Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors
BBox-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors
Chol-Hui Yun , Mi-Kyong Ri
2) 1), 2)
Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea * [email protected] Abstract: We estimate the bounds of box-counting dimension of hidden variable fractal interpolation functions (HVFIFs) and hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors and present analytic properties of HVFIFs which are constructed to ensure more flexibility and diversity in modeling natural phenomena. Firstly, we construct the HVFIFs and analyze their smoothness and stability. Secondly, we obtain the lower and upper bounds of box-counting dimension of the HVFIFs. Finally, in the similar way, we get the lower and upper bounds of box-counting dimension of HVBFIFs constructed in [21]. Keywords: Iterated function system (IFS), Fractal interpolation function, Smoothness, Stability, Hidden variable, Function contractivity factor, Box-counting dimension. AMS Subject Classification: 28A80, 81Q35, 41A05, 97N50 Introduction
Fractal interpolation function (FIF) is an interpolation function whose graph is a fractal set. In 1986, M. F. Barnsley [1] introduced a concept of FIF to model better natural phenomena which are irregular and complicated and the FIFs have been widely studied ever since in many papers. [2-15, 17-22] In general, to get the FIF, we construct an iterated function system (IFS) on the basis of a given data set and then define a Read-Bajraktarevic operator on some space of continuous functions. A fixed point of the operator is an interpolation function of the data set and its graph is an attractor of the constructed IFS. The FIFs are not differentiable and fractal dimensions of their graphs are not integers. The fractal dimensions [10, 14, 15, 19 and 20], smoothness [5, 10, 11, 12 and 22] and stability [6, 7, 11, 13, 21 and 22] of the FIFs have been studied. M.F. Barnsley et al. [2] introduced a concept of hidden variable fractal interpolation function (HVFIF) which is more complicated, diverse and irregular than the FIF for the same set of interpolation data. P. Bouboulis and L. Dalla [3] constructed hidden variable vector valued FIFs on grids in R . Many researchers have studied the HVFIFs. [2-6, 12, 13, 15, 17, 18, 21 and 22] The idea of the construction of the HVFIF is to extend the given data set on R into a data set on R , make a vector valued fractal interpolation function for the extended data set and then project the vector valued function onto R , which gives the HVFIF. It is usually non self-affine, because the HVFIF is the projection of a vector valued function. In [2], authors studied the HVFIF with three free parameters: free variable, constrained free variable, and free hidden variable (they are also called ontractivity factors). Changing hidden variables, we can control shapes and fractal dimensions of the graphs of HVFIFs more flexibly. A. K. B. Chand and G. P. Kapoor [5] studied a coalescence hidden variable fractal interpolation function using the IFS with free parameters and constrained free parameters. In many articles, authors have studied smoothness [5, 12 and 22], stability [6, 13, 21 and 22] and fractal dimension [5, 15] of HVFIFs. The constructions of FIFs and HVFIFs with constant contractivity factors lack the flexibility which is necessary to model complicated and irregular natural phenomena.[1-7, 10, 12, 13, 15 and 18] A lot of fractal objects in nature have different contractivity factors at each point. Therefore, using function contractivity factors, we can model better mathematically fractal objects in nature. In many papers, the construction and analytic properties of FIFs [9, 11, 14, 19, and 20] and HVFIFs [16, 21, and 22] with function contractivity factors have been studied. The authors in [5, 15] estimated the lower and upper bounds of fractal dimension of HVFIF with three constant contractivity factors. In order to ensure the flexibility and diversity of the methods of constructing FIFs, R. Uthayakumar and M. Rajkumar [17] presented a construction of HVFIF with function contractivity factors, where only one free variable is a function and the constrained free variable and free hidden variable are all constants. C.H. Yun and M.K. Li introduced a construction of hidden variable bivariate fractal interpolation functions (HVBFIFs) with four function contractivity factors [21] and studied their analytic properties. [22] The main aim of the present paper is to find a lower and upper bounds of the fractal dimension of HVFIFs and HVBFIFs (constructed in [21]) with four function contractivity factors. To do this, first of all, we construct HVFIFs with four function contractivity factors and analyze their smoothness and stability, which are similar to ones in [21, 22]. Next, we estimate the box-counting dimension of the HVFIFs and HVBFIFs. The remainder of this paper is organized as follows: In section 2, IFSs with function contractivity factors for the extended data set are constructed (Theorem 1) and using the Read-Bajraktarebic operator, vector valued fractal interpolation functions whose graphs are attractors of the IFSs are constructed. (Theorem 2) Then projecting them onto R , HVFIFs for the given data set are constructed. In section 3, we consider smoothness (Theorem 3) and stability (Theorem 4-7) of the constructed HVFIFs. In section 4, the lower and upper bounds of fractal dimension of the HVFIFs are given (Theorem 8). In section 5, we introduce the result for the box-counting dimension of HVBFIFs constructed in [21].
2. Construction of hidden variable fractal interpolation function(HVFIF).
In this section, we construct hidden variable fractal interpolation functions with four function contractivity scaling factors. Let a data set P in R be given by )(},,,1,0;R),{( nii xxxniyxP . To construct a hidden variable fractal interpolation function of this data set, we extend the data et P into a data set P in R as follows: )(},,,1,0;R),(),,{( niiiii xxxniyxzyxP , where ),( iii zyy and i z are parameters. Let us denote },,2,1{ nN n , ],[ iii xxI and ],[ n xxI . Let mappings ],[],[: iini xxxxL for n Ni be contraction homeomorphisms such that they map the end points of the interval I into the end points of the subinterval i I , i.e. },{}),({ iini xxxxL . We denote Lipschitz constant of Lipschitz mapping f by )( ff cL . We define mappings RR: IF i for },,1{ ni by )(~))((~))((~ )())(())((),( xqzxLsyxLs xqzxLsyxLsyxF iiiii iiiiii = )(~ )())((~))((~ ))(())(( xq xqzyxLsxLs xLsxLs iiiiii iiii , where functions RIssss iiiii :~,~,, are arbitrary Lipschitz functions whose absolute values are less than 1 and i q , i q ~ : RI are Lipschitz functions defined as follows: ))(()())(()())(()( xLhxgxLsxgxLsxq iiiiiiiii , ))((~)())((~)())((~)(~ xLhxgxLsxgxLsxq iiiiiiiii , where i g , i g : RI and i h , i h ~ : RI i are Lipschitz functions such that yxg i )( , zxg i )( , },0{ n , aai yxh )( , aai zxh )(~ , },1{ iia , respectively. For example, they can be constructed as the following Lagrange interpolation polynomials: )( yxx xxyxx xxxg nnnni , )( zxx xxzxx xxxg nnnni , (1) )( iii iiii ii yxx xxyxx xxxh , )(~ iii iiii ii zxx xxzxx xxxh . Then, we have ))((~))())(((~))())(((~ ))(())())((())())(((),( xLhxgzxLsxgyxLs xLhxgzxLsxgyxLsyxF iiiiiiii iiiiiiiii and for },0{ n and },1{ iia such that ai xxL )( , we get ai yyxF ),( . It is obvious that ),( yxF i are Lipschitz mappings. Let us denote as follows: )())(())((),,( xqzxLsyxLszyxF iiiiii , )(~))((~))((~),,( xqzxLsyxLszyxF iiiiii . Let us denote )(),( xQySyxF iii , where ))((~))((~ ))(())(()( xLsxLs xLsxLsxS iiii iiiii , )(~ )()( xq xqxQ iii , zyy . Suppose that RE is an enough large bounded domain containing i y , ni ,,1 . Now we define transformations R: ii IEIW , ni ,,1 by )),(),((),( yxFxLyxW iii , ni ,,1 . Then we can prove that i W map the data points of P on the end of interval I into the data points of P on the end of interval I i . For a function f , let us denote |)(|sup xff x . We denote },,1;~,~max{ nissssS iiii . The following theorem gives conditions for i W to be contraction transformations. Theorem 1. If S , then there exists a distance equivalent to the Euclidean metric on R such that i W are contraction mappings with respect to the distance . Proof.
We take as a positive number satisfying Qs L
LL c , where },,1;max{ nicc i LL , },,1;,max{ ~~ nicLcLcLcLL iiiiiiii LsLsLsLsS , ||||sup y Dy , },,1;max{ ~ niLLL ii qqQ . |||| is a norm on R . Let us define a distance on R by ||||||||)),(),,(( yyxxyxyx , R),(),,( yxyx . It is clear that the is equivalent to Euclidean metric on R . For DIyxyx ),(),,( , we have ||),(),(||||)()(||)),(),,(( yxFyxFxLxLyxWyxW iiiiii
11 11 ||)()()()(||||)()(|| ||),(),(||||)()(|| xQyxSxQyxSxLxL yxFyxFxLxL iiiiii iiii ||))()(())()(())()((||||)()(|| xQxQyxSyxSyxSyxSxLxL iiiiiiii ||||||)((||||)()(|| yyxSxLxL iii ||||||)()(|| yxSxS ii )||)()(|| xQxQ ii |||||||| yySxxc L |||| xxL S |||| xxL Q = ||||||||))(( yySxxLLc QSL )||||||}(||),(max{ yyxxSLLc QSL = )),(),,(( yxyxc . From the hypothesis of the theorem and the condition on , it follows that L cc max{ ( QS LL ), } S <1. Therefore, i W are contraction transformations. □ Remark 1 . Even in the case when || || in the definition of is changed into || || , we get the similar result to Theorem 1. herefore, },,1,;R{ niW i is a hyperbolic iterated function system (IFS) corresponding to the extended data set P . Let us denote by A an attractor of the IFS. For the IFS, we have the following theorem. Theorem 2.
There is a continuous interpolation function f of the extended data set P such that the graph of f is the attractor of the above constructed IFS. Proof.
Let us define a set )( IC as follows: hRIhIC ;:{)( interpolates the data set P and is continuous.}. Then, it can be easily proved that the set )( IC is a complete metric space with respected to a norm |||| . For ))(( ICh , we define a mapping hT on I as follows: iiii IxxLhxLFxhT ))),((),(())(( . Then, it follows that )( IChT . In fact, for any },,1,0{ ni , there is },0{ n such that iiiiiiii yyxFxLhxLFxhT ),()))((),(())(( , where ii xxL )( and )()( xLxL . Therefore, an operator )()(: ECECT is well defined on )( EC . Moreover, it is a contraction operator, because ||)))((),(()))((),((||||))(())((|| xLhxLFxLhxLFxhTxhT iiiiii ||))(())(()())(())(()(|| xLQxLhxSxLQxLhxS iiiiiiii .|||| ||))(())((||||)(|| hhS xLhxLhxS iii Hence, from the fixed point theorem in the complete metric space, it follows that T has a unique fixed point ))(( ICf . Then, we have )))((),(()( xLfxLFxf iii . Therefore, for the graph )( fGr of f , we have nj j fGrWfGr ))(()( . This means that )( fGr is the attractor of the constructed IFS. Therefore, from the uniqueness of an attractor, we have )( fGrA . □ Let us denote the vector valued function : RIf by ))(),(( xfxff , where RIf : interpolates the given data set P and is called a hidden variable fractal interpolation function HVFIF) of the data set P . Then, a set }:))(,{( Ixxfx is a projection of A onto R . Since a projection of the attractor is not always self-affine set, the hidden variable fractal interpolation function is not self-affine fractal interpolation function in general. The function )( xf interpolates a set Pzyxzx iiiii ),,();,,{( , },,1,0 ni . As we can know from the above proof, we get iiii IxxLfxLFyxf ))),((),((),( , i.e. iiiii IxxLfxLfxLFyxf ))),(()),((),((),( . Therefore, for all Ix , we have ))(())(()())(()()( xLqxLfxsxLfxsxf iiiiii , (2) ))((~))(()(~))(()(~)( xLqxLfxsxLfxsxf iiiiii . (3) Remark 2.
The contractivity factors iii sss ~,, of the above IFS are constants and i s in [3- 6, 12, 13]. In [17], i s are functions and ii ss ~, are constants and i s . Example. Fig.1 shows the graphs of HVFIFs constructed from a data set P ={(1, 20), (0.25, 30), (0.5, 10), (0.75, 50), (1, 40)} with different contractivity factor functions. The contractivity factors { ,,, ssss }, }~,~,~,~{ ssss , },,,{ ssss , }~,~,~,~{ ssss are as follows: (a) {0.3, 0.85, 0.8, 0.5}, {0, 0, 0, 0}, {0.8, 0.6, 0.4, 0.5 }, {0.19, 0.37, 0.48, 0.43}, (b) {0.3, 0.85, 0.8, 0.5}, {0.64, 0.14, 0.19, 0.49}, {0.8, 0.6, 0.4, 0.5 }, {0.19, 0.37, 0.48, 0.43}, (c) {sin( x ), cos(30 x ), sin( x ), cos(5 x )}, {0, 0, 0, 0}, {2.9x, 1.9 x , x , x }, {0.9-2.9 x , 0.95-1.9 x , 0.9- x , 0.99- x }, (d) {sin( x ), cos(30 x ), sin( x ), cos(5 x )}, {0.9-| sin( x )|, 0.89-| cos(30 x )|, 0.94-| sin( x )|, 0.9-| cos(5 x )|}, {2.9 x , 1.9 x , x , x }, {0.9-2.9 x , 0.95-1.9 x , 0.9- x , 0.99- x }.
500 1000 1500 200040202040
500 1000 1500 20005050 (a) (b)
500 1000 1500 20002020406080
500 1000 1500 20002020406080 (c) (d)
Fig.1. Hidden variable fractal interpolation functions.
3. Smoothness and stability of the HVFIFs.
In this section, we analyze the smoothness and stability of the constructed HVFIF. Let ]1,0[ I and ii IIL : be similarities with Lipschitz constants i L L . Let us denote as follows: |)(|max xgg x , }{max i sis LL , }{max i sis LL , }{max ~ i sis LL , }{max ~ i sis LL , }{max i qiq LL , }{max ~~ i qiq LL , |}{|min|| min ii II , |}{|max|| max ii II , |})(||,)({|max xsxs kkIxk , |})(~||,)(~{|max~ xsxs kkIxk , }~{max~},{max kkkk . Lemma 1 . ([21]) If , then for x obeying x , exx . Theorem 3.
Let )( xf be the HVFIF constructed in the Theorem 2 and ||2 ||}~,max{ maxmin II . Then there exist a positive constant L and with such that |||)()(| xxLxfxf . (4) Proof.
Let us denote )( ... ILLLI rrrrrr mmm , II . Then, for any x , ]1,0[ x with xx , there is Nm such that ],[ ... xxI m rrr mk rr I ... , kNrrr nm and we get |)))((()))(((||)()(| xLLfxLLfxfxf mmmm rrrr |)))(())(()( ))(()(()))(())(()())(()((|
112 1111211 xLqxLfxs xLfxsxLqxLfxsxLfxs mmmm mmmmmmmm rrrr rrrrrrrr |,))(())((| |))(()())(()(||))(()())(()(|
11 12121111 xLqxLq xLfxsxLfxsxLfxsxLfxs mmmm mmmmmmmm rrrr rrrrrrrr (5) and |))(()())(()(| xLfxsxLfxs mmmm rrrr |))(()())(()(||))(()())(()(| xLfxsxLfxsxLfxsxLfxs mmmmmmmm rrrrrrrr |))((||)()(||))(())((||)(| xLfxsxsxLfxLfxs mmmmmm rrrrrr |||||||))(())((||)(| fxxLxLfxLfxs mrmmm srrr (6) |))(()())(()(| xLfxsxLfxs mmmm rrrr |))(()())(()(||))(()())(()(| xLfxsxLfxsxLfxsxLfxs mmmmmmmm rrrrrrrr |))((||)()(||))(())((||)(| xLfxsxsxLfxLfxs mmmmmm rrrrrr |||||||))(())((||)(| fxxLxLfxLfxs mrmmm srrr (7) |))(())((| xLqxLq mmmm rrrr |)()(| xLxLL mmmr rrq = |||| xxIL mmr rq . (8) Therefore, by (6), (7) and (8), it follows that |||||||||||||)()(| xxILfLfLxfxf mmrmrmr rqss |))(())((||)(||))(())((||)(| xLfxLfxsxLfxLfxs mmmmmm rrrrrr = || xxM m r |)))(())((||))(())(((| xLfxLfxLfxLf mmmmm rrrrr , where k M = |||||||||| kqss ILfLfL kkk . (9) Since mkm rrrrr IxxI ...... ],[ , we have mrrrrrm IIIIxxIII mmk |||||||||||||||| min2max . (10) Let us denote as follows: m rrr L := m rrr LLL , || }~,max{2max: k kkk I , k M ~ := |||||||||| ~2~1~ kqss ILfLfL kkk , }~,{max: kkk MMM . (11) Then, by induction, we have |)))((()))(((||))(())((| xLLfxLLfxLfxLf mmmmmmmm rrrrrrrr |)))(())((||))(())(((|)()( xLfxLfxLfxLfxLxLM mmmmmmmmmmmm rrrrrrrrrrrr |).))(())((||))(())(((||| xLfxLfxLfxLfxxIM mmmmmmmmmmm rrrrrrrrrrr Similarly, we get .|)))(())((||))(())(((|~||~ |))(())((| xLfxLfxLfxLfxxIM xLfxLf mmmmmmmmmmm mm rrrrrrrrrrr rr
Then, |)))(())((||))(())(((|~||~ |)))(())((||))(())(((||||| |)()(| xLfxLfxLfxLfxxIM xLfxLfxLfxLfxxIMxxM xfxf mmmmmmmmmmm mmmmmmmmmmmmm rrrrrrrrrrr rrrrrrrrrrrrr )|))(())((| |))(())(()(|~(||||2||
11 1111 xLfxLf xLfxLfxxMIxxM mmmm mmmmmmmm m rrrr rrrrrrrr r |))(())((||))(())(((| )~)(~(|||| ~||2||2 xLfxLfxLfxLf xxMIIMIM mmmmmmmmmmmm mmmmmm mmm mm m rrrrrrrrrrrr rrrrrr rrr rr r |||| ~~2|| ~||2||2 xxMIMIIMIM mk r rrrr rr rrr rr r k kkmm mm mmm mm m |)))(())((||))(())(((|)~( xLfxLfxLfxLf mmmmmkk rrrrrrrrrmk rr )||||2||||2()~(||)1( ffxxM mkk rmk rrm ||)||||||(||2||)1( )~()||||||(||2||)1( mmk rm mk rrm k kk IffxxM ffxxM |||||| )||||||(||2||)1( xxII ffxxM mrrm || xxD mk , (12) where )||||||(||2,max L l ffMD . (1) If , then by (12), we have ||1 1||11|| xxDxxDxxD mmk . Hence, ||1 1|)()(| xxDxfxf . Therefore, let us denote DL , , then |||)()(| xxLxfxf . (2) If , then )1( mDD mk by (12) and m Ixx by (10). Therefore, ||ln ||ln2 max
I xxm . This gives ||||ln ||ln1||)1(|)()(| max11 xxI xxDxxmDxfxf |||||ln| ||ln|||| xxI xxxxxxD , where . Since xx , by Lemma1, we have |)()(| xfxf |||||ln||| xxIe DxxD . Let us denote |||ln| 11 max1 IeDL , . Then |||)()(| xxLxfxf . 3) If , then mmmmmk DDDD . (13) Let us choose such that L L . Then |||| xxxx m . (14) In fact, we get xxmxx xx m and by (10), it follows that L Lmxx ln)2(||ln . Hence, maxmax
IIm mxxm , ||ln |)|ln( xx xx m max I . Therefore, by (12), (13) and (14), choosing )1/( DL and gives |||)()(| xxLxfxf . □ Remark 3 . For the interpolation function )( xf , we can prove similarly that (4) holds true, i.e. there exist positive number L and such that for any ,, Ixx |||)()(| xxLxfxf Next let us consider the stability of the HVFIF. Firstly, we mention the stability of the HVFIF according to perturbation of each coordinate of points in the data set P , and then all coordinates. Let ]1,0[ I . For nn xxxxx **1*00 , let us denote ],[ ** 1* iii xxI and define mapping IIR : as follows: )()(
11* 1** 1 iii iii xxxx xxxxR , for i Ix . Then * )( ii IIR . Now, let us denote as follows: .~~,,~~ ,~~,,, RqqRqqRss RssRssRssLRL iiiiii iiiiiiii . Let us denote by *1 f the HVFIF constructed using the construction in section 2 with (1) and an extended data )}(,,1,0;R),,{( **1*003* * nniiix xxxxxnizyxP . Theorem 4 . Let f , *1 f be HFIF for the extended data sets P and * x P , respectively, and |2 ||}~,max{ maxmin II . Then },max{,||max~1 )~1()~1(|||| iiiqq xxLLLLff , where L and are constants in Theorem 3 and L and are constants in Remark 3. We abbreviate the proof of Theorem 5. This can be proved in the similar way of Theorem 3 in [21]. Let us denote by *1 f the HVFIF constructed using the construction in section 2 for an extended data set },,1,0;R),,{( * nizyxP iiiy . Theorem 5.
Let f , *1 f be HFIF for the extended data sets P and * y P , respectively, and ||2 ||}~,max{ maxmin II . Then ||max~1 ~21|||| **11 iii yyff . Let us denote by *1 f the HVFIF constructed using the construction in section 2 for an extended data set },,1,0;R),,{( * nizyxP iiiz .Then, the following theorem holds. Theorem 6 . Let f , *1 f be HFIF for the extended data sets P and * z P , respectively, and ||2 ||}~,max{ maxmin II . Then ||max~1 ~21|||| **11 iii zzff . Let us denote by *1 f the HVFIF constructed using the construction in section 2 for an extended data set },,1,0;R),,{( nizyxP iii . Then from the theorem 4, 5 and 6, we have the following theorem. Theorem 7.
Let f , *1 f be HFIF for the extended data sets P and * P , respectively, and ||2 ||}~,max{ maxmin II . Then ~1 |)|max||max)(~21(||max)]())(~1[(|||| ***~21*11 iiiiiiiiiqq zzyyxxLLLLff . . Box-counting dimension of the HVFIF In this section, we get a lower and upper bounds for the box-counting dimension of the graph of HVFIF in the case where the data set is },,1,0;R),{( niyin xxxP in , the extended data set is },,1,0;R),,{( nizyin xxxP iin and xsxsxsxs iiii , niIx ,2,1, in (2) and (3) in section 2. Let us denote the graph of the HVFIF f by )( fGr . As usual, the box-counting dimension of the set A by A B dim is defined by log )(loglimdim ANA B (if this limit exists), where )( AN is any of the following(see [7]): (i) the smallest number of closed balls of radius that cover the set A ; (ii) the smallest number of cubes of side that cover the set A ; (iii) the number of -mesh cubes that intersect the set A ; (iv) the smallest number of sets of diameter at most that cover the set A ; (v) the largest number of disjoint balls of radius with centers in the set A . Lemma 2 (Perron-Frobenius Theorem). ([16])
Let A be an irreducible square matrix. Then we have the following two statements. (1) The spectral radius )( A of A is an eigenvalue of A and it has strictly positive eigenvector y (i.e., i y for all i ). (2) )( A increases if any element of A increases. Let us denote as follows: |})(||,)({|min xsxs kkIxk , |})(~||,)(~{|min~ xsxs kkIxk , ni ii )~( , ni ii )~( , where k ~ and k are given in section 3. For a set RD and a function f defined on D , denote as follows: },|:)()(sup{|][ DxxxfxfDR f . Theorem 8.
Let )( xf be the HVFIF in the Theorem 3 . Suppose that there exist three interpolation points ),( yx , ),( yx , ),( Pyx )( xxx which are not collinear and tha+t take , zz and z such that )( ji yy ji zz , jiji ,3,2,1, and three points ),( zx , ),( zx and ),( zx are not collinear. Then the box-counting dimension of the graph of )( xf is as follows: ( a ) If , then nBn fGr log1)(dimlog1 , ( b ) If , then fGr B . Proof.
Firstly, we prove ( a ). We denote the y-axis vertical distance from the point ),( yx to the line through the points ),( yx and ),( yx by H and the z-axis vertical distance from he point ),( zx to the line through the points ),( zx and ),( zx by h . Then we have hH . We apply i W to the interpolation points in I one time. Then we have ).))(((~)))(((~),,(),,( ),))((()))((( ))())(())((())())(())((( ),,(),,(
22 11 zzxLsyyxLszyxFzyxF zzxLsyyxLs xqzxLsyxLsxqzxLsyxLs zyxFzyxF iiiiii iiii iiiiiiiiii ii (15) Since )( fGr is the graph of a continuous function defined on I , the smallest number of r -mesh squares that cover )( fGrRI i is greater than the smallest number of r - mesh squares necessary to cover the vertical line whose length is )( hH i and less than the smallest number of r -mesh squares necessary to cover the rectangle ][ ifi IRI . By (5), (7) and (8) in section 3, we can easily prove that .1)(])[][(~][ ,1)(])[][(][ ~2~1~ 21
212 211 nnLfLfLIRIRIR nnLfLfLIRIRIR iii iii qssffiif qssffiif Then by (9) and (11) in section 3, we have .1])[][(~][,1])[][(][
MnIRIRIRMnIRIRIR ffiifffiif (16) Therefore, by (16), we get ni rrffirni ri nnMIRIRNhH , ni rrffirni ri nnMIRIRNhH , rrrr nnNn , where )),,,(()( Tnni i aaaaaa , Tn hHhHhH ))(,),(),((H , .2])[][)(~( ,,2])[][)(~(,2])[][)(~(U
21 2121
Tffnn ffff nMIRIR nMIRIRnMIRIR Let us apply j W once agian. Then we have n subintervals in every i I . By (15), we get ),))](((~))(())(())(([ )))](((~))(())(())(([ ))]((),,())((),,())(([ ))]((),,())((),,())(([),,(),,(
21 2111 zzxLsxLLsxLsxLLs yyxLsxLLsxLsxLLs xLqzyxfxLLszyxfxLLs xLqzyxfxLLszyxfxLLszyxFzyxF iiijjiiijj iiijjiiijj iiiijjiijj iiiijjiijjijij .))](((~))((~))(())(([ )))](((~))((~))(())((~[),,(),,( zzxLsxLLsxLsxLLs yyxLsxLLsxLsxLLszyxFzyxF iiijjiiijj iiijjiiijjijij By (16), we get .1])[][(~][,1])[][(][ MnIRIRIRMnIRIRIR jfifiijfififiijf (17) Since the smallest number of r -mesh squares that cover )( fGrRI ij is greater than the smallest number of r - mesh squares necessary to cover the vertical line whose length is ))(~( hH iij and less than the smallest number of r -mesh squares necessary to cover the rectangle ][ ijfij IRI , by (17), we have ni nj rr ififirni nj riij nnMIRIRNhH ni nj rr ififirni nj riij nnMIRIRNhH (18) Let us denote as follows: )~,,~,~(diag),~,,~,~(diag nnnn SS and C is nn matrix whose entries are all 1. Then we can rewrite (18) as follows: rrrr nnNn , where HH CS and nIMCS /2UU . Suppose that rkr nn . (19) If we apply i W to I k times, we have rkkr krkr k nnNn , (20) where HH kk CS and nIMCS kk /2UU . Then H)(H)(HH kkkk
CSCSCS , (21) InMCISnMCSInMCS kkk
InMICSnMICSnMCS kkk . (22) Since CS and CS are non-negative irreducible matrices, by Perron-Frobenius theorem, there exist strictly positive eigenvectors e and e of CS and CS which correspond to eigenvalues ni ii )~( and ni ii )~( , respectively, such that He , Ue , InM . Then by (19), (20) and (21), we get kr kkrkkr kr nCSnCSnN )e)(()H)(()H()( krkkrk n and since , we have r kr krrr nkkkN log )/1)e(log(loglog)1(1log )/1)e(log(log log)1(1log )(log , nr krr nnkkN rr log1loglog1log )/1)e(log(loglog)1(1limlog )(loglim . (23) Since ,we get . Then by (19), (20) and (22), we obtain
321 3211 rkkr kkk rkkr kkk rkkr kr nnCSCSCS nnInMICSnMICSnMCS nnN = rkkrkk nn (24) ,1/11 /11)e()1()1(1/11 )/11()e( 11/11 )/11()e( rkkkrkrkkr rkkkkr nnnn where kk n . Therefore we have rrrrr nkkkN loglogloglog)1(1logloglog log)1(1log )(log , nrrr nnkkN rr log1loglog1loglogloglog)1(1limlog )(loglim . (25) By (23) and (25), we get nBn fGr log1)(dimlog1 . Secondly, we prove (b). Since , by (24), we have rrkrrkkrkkr nnnN )1(111)e(11)e()1()( , here n k . Then we get rrr N loglog1log )(log and rrrB rr NfGr . Since fGr B , we get fGr B . □
5. Box-counting dimension of the HVBFIF
As mentioned in section 1, the construction and smoothness of HVFIFs are similar to ones of HVBFIFs in [21, 22]. Therefore, we present only the result for the box-counting dimension of HVBFIFs. 5.1. Construction of HVBFIFs. (see [21]) Let P denote the following data set in R : ),(},,1,0,,,1,0;R),,{( mnijji yyyxxxmjnizyxP We extend the data set P to a data set P in R as follows: ),(},,1,0,,,1,0;R),,,{( mnijijji yyyxxxmjnitzyxP where ij t are parameters. Let us denote ),( jiij yxx , ),( ijijij tzz , },,1{},,1{ mnN nm , ],[ iix xxI i , ],[ jjy yyI j , ],[ nx xxI , ],[ my yyI , ],[],[ mn yyxxE and ji yxij IIE . We define mappings ii xxx IIL : , jj yyy IIL : for nm Nji ),( to be contraction homeomorphisms satisfying the following conditions: },,{}),({ iinx xxxxL i }.,{}),({ jjmy yyyyL j Next, we define transformations ijij
EEL : by ))(),(()( yLxLxL ji yxij . Then, ij L maps the end points of E to the end points of ij E , i.e. abij xxL )( , },1{ iia , },1{ jjb , },0{ n , },0{ m . We define mappings mjniEF ij ,,1,,,1,RR: as follows: )(~ )())((~))((~ ))(())(()(~))((~))((~ )())(())((),( xq xqtzxLsxLs xLsxLsxqtxLszxLs xqtxLszxLszxF ijijijijijij ijijijijijijijijij ijijijijijij (26) where ),(),,( tzzyxx and REssss ijijijijij :~,~,, are arbitrary Lipschitz functions whose absolute values are less than 1 and ij q , ij q ~ : RE are Lipschitz functions satisfying the following onditions: If },0{ n , },0{ m , },1{ iia , },1{ jjb , ax xxL i )( , )( yL j y b y , then abij zzxF ),( . We assume that RD is an enough large bounded domain containing ij z , ni ,,1 , mj ,,1 . We define transformations R: ijij EDEW as follows: )),(),((),( zxFxLzxW ijijij , mjni ,,1,,,1 . For a function f , let us denote |)(|sup xff x and nissssS ijijijij ,,1;~,~max{ , },,1 mj . If S , then there exists a distance equivalent to the Euclidean metric on R such that ij W , ni ,,1 , mj ,,1 are contraction transformations with respect to the distance . (See Theorem 2.1 in [21].) Therefore, we have an hyperbolic iterated function system ij WR ;{ , ni ,,1 , },,1 mj corresponding to the extended data set P . Then there exists a continuous interpolation function f of the extended data set P such that the graph of f is the attractor of the IFS. (See Theorem 3.1 in [21].) We denote the vector valued function : REf by )),(),,(( yxfyxff , where REf : interpolates the given data set P and the function ),( yxf interpolates the set ;R),(),,{( ijijijji txtyx ,,,1,0 ni },,1,0 mj . Then, we have ,),())),,(()),,((),,(())),((),,((),( ijijijijijijijij EyxyxLfyxLfyxLFyxLfyxLFyxf )),(()),((),()),((),(),( yxLqyxLfyxsyxLfyxsyxf ijijijijijij , )),((~)),((),(~)),((),(~),( yxLqyxLfyxsyxLfyxsyxf ijijijijijij . .2. Box-counting dimension of HVBFIFs Let the data set be njizjn yyyin xxxP ijnn ,,1,0,;R,, and the extended data set njitzjn yyyin xxxP ijijnn ,,1,0,;R,,, and yxsyxs ijij , yxsyxs ijij , njiEyx ,,2,1,,),( in (26). Let us denote as follows: nnjzjn yyyn xxxP jnnx ,...,1,0,,,1,0;R,, , nnizn yyyin xxxP inny ,...,1,0,,,1,0;R,, , |}),(||,),({|min ),( yxsyxs ijijEyxij ij , |}),(~||,),(~{|min~ ),( yxsyxs ijijEyxij ij , |}),(||,),({|max ),( yxsyxs ijijEyxij ij , |}),(~||,),(~{|max~ ),( yxsyxs ijijEyxij ij , nji ijij )~( , nji ijij )~( . Theorem 9.
Let ),( yxf be the HVBFIFs constructed above. Suppose that there exist three interpolation points ),,( jj zyx , ),,( jj zyx , xjj Pzyx ),,( )( jjj yyy (or ),,( ii zyx , yiiii Pzyxzyx ),,(),,,( )( iii xxx ) which are not collinear and that take j t , j t and j t (or i t , i t and i t ) such that )( lk jj zz lk jj tt (or )( lk ii zz lk ii tt ), lklk ,3,2,1, and three points ),,( jj tyx , ),,( jj tyx , ),,( jj tyx (or ),,( ii tyx , ),,( ii tyx , ),,( ii tyx ) are not collinear. Then the box-counting dimension of the graph of ),( yxf is as follows: ( a ) If n , then nBn fGr log1)(dimlog1 , ( b ) If n , then fGr B .
6. Conclusion