9 0.95 0.9 0.9 0.2 0.45 0.8 0.7 0.6 (s ) = 0 0 00.5 0.95 . ij 0≤i,j≤4 0.4 0.2 0 0.3 0.6 − − 0.8 0.4 0 0.1 0.25 − − By above discussion, we can check that vertical scaling factors are steady and have uniform sums under B1,B2,B3 with γ12 =1.8, γ21 =2.8, γ31 =1.2 and γ32 =0.8. Assume that { } 23212 22313 (z ) = 13231 . ij 0≤i,j≤4 32420 23244 The corresponding bilinear RFIS is shown in Figure 1. Furthermore, we can check that V1 = 1, 2 is the unique non-degenerate con- nected component of 1, 2, 3 with { } { } 0 1.8 G V = . | 1 2.8 0
Thus d∗ = log 5.04/(2 log 2) so that dimB(Γf) = 1 + log 5.04/(2 log 2). 20 ZHENLIANGANDHUO-JUNRUAN
Acknowledgments
The authors wish to thank Dr. Qing-Ge Kong and Professor Yang Wang for helpful discussions.
References 1. B. Bar´ any,´ M. Rams and K. Simon, Dimension of the repeller for a piecewise expanding affine map, arXiv: 1803.03788v2. 2. M. F. Barnsley, Fractal functions and interpolation, Constr. Approx., 2 (1986), 303–329. 3. M. F. Barnsley, Fractals Everywhere, second ed., Academic Press Professional, Boston, 1993. 4. M. F. Barnsley, J. H. Elton, and D. P. Hardin, Recurrent iterated function systems, Constr. Approx., 5 (1989), 3–31. 5. M. F. Barnsley, B. Hardin, A. Vince, and P. Viswanathan, Approximation of rough functions, J. Approx. Theory, 209 (2016), 23–43. 6. M. F. Barnsley and P. R. Massopust, Bilinear fractal interpolation and box dimension, J. Approx. Theory, 192 (2015), 362–378. 7. P. Bouboulis and L. Dalla, A general construction of fractal interpolation functions on grids of Rn, Euro. J. Appl. Math., 18 (2007), 449–476. 8. P. Bouboulis, L. Dalla and V. Drakopoulos, Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension, J. Approx. Theory, 141 (2006), 99–117. 9. A. K. B. Chand, S. K. Katiyar, and P. V. Viswanathan, Approximation using hidden variable fractal interpolation function, J. Fractal Geom., 2 (2015), 81–114. 10. L. Dalla, Bivariate fractal interpolation functions on grids, Fractals, 10 (2002), 53–58. 11. K. J. Falconer, Fractal Geometry: Mathematical Foundation and Applications, New York, Wiley, 1990. 12. Z. Feng, Variation and Minkowski dimesnsion of fractal interpolation surfaces, J. Math. Anal. Appl., 345 (2008), 322–334. 13. Q.-G. Kong, H.-J. Ruan and S. Zhang, Box dimension of bilinear fractal interpolation surfaces, Bull. Aust. Math. Soc., 98 (2018), 113–121. 14. Z. Liang and H.-J. Ruan, Recurrent fractal interpolation surfaces on triangular domains, Fractals, 27 (2019), 1950085, 12 pages. 15. R. Malysz , The Minkowski dimension of the bivariate fractal interpolation surfaces, Chaos Solitons Fractals, 27 (2006), 1147–1156. 16. P. R. Massopust, Fractal surfaces, J. Math. Anal. Appl., 151 (1990), 275–290. 17. D. S. Mazel and M. H. Hayes, Using iterated function systems to model discrete sequences, IEEE Trans. Signal Process, 40 (1992), 1724–1734. 18. W. Metzler and C. H. Yun, Constuction of fractal interpolation surfaces on rectangular grids, Internat. J. Bifur. Chaos, 20 (2010), 4079–4086. 19. M. A. Navascues´ , Fractal approximation, Complex Anal. Oper. Theory, 4 (2010), 953–974. 20. H.-J. Ruan, Z. Sha and M.-D. Ye, Correction of a theorem of Barnsley-Elton-Hardin (in Chinese), Appl. Math. J. Chinese Univ. Ser. A, 15 (2000), 157–162. 21. H.-J. Ruan, W.-Y. Su and K. Yao, Box dimension and fractional integral of linear fractal interpolation functions, J. Approx. Theory, 161 (2009), 187–197. 22. H.-J. Ruan and Q. Xu, Fractal interpolation surfaces on rectangular grids, Bull. Aust. Math. Soc., 91 (2015), 435–446. 23. R.S. Varga, Matrix iterative analysis (second edition), Springer-Verlag, Berlin, 2000. 24. S. Verma and P. Viswanathan, A fractal operator associated to bivariate fractal interpola- tion functions, arXiv: 1810.09701. 25. J. L. Vehel,´ K. Daoudi and E. Lutton, Fractal modeling of speech signals, Fractals, 2 (1994), 379–382. 26. H. Y. Wang, and J. S. Yu, Fractal interpolation functions with variable parameters and their analytical properties, J. Approx. Thoery, 175 (2013), 1–18. 27. C.-H. Yun, H.-C. Choi and H.-C. O, Construction of recurrent fractal interpolation surfaces with function scaling factors and estimation of box-counting dimension on rectangular grids, Fractals, 23 (2015), 1550030, 10 pp. RECURRENT FRACTAL INTERPOLATION SURFACES 21
Department of Statistics, Nanjing Audit University, Nanjing 211815, China E-mail address: [email protected]
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China E-mail address: [email protected]