Box dimension of fractal functions on attractors

Abstract

We study a wide class of fractal interpolation functions in a single platform by considering the domains of these functions as general attractors. We obtain lower and upper bounds of the box dimension of these functions in a more general setup where the interpolation points need not be equally spaced, the scale vectors can be variables and the maps in the corresponding IFS can be non-affine. In particular, we obtain the exact value of the box dimension of non-affine fractal functions on general m-dimensional cubes and Sierpiński Gasket.

R. Pasupathi

Sobolev Institute of Mathematics SB RAS, 630090, Novosibirsk, Russia

Email: pasupathi4074@gmail.com

Keywords: Fractal interpolation function, Attractor, Box dimension, Sierpiński Gasket

MSC Classification: 28A80, 41A05, 37E05

1 Introduction

In 1986, Barnsley [2] proposed the concept of a fractal interpolation function (FIF) on intervals using the notion of iterated function system. This work was extended to many different domains such as triangles [19], $m$ -dimensional cubes [22], Sierpiński Gasket [8], post critically finite (p.c.f.) self-similar sets [24] etc.

The graph of a function, and its box and Hausdorff dimensions, have been of qualitative interest for many researchers since the past few decades. Several theories concerning the box dimension of fractal functions have been explored in the literature. Some of recent works on FIF and dimension theory can be found in [7, 17, 15, 21, 23].

Hardin and Massopust [13] have estimated the value of the box dimension of graph of FIF on an interval when the maps in the corresponding IFS are affine and the interpolation points are equally spaced. Barnsley and Massopust [3] studied the box dimension of bilinear FIF on an interval in the case of equally spaced data points. Nasim et al. [1] obtained the box dimension of non-affine FIF in the case of equally spaced data points by using Hölder exponent. Feng and Sun [11] studied the box dimension of FIFs on a rectangle derived from affine FIFs on an interval with arbitrary interpolation nodes. Geronimo and Hardin [12] have estimated the box dimension of self-affine FIF on polygonal regions. Bouboulis et al. [4] introduced recurrent FIF (RFIF) on a rectangle as the invariant set of the recurrent IFS (RIFS) in order to gain more flexibility, and studied the value of the box dimension of RFIF when the maps in the corresponding RIFS are affine with respect to each variable and the interpolation points are equally spaced. Bouboulis and Dalla [5] generalized the theory of RFIF and its box dimension to higher dimensions.

In the literature, we found that, in most of the cases, the authors assumed that the interpolation points are equally spaced and the maps in the corresponding IFS are affine when the box dimension of FIF was considered, and there was no discussion about the box dimension of FIFs on p.c.f. self-similar sets except the Sierpiński Gasket. Even in the case of Sierpiński Gasket also, the authors [25] obtained only a non-trivial upper bound of the box dimension of FIFs by using Hölder exponent. The authors always assumed all the scale variables to be constants for estimating a non-trivial lower bound of the box dimension of fractal functions.

In this paper, we consider FIFs on attractors and we study lower and upper bounds of the box dimension of these functions without assuming that the interpolation points are equally spaced, the maps in the corresponding IFSs are affine and all the scale vectors are constants. We derive upper bounds of the box dimension of fractal functions using a function space called the oscillation space, which contains the collection of Hölder continuous functions. We provide non-trivial lower bounds of the box dimension of FIF on $m$ -dimensional cubes and Sierpiński Gasket.

2 Preliminaries

Definition 2.1.

An iterated function system (IFS) consists of a complete metric space $(X,d)$ together with a finite set of continuous mappings $f_{i}:X\to X,i=1,2,\dots,N$ . We denote it as $\mathcal{S}=\{(X,d),(f_{i})_{i=1}^{N}\}$ .

For an IFS, we can define an operator (called Hutchinson operator) $\mathcal{F}_{\mathcal{S}}:(\mathcal{H}(X),h)\to(\mathcal{H}(X),h)$ by

\mathcal{F}_{\mathcal{S}}(B)=\underset{i=1}{\overset{N}{\cup}}f_{i}(B)\;\;\;\text{for}\;B\in\mathcal{H}(X),

where $(\mathcal{H}(X),h)$ is the Hausdorff metric space on $X$ i.e., $\mathcal{H}(X)$ is the collection of all non-empty compact subsets of $X$ and $h$ is the Hausdorff distance.
If $\mathcal{F}_{\mathcal{S}}$ has a unique fixed point $A_{\mathcal{S}}$ (say) and $\underset{n\to\infty}{\lim}\mathcal{F}_{\mathcal{S}}^{[n]}(B)=A_{\mathcal{S}}$ for every $B\in\mathcal{H}(X)$ , then $A_{\mathcal{S}}$ is called the attractor of the IFS, where by $f^{[n]}$ , we mean the composition of a function $f$ with itself $n$ times.

Definition 2.2.

Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be a function. If there is a constant $r\in[0,1)$ such that:

$-$

$d(f(x),f(y))\leq r~{}d(x,y)~{}~{}\text{for}~{}x,y\in X,$

then $f$ is called contraction;
$-$

$d(f(x),f(y))=r~{}d(x,y)~{}~{}\text{for}~{}x,y\in X,$

then $f$ is called similarity.

The constant $r$ is called the contractivity factor of $f$ .

Remark 2.1.

[14] An IFS $\{(X,d),(f_{i})_{i=1}^{N}\}$ has an attractor provided that $f_{i}$ ’s are contractions.

Remark 2.2.

If a function $f:\mathbb{R}^{k}\to\mathbb{R}^{k}$ is similarity, then it is an affine map.

For an IFS $\mathcal{S}=\{(X,d),(f_{i})_{i=1}^{N}\}$ , we denote $\mathcal{N}=\{1,2,\dots,N\}$ , $\mathcal{N}^{k}=\left\{(\omega_{1},\omega_{2},\dots,\omega_{k}):\omega_{i}\in\mathcal{N}\right\}$ and $\mathcal{N}^{\infty}=\left\{(\omega_{i})_{i=1}^{\infty}:\omega_{i}\in\mathcal{N}\right\}$ , and we define

f_{\omega}=f_{\omega_{1}}\circ f_{\omega_{2}}\circ\dots\circ f_{\omega_{k}}\;\;\;\text{for}\;\omega=(\omega_{1},\omega_{2},\dots,\omega_{k})\in\mathcal{N}^{k},k\in\mathbb{N}.

Let $\pi:\mathcal{N}^{\infty}\to A_{\mathcal{S}}$ be defined by

\pi(\omega)=\underset{k\in\mathbb{N}}{\bigcap}f_{\omega_{1}\omega_{2}\cdots\omega_{k}}(A_{\mathcal{S}})~{}~{}~{}~{}\text{for}~{}~{}\omega\in\mathcal{N}^{\infty}.

Following Kigami [16], we define the critical set $\mathcal{C}$ and the post critical set $\mathcal{P}$ of $A_{\mathcal{S}}$ by

\mathcal{C}=\pi^{-1}\left(\underset{\begin{subarray}{c}i,j\in\mathcal{N}\\ i\neq j\end{subarray}}{\bigcup}\left(f_{i}(A_{\mathcal{S}})\cap f_{j}(A_{\mathcal{S}})\right)\right)~{}~{}~{}\text{and}~{}~{}~{}\mathcal{P}=\underset{k\in\mathbb{N}}{\bigcup}\tau^{[k]}(\mathcal{C}),

where $\tau$ is the left shift operator on $\mathcal{N}^{\infty}$ . If $\mathcal{P}$ is a finite set, then $A_{\mathcal{S}}$ is called a post critical finite (p.c.f.) self-similar set. The boundary of $A_{\mathcal{S}}$ is defined by $V_{0}^{*}=\pi(\mathcal{P})$ and we define

V_{k}=\underset{\omega\in\mathcal{N}^{k}}{\bigcup}f_{\omega}(V_{0}^{*})~{}~{}~{}\text{for}~{}k\in\mathbb{N}.

Remark 2.3.

[16]

(i)

$V_{0}^{*}\subseteq V_{1}\subseteq V_{2}\subseteq\cdots V_{k-1}\subseteq V_{k}\subseteq\cdots$ .

(ii)

For $\omega,\omega^{\prime}\in\mathcal{N}^{k},k\in\mathbb{N}$ with $\omega\neq\omega^{\prime}$ , we get

~{}~{}~{}~{}f_{\omega}(A_{\mathcal{S}})\cap f_{\omega^{\prime}}(A_{\mathcal{S}})=f_{\omega}(V_{0}^{*})\cap f_{\omega^{\prime}}(V_{0}^{*}).

Definition 2.3.

For a nonempty bounded subset $F$ of $\mathbb{R}^{k}$ , the lower and upper box dimensions are defined by

\underline{\dim}_{B}(F)=\liminf\limits_{\delta\to 0^{+}}\frac{\log{N}_{\delta}(F)}{\log\left(\frac{1}{\delta}\right)}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}~{}\overline{\dim}_{B}(F)=\limsup\limits_{\delta\to 0^{+}}\frac{\log{N}_{\delta}(F)}{\log\left(\frac{1}{\delta}\right)},

where ${N}_{\delta}(F)$ is the minimum number of boxes with side length $\delta$ and sides parallel to the axes, whose union contains $F$ . If $\underline{\dim}_{B}(F)=\overline{\dim}_{B}(F)$ , this common value is denoted by $\dim_{B}(F)$ and is called the box dimension or Minkowski dimension of $F$ .

Remark 2.4.

[10] For a nonempty bounded subset $F$ of $\mathbb{R}^{k}$ and a continuous function $f:F\to\mathbb{R}$ , we have the following inequalities

\dim_{H}(F)\leq\underline{\dim}_{B}(F)\leq\overline{\dim}_{B}(F)\;\;\text{and}\;\;\dim_{H}(F)\leq\dim_{H}(G_{f}),

where $\dim_{H}(F)$ means the Hausdorff dimension of $F$ .

Definition 2.4.

Let $(X,d_{X})$ and $(Y,d_{Y})$ be metric spaces. A function $f:X\to Y$ is called Hölder continuous with exponent $\eta$ if $\eta\in(0,1]$ and there exists $H\in[0,\infty)$ such that

d_{Y}(f(x),f(x^{\prime}))\leq H~{}d_{X}(x,x^{\prime})^{\eta}~{}~{}\text{for}~{}x,x^{\prime}\in X.

3 Fractal interpolation function

Let $p:V:=\underset{i\in\mathcal{N}}{\bigcup}l_{i}(V_{0})\to\mathbb{R}$ be a given function (data), where

$-$

$V_{0}=\{k_{1},k_{2},\dots,k_{r}\}\subseteq V\subseteq K$ ,
$-$

$K$ is an attractor of an IFS $\{(\mathbb{R}^{m},\|.\|_{2}),(l_{i})_{i\in\mathcal{N}}\}$ , $\|.\|_{2}$ is the Euclidean metric on $\mathbb{R}^{m}$ , $\mathcal{N}=\{1,2,\dots,N\}$ and $l_{i}$ is a similarity map on $\mathbb{R}^{m}$ with the contractivity factor $r_{i}$ for $i\in\mathcal{N}$ .

Let us consider $g_{i}:K\times\mathbb{R}\to\mathbb{R}$ defined by

g_{i}(x,z)=s_{i}(x)z+q_{i}(x)\;\;\;\text{for}\;(x,z)\in K\times\mathbb{R},i\in\mathcal{N},

(3.1)

where $s_{i}$ is a continuous function with $\|s\|_{\infty}:=\max\{\|s_{i}\|_{\infty}:~{}i\in\mathcal{N}\}<1$ , $\|.\|_{\infty}$ is the uniform metric and $q_{i}$ is a continuous function which satisfy the following ‘join-up’ conditions

q_{i}(k_{j})=p(l_{i}(k_{j}))-s_{i}(k_{j})p(k_{j})\;\;\;\;\;\text{for}\;j\in\{1,2,\dots,r\}.

(3.2)

We consider the IFS $\mathcal{S}=\left\{(K\times\mathbb{R},\|.\|_{2}),(f_{i})_{i\in\mathcal{N}}\right\}$ , where

f_{i}(x,z)=(l_{i}(x),g_{i}(x,z))~{}\;\text{for}~{}(x,z)\in K\times\mathbb{R}.

Let us suppose that the map $T:\mathcal{C}^{*}\to\mathcal{C}$ given by

T(f)(x)=g_{i}(l_{i}^{-1}(x),f(l_{i}^{-1}(x))~{}\;\;\text{for}~{}x\in l_{i}(K),i\in\mathcal{N},f\in\mathcal{C}^{*}

is well-defined, where $\mathcal{C}=\left\{f:K\to\mathbb{R}~{}:~{}f~{}\text{is continuous}\right\}$ and $\mathcal{C}^{*}=\\ \left\{f\in\mathcal{C}~{}:f|_{V_{0}}=p|_{V_{0}}\right\}$ .

Lemma 3.1.

T(\mathcal{C}^{*})\subseteq\mathcal{C}^{**}\subseteq\mathcal{C}^{*},

where $\mathcal{C}^{**}=\left\{f\in\mathcal{C}~{}:f|_{V}=p|_{V}\right\}$ .

Proof.

Since

T(f)(l_{i}(k_{j}))=g_{i}(k_{j},f(k_{j}))=g_{i}(k_{j},p(k_{j}))\overset{(\ref{0702e1})\&(\ref{1324e2})}{=}p(l_{i}(k_{j})),

(3.3)

for $j\in\{1,2,\dots,r\},i\in\mathcal{N}$ and $f\in\mathcal{C}^{*}$ , we get the proof. ∎

Theorem 3.1.

$T$ is a contraction map on the complete space $(\mathcal{C}^{*},\|.\|_{\infty})$ .

Proof.

For $f_{1},f_{2}\in\mathcal{C}^{*}$ , we get

	$\displaystyle\\|T(f_{1})-T(f_{2})\\|_{\infty}$	$\displaystyle\leq\underset{i\in\mathcal{N}}{\max}~{}\underset{x\in K}{\max}~{}\|g_{i}(x,f_{1}(x)-g_{i}(x,f_{2}(x)\|$
		$\displaystyle=\underset{i\in\mathcal{N}}{\max}~{}\underset{x\in K}{\max}~{}\|s_{i}(x)\|\|f_{1}(x)-f_{2}(x)\|=\\|s\\|_{\infty}\\|f_{1}-f_{2}\\|_{\infty}.$

∎

Corollary 3.1.

There exists a unique continuous function $f^{*}:K\to\mathbb{R}$ such that $f^{*}|_{V}=p|_{V}$ and

f^{*}\left(l_{i}(x)\right)=s_{i}(x)f^{*}(x)+q_{i}(x)\;\;\;\text{for}~{}x\in K,i\in\mathcal{N}.

(3.4)

Proof.

From Theorem 3.1 and the Banach contraction principle, we conclude that there exists $f^{*}\in\mathcal{C}^{*}$ such that $f^{*}=T(f^{*})\overset{\text{Lemma}\ref{1706r1}}{\in}\mathcal{C}^{**}$ . ∎

We call this unique map $f^{*}$ fractal interpolation function (FIF) on $K$ .

Proposition 3.1.

$G_{f^{*}}$ is a fixed point of the Hutchison operator $\mathcal{F}_{\mathcal{S}}$ of $\mathcal{S}$ , where $G_{f^{*}}$ denotes the graph of $f^{*}$ .

Proof.

Since $G_{f^{*}}\in\mathcal{H}(K\times\mathbb{R})$ and

	$\displaystyle G_{f^{*}}$	$\displaystyle=\underset{i=1}{\overset{N}{\cup}}\{\left(x,f^{}(x)\right):x\in l_{i}(K)\}\overset{f^{}=T(f^{})}{=}\underset{i=1}{\overset{N}{\cup}}\{(l_{i}(x),g_{i}(x,f^{}(x))):x\in K\}$
		$\displaystyle=\underset{i=1}{\overset{N}{\cup}}f_{i}(G_{f^{*}}),$		(3.5)

we get the result. ∎

Theorem 3.2.

If $\mathcal{S}$ has an attractor $A_{\mathcal{S}}$ (say), then

A_{\mathcal{S}}=G_{f^{*}},

Proof.

Since $A_{\mathcal{S}}$ is the unique fixed point of the Hutchison operator of $\mathcal{S}$ , from Proposition 3.1, we get the result. ∎

Lemma 3.2.

$f_{i}:K\times[-M,M]\to K\times[-M,M],i\in\mathcal{N}$ are well-defined operators, where $M:=\frac{\underset{i\in\mathcal{N}}{\max}~{}\|q_{i}\|_{\infty}}{1-\|s\|_{\infty}}$ .

Proof.

For $(x,z)\in K\times[-M,M]$ and $i\in\mathcal{N}$ , we get

\displaystyle|g_{i}(x,z)|\leq\|s_{i}\|_{\infty}z+\|q_{i}\|_{\infty}\leq\|s\|_{\infty}M+\underset{i\in\mathcal{N}}{\max}~{}\|q_{i}\|_{\infty}=M.

∎

Proposition 3.2.

If $s_{i}$ ’s are Hölder continuous, then $\mathcal{S}$ has an attractor.

Proof.

From (3.4), we get

\|f^{*}\|_{\infty}\leq\|s\|_{\infty}\|f^{*}\|_{\infty}+\underset{i\in\mathcal{N}}{\max}~{}\|q_{i}\|_{\infty}~{}~{}\Rightarrow~{}~{}\|f^{*}\|_{\infty}\leq M.

(3.6)

From the assumption, there exist $H_{i}\in[0,\infty)$ for $i\in\mathcal{N}$ and $\eta\in(0,1]$ such that

|s_{i}(x)-s_{i}(x^{\prime})|\leq H_{i}\|x-x^{\prime}\|_{2}^{\eta}\;\;\;\text{for}\;x,x^{\prime}\in K,i\in\mathcal{N}.

(3.7)

For $\theta=\frac{1-\underset{i\in\mathcal{N}}{\max}~{}r_{i}^{\eta}}{2M\underset{i\in\mathcal{N}}{\max}~{}H_{i}+1}>0,$ let us consider the metric $d$ on $K\times\mathbb{R}$ , given by

d((x,z),(x^{\prime},z^{\prime}))=\|x-x^{\prime}\|_{2}^{\eta}+\theta|(z-f^{*}(x))-(z^{\prime}-f^{*}(x^{\prime}))|,

for $(x,z),(x^{\prime},z^{\prime})\in K\times\mathbb{R}$ , which is equivalent to the Euclidean metric.
For $i\in\mathcal{N}$ and $(x,z),(x^{\prime},z^{\prime})\in K\times[-M,M]$ , we have

	$\displaystyle d(f_{i}(x,z),f_{i}(x^{\prime},z^{\prime}))$
	$\displaystyle=\\|l_{i}(x)-l_{i}(x^{\prime})\\|_{2}^{\eta}+\theta\|s_{i}(x)z+q_{i}(x)-f^{}(l_{i}(x)))-(s_{i}(x^{\prime})z^{\prime}+q_{i}(x^{\prime})-f^{}(l_{i}(x^{\prime})))\|$
	$\displaystyle\overset{(\ref{2101e1})}{=}r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\|s_{i}(x)(z-f^{}(x))-s_{i}(x^{\prime})(z^{\prime}-f^{}(x^{\prime})))\|$
	$\displaystyle\leq r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\\|s_{i}\\|_{\infty}\|(z-f^{}(x))-(z^{\prime}-f^{}(x^{\prime}))\|$
	$\displaystyle~{}~{}~{}+\theta\|(z^{\prime}-f^{*}(x^{\prime}))\|\|s_{i}(x)-s_{i}(x^{\prime})\|$
	$\displaystyle\overset{(\ref{0702e2})\&(\ref{0702e11})}{\leq}r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\\|s\\|_{\infty}\|(z-f^{}(x))-(z^{\prime}-f^{}(x^{\prime}))\|+\theta 2MH_{i}\\|x-x^{\prime}\\|_{2}^{\eta}$
	$\displaystyle\leq c_{i}d((x,z),(x^{\prime},z^{\prime})),$

where $c_{i}=\max\left\{r_{i}^{\eta}+\theta 2MH_{i},\|s\|_{\infty}\right\}<1$ .
Thus, $f_{i}$ ’s are contractions on $(K\times[-M,M],d)$ .
From Remark 2.1, we get the result. ∎

Case 1.

If $K$ is a p.c.f. self-similar set and $V_{0}$ is the boundary of $K$ , then by Remark 2.3 (ii), we get

l_{i}(K)\cap l_{i^{\prime}}(K)=l_{i}(V_{0})\cap l_{i^{\prime}}(V_{0})~{}~{}\text{for}~{}~{}i,i^{\prime}\in\mathcal{N}~{}\text{with}~{}i\neq i^{\prime}.

(3.8)

From equation (3.3), we get

T(f)|_{l_{i}(V_{0})}=p|_{l_{i}(V_{0})}~{}~{}\text{for}~{}f\in\mathcal{C}^{*},i\in\mathcal{N}.

(3.9)

Let $f\in\mathcal{C}^{*}$ and $x\in l_{i}(K)\cap l_{i^{\prime}}(K)$ for some $i,i^{\prime}\in\mathcal{N}$ with $i\neq i^{\prime}$ .
From equation (3.8) and (3.9), we get

T(f)(x)=p(x)

by viewing $x$ as an entity belonging to $l_{i}(K)$ and $l_{i^{\prime}}(K)$ .
Therefore $T(f)$ is a continuous function. Thus, $T$ is well defined.

Remark 3.1.

If $K$ is a p.c.f. self-similar set, then for any given data on it’s boundary $V_{0}$ , there exists a unique harmonic function on $K$ such that it satisfies the given data (see [16]). This gives the guarantee for the existence of $q_{i}$ ’s as in (3.2).

Remark 3.2.

If $K$ is a p.c.f. self-similar set and $V_{0}$ is its boundary with respect to the IFS $\{(\mathbb{R}^{m},\|.\|_{2}),(l_{i})_{i=1}^{N}\}$ , then for any $n\in\mathbb{N}$ , $K$ $(V_{0})$ is again a p.c.f. self-similar set (boundary of $K$ ) with respect to the IFS $\{(\mathbb{R}^{2},\|.\|_{2}),\{l_{\omega}\}_{\omega\in\mathcal{N}^{n}}\}$ . So, for any $n\in\mathbb{N},$ we can get a FIF $f^{*}$ for the given data on $(V=)V_{n}:=\underset{\omega\in\mathcal{N}^{n}}{{\cup}}l_{\omega}(V_{0})$ . Note that $V_{n}\to K$ as $n\to\infty$ .

Remark 3.3.

Sierpiński Gasket (SG), Sierpiński sickle, Koch curve, Hata’s tree-like set are some of the examples of p.c.f. self-similar sets.
The IFS of SG is $\{(\mathbb{R}^{2},\|.\|_{2}),\{l_{i}\}_{i=1}^{3}\}$ , where $l_{i}(x)=\frac{1}{2}(x+k_{i})$ for $i\in\{1,2,3\}$ and $\{k_{1},k_{2},k_{3}\}$ are the vertices of an equilateral triangle. In this case, the boundary of $K$ is $V_{0}=\{k_{1},k_{2},k_{3}\}$ .

Case 2.

V=\left\{(x_{1i_{1}},x_{2i_{2}},\dots,x_{mi_{m}})\in\mathbb{R}^{m}:i_{u}\in\{0,1,\dots,n_{u}\},u\in\{1,2,\dots,m\}\right\}

with $x_{u0}<x_{u1}<\dots<x_{un_{u}}$ for $u\in\{1,2,\dots,m\}$ , then we define $l_{i}:\mathbb{R}^{m}\to\mathbb{R}^{m}$ to be

l_{i}(x)=\left(l_{1i_{1}}(x_{1}),l_{2i_{2}}(x_{2}),\dots,l_{mi_{m}}(x_{m})\right),

for $x=(x_{1},x_{2},\dots,x_{m})\in\mathbb{R}^{m}$ and $i=(i_{1},i_{2},\dots,i_{m})\in\mathcal{N},$ where

$-$

$\mathcal{N}=\{(i_{1},i_{2},\dots i_{m}):i_{u}\in\{1,2,\dots,n_{u}\},u\in\{1,2,\dots,m\}\}$ ,
$-$

$l_{ui_{u}}(t)=\left(\frac{x_{u(i_{u}-\epsilon_{ui_{u}})}-x_{u(i_{u}-1+\epsilon_{ui_{u}})}}{x_{un_{u}}-x_{u0}}\right)t+\left(\frac{x_{u(i_{u}-1+\epsilon_{ui_{u}})}x_{un_{u}}-x_{u(i_{u}-\epsilon_{ui_{u}})}x_{u0}}{x_{un_{u}}-x_{u0}}\right)$ for $t\in\mathbb{R},u\in\{1,2,\dots,m\}$ and
$-$

$\epsilon_{u}=(\epsilon_{u1},\epsilon_{u2},\dots,\epsilon_{un_{u}})\in\{0,1\}^{n_{u}}$ for $u\in\{1,2,\dots,m\}$ (which is called signature).

We defined $l_{ui_{u}}$ for $u\in\{1,2,\dots,m\}$ such that

l_{ui_{u}}([x_{u0},x_{un_{u}}])=[x_{u(i_{u}-1)},x_{ui_{u}}]

and

l_{ui_{u}}(x_{u0})=x_{u(i_{u}-1+\epsilon_{ui_{u}})}~{}~{}\text{and}~{}~{}~{}l_{ui_{u}}(x_{un_{u}})=x_{u(i_{u}-\epsilon_{ui_{u}})}.

From our construction, we get

$-$

$V_{0}=\left\{(x_{1i_{1}},x_{2i_{2}},\dots,x_{mi_{m}})\in\mathbb{R}^{m}:i_{u}\in\{0,n_{u}\},u\in\{1,2,\dots,m\}\right\}$ and
$-$

$K=[x_{10},x_{1n_{1}}]\times[x_{20},x_{2n_{2}}]\times\dots\times[x_{m0},x_{mn_{m}}]$ .

(i)

If $m=1$ , then $K$ is a p.c.f. self-similar set and $V_{0}$ is its boundary. So, $T$ is well-defined.
In this case, we obtain the fractal interpolation function of the zipper on an interval (see [7]).
If we choose $\varepsilon_{u}=0$ for $u\in\{1,2,\dots,m\}$ and $s_{i}$ ’s are constant functions, we get the standard fractal interpolation function on an interval (see [2]).

(ii)

If $m>1$ , then $T$ is well-defined provided that

\varepsilon_{u}=(0,1,0,1,\dots)~{}\text{or}~{}\varepsilon_{u}=(1,0,1,0,\dots)~{}\text{for}~{}u\in\{1,2,\dots,m\}

and

F_{i}(x,z)=F_{(i_{1}\dots i_{j-1},i_{j}+1,i_{j+1}\dots i_{m})}(x,z),

for $i=(i_{1},i_{2}\dots i_{m})\in\mathcal{N},j\in\{1,2\dots m\}$ with $i_{j}\in\{1,2\dots n_{j}-1\}$ ,
$x=(x_{1}\dots x_{j-1},x_{j}^{*},x_{j+1}\dots x_{m})\in K$ with $x_{j}^{*}=l_{ji_{j}}^{-1}(x_{ji_{j}})=l_{j(i_{j}+1)}^{-1}(x_{ji_{j}})$ and $z\in\mathbb{R}$ .
For more details ref. [20, 22, 17]. In this case, we call $f^{*}$ multivariate FIF.
In particular, if $\varepsilon_{u}=(0,1,0,1,\dots)$ for $u\in\{1,2,\dots,m\}$ , $s_{i}$ ’s are equal constants i.e., there exists unique $s\in(-1,1)$ such that $s_{i}(x)=s$ for $x\in K$ and $i\in\mathcal{N}$ , and

q_{i}(x)=\underset{J\subseteq\{1,2,\dots,m\}}{\sum}e_{i,J}\underset{j\in J}{\prod}x_{j}~{}~{}~{}\text{for}~{}x=(x_{1},x_{2},\dots,x_{m})\in K,i\in\mathcal{N},

where $e_{i,J}$ ’s are constants, then $T$ is well-defined (see [18]).

4 Box dimension of fractal functions

For a continuous function $f:K\to\mathbb{R}$ and $\omega\in\mathcal{N}^{k}$ , we define the oscillation of $f$ over $l_{\omega}(K)$ by

\displaystyle\text{Osc}_{\omega}(f)=\underset{x\in l_{\omega}(K)}{\sup}f(x)-\underset{x\in l_{\omega}(K)}{\inf}f(x)=\underset{x,x^{\prime}\in l_{\omega}(K)}{\sup}|f(x)-f(x^{\prime})|

and total oscillation of order $k$ by

\text{Osc}(k,f)=\underset{\omega\in\mathcal{N}^{k}}{\sum}\text{Osc}_{\omega}(f).

We define oscillation space on $K$ , for $\eta\in\left[0,\log_{\Lambda}N\right]$ , as follows (Ref. [9])

\mathcal{C}^{\eta}(K)=\{f:K\to\mathbb{R}:f~{}\text{is continuous and}~{}[f]_{{\eta}}<\infty\},

where $[f]_{{\eta}}=\underset{k\in\mathbb{N}}{\sup}\frac{\text{Osc}(k,f)}{\Lambda^{k\left(\log_{\Lambda}N-\eta\right)}}$ and $\Lambda^{-1}=\underset{i\in\mathcal{N}}{\max}~{}r_{i}$ .

Remark 4.1.

If $m=1$ and $\eta=\log_{\Lambda}N,$ then $\mathcal{C}^{\eta}(K)$ is the collection of continuous bounded variation functions on $K$ .

In the following proposition, we discuss the relation between Hölder continuous functions and oscillation spaces, draws inspiration from [6].

Proposition 4.1.

If $f:K\to\mathbb{R}$ is a Hölder continuous function with exponent $\eta\in(0,1],$ then $f\in\mathcal{C}^{\eta}(K).$

Proof.

By assumption, there exists a constant $H$ such that

|f(x)-f(x^{\prime})|\leq H\|x-x^{\prime}\|_{2}^{\eta}\;\;\;\text{for}~{}~{}x,x^{\prime}\in K.

Thus, we have

	$\displaystyle\text{Osc}(k,f)$	$\displaystyle=\underset{\omega\in\mathcal{N}^{k}}{\sum}\underset{x,x^{\prime}\in l_{\omega}(K)}{\sup}\|f(x)-f(x^{\prime})\|\leq H\underset{\omega\in\mathcal{N}^{k}}{\sum}\underset{x,x^{\prime}\in K}{\sup}\\|l_{\omega}(x)-l_{\omega}(x^{\prime})\\|_{2}^{\eta}$
		$\displaystyle\leq H\underset{\omega\in\mathcal{N}^{k}}{\sum}\left(\Lambda^{-k}\right)^{\eta}\underset{x,x^{\prime}\in K}{\sup}\\|x-x^{\prime}\\|_{2}^{\eta}=HN^{k}\Lambda^{-k\eta}\|K\|^{\eta}$
		$\displaystyle=H\|K\|^{\eta}\Lambda^{k\left(\log_{\Lambda}N-\eta\right)},$

for all $k\in\mathbb{N}$ , where $|K|$ denotes the diameter of $K$ . ∎

Theorem 4.1.

Let $s_{i},q_{i}\in\mathcal{C}^{\eta}(K)$ for $i\in\mathcal{N}$ . If:

(i)

$\gamma\leq\frac{N}{\Lambda^{\eta^{\prime}}},$ then

\dim_{H}(K)\leq\dim_{H}\left(G_{f^{*}}\right)\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1-\eta^{\prime}+\log_{\Lambda}N;

(ii)

$\gamma>\frac{N}{\Lambda^{\eta^{\prime}}},$ then

\dim_{H}(K)\leq\dim_{H}\left(G_{f^{*}}\right)\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1+\log_{\Lambda}\gamma,

where $\gamma:=\underset{i\in\mathcal{N}}{\sum}\|s_{i}\|_{\infty}$ and $\eta^{\prime}:=\min\{1,\eta\}$ .

Proof.

Let ${N}(k)$ and ${N}(k,\omega)$ denote the minimum number of cubes of size $\frac{|K|}{\Lambda^{k}}\times\frac{|K|}{\Lambda^{k}}\times\dots\frac{|K|}{\Lambda^{k}}$ which covers $G_{f^{*}}$ and $G_{f^{*},\omega}$ respectively, where $G_{f^{*},\omega}=\{(x,f^{*}(x)):x\in l_{\omega}(K)\}$ for $\omega\in\mathcal{N}^{k}$ .
For $i\in\mathcal{N},\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ and $x,x^{\prime}\in l_{\omega}(K)$ , we have

	$\displaystyle\|f^{}(l_{i}(x))-f^{}(l_{i}(x^{\prime}))\|$
	$\displaystyle\overset{(\ref{2101e1})}{\leq}\\|s_{i}\\|_{\infty}\|f^{}(x)-f^{}(x^{\prime})\|+\\|f^{*}\\|_{\infty}\|s_{i}(x)-s_{i}(x^{\prime})\|+\|q_{i}(x)-q_{i}(x^{\prime})\|$
	$\displaystyle\leq\\|s_{i}\\|_{\infty}\frac{{N}(k,\omega)\|K\|}{\Lambda^{k}}+\\|f^{*}\\|_{\infty}\text{Osc}_{\omega}(s_{i})+\text{Osc}_{\omega}(q_{i}).$

For $i\in\mathcal{N},\omega\in\mathcal{N}^{k}$ and $k\in\mathbb{N}$ , we get $G_{f^{*},(i,\omega)}$ is contained in a cuboid of size $\frac{|K|}{\Lambda^{k+1}}\times\frac{|K|}{\Lambda^{k+1}}\times\dots\frac{|K|}{\Lambda^{k+1}}\times\frac{\|s_{i}\|_{\infty}{N}(k,\omega)|K|}{\Lambda^{k}}+\|f^{*}\|_{\infty}\text{Osc}_{\omega}(s_{i})+\text{Osc}_{\omega}(q_{i})$ .
Since $\Lambda>1$ and

G_{f^{*}}\overset{(\ref{47e4})}{=}\underset{\omega\in\mathcal{N}^{k}}{\cup}G_{f^{*},\omega}~{}\;\;\text{for}~{}k\in\mathbb{N},

(4.10)

we get

	$\displaystyle{N}(k+1)\leq\underset{\omega\in\mathcal{N}^{k+1}}{\sum}{N}(k+1,\omega)=\underset{i\in\mathcal{N}}{\sum}\underset{\omega\in\mathcal{N}^{k}}{\sum}{N}(k+1,(i,\omega))$
	$\displaystyle~{}~{}\leq\underset{i\in\mathcal{N}}{\sum}\underset{\omega\in\mathcal{N}^{k}}{\sum}\left(\frac{\Lambda^{k+1}}{\|K\|}\left(\frac{\\|s_{i}\\|_{\infty}{N}(k,\omega)\|K\|}{\Lambda^{k}}+\\|f^{*}\\|_{\infty}\text{Osc}_{\omega}(s_{i})+\text{Osc}_{\omega}(q_{i})\right)+1\right)$
	$\displaystyle~{}~{}\leq\Lambda\gamma{N}(k)+\frac{\Lambda^{k+1}}{\|K\|}\underset{i\in\mathcal{N}}{\sum}\left(\\|f^{*}\\|_{\infty}[s_{i}]_{\eta}+[q_{i}]_{\eta}\right)\Lambda^{k\left(\log_{\Lambda}N-\eta\right)}+N^{k+1}$
	$\displaystyle~{}~{}\leq\Lambda\gamma{N}(k)+N^{k}\Lambda^{k(1-\eta^{\prime})}C,$

for $k\in\mathbb{N}$ , where $C=\frac{\Lambda}{|K|}\underset{i\in\mathcal{N}}{\sum}\left(\|f^{*}\|_{\infty}[s_{i}]_{\eta}+[q_{i}]_{\eta}\right)+N$ .
Via the mathematical induction method, for $k\in\mathbb{N},$ we get

		$\displaystyle{N}(k+1)\leq(\Lambda\gamma)^{2}{N}(k-1)+\Lambda\gamma\left(N\Lambda^{1-\eta^{\prime}}\right)^{k-1}C+\left(N\Lambda^{1-\eta^{\prime}}\right)^{k}C$
		$\displaystyle~{}\leq(\Lambda\gamma)^{3}{N}(k-2)+C\left((\Lambda\gamma)^{2}\left(N\Lambda^{1-\eta^{\prime}}\right)^{k-2}+\Lambda\gamma\left(N\Lambda^{1-\eta^{\prime}}\right)^{k-1}+\left(N\Lambda^{1-\eta^{\prime}}\right)^{k}\right)$
		$\displaystyle\leq(\Lambda\gamma)^{k}{N}(1)+C\left((\Lambda\gamma)^{k-1}N\Lambda^{1-\eta^{\prime}}+(\Lambda\gamma)^{k-2}\left(N\Lambda^{1-\eta^{\prime}}\right)^{2}+\dots+\left(N\Lambda^{1-\eta^{\prime}}\right)^{k}\right).$

$\mathit{Case}~{}(i).$ Consider $\gamma\leq\frac{N}{\Lambda^{\eta^{\prime}}}$ , then for $k\in\mathbb{N}$ , we have

	$\displaystyle{N}(k+1)\leq(\Lambda\gamma)^{k}{N}(1)+C\left(N\Lambda^{1-\eta^{\prime}}\right)^{k}\left(1+\frac{\Lambda^{\eta^{\prime}}\gamma}{N}+\dots+\left(\frac{\Lambda^{\eta^{\prime}}\gamma}{N}\right)^{k-1}\right)$
	$\displaystyle~{}~{}\leq\Lambda^{k}\left(\frac{N}{\Lambda^{\eta^{\prime}}}\right)^{k}{N}(1)+C\left(N\Lambda^{1-\eta^{\prime}}\right)^{k}k$
	$\displaystyle~{}~{}\leq\left(N\Lambda^{1-\eta^{\prime}}\right)^{k+1}(k+1)(N(1)+C).$

Therefore, we get

\displaystyle\overline{\dim}_{B}(G_{f^{*}})=\limsup_{k\to\infty}\frac{\log{N}(k+1)}{\log\left(\frac{\Lambda^{k+1}}{|K|}\right)}\leq 1-\eta^{\prime}+\frac{\log N}{\log\Lambda}.

$\mathit{Case}~{}(ii).$ Consider $\gamma>\frac{N}{\Lambda^{\eta^{\prime}}},$ then for $k\in\mathbb{N}$ , we have

	$\displaystyle{N}(k+1)\leq(\Lambda\gamma)^{k}{N}(1)+C(\Lambda\gamma)^{k-1}\Lambda\left(\frac{N}{\Lambda^{\eta^{\prime}}}\right)\left(1+\frac{N}{\Lambda^{\eta^{\prime}}\gamma}+\dots+\left(\frac{N}{\Lambda^{\eta^{\prime}}\gamma}\right)^{k-1}\right)$
	$\displaystyle\leq(\Lambda\gamma)^{k}{N}(1)+C(\Lambda\gamma)^{k-1}\Lambda\gamma\frac{1}{1-\frac{N}{\Lambda^{\eta^{\prime}}\gamma}}$
	$\displaystyle\leq(\Lambda\gamma)^{k+1}\left({N}(1)+\frac{C}{1-\frac{N}{\Lambda^{\eta^{\prime}}\gamma}}\right).$

Hence

\displaystyle\overline{\dim}_{B}(G_{f^{*}})\leq 1+\log_{\Lambda}\gamma.

∎

Let us denote $\Lambda_{0}^{-1}=\underset{i\in\mathcal{N}}{\min}~{}r_{i}$ , and for $r\in\{0,1,2,\dots,m\}$ , we denote:

-

$\gamma_{1,r}=\underset{i\in\mathcal{N}}{\sum}s_{i,1,r}$ , where

s_{i,1,r}=\begin{cases}|s_{i}|,&\quad\text{if~{}}r=0,s_{i}\text{~{}is a constant and~{}}q_{i}\text{~{}is affine}\\ |s_{i}|,&\quad\text{if~{}}r\neq 0,s_{i}\text{~{}is a constant and~{}}\\ &\quad q_{i}\text{~{}is affine with respect to the $r^{\text{th}}$ co-ordinate ~{}}\\ 0,&\quad\text{~{}~{}otherwise;}\end{cases}

-

$\gamma_{2,r}=\underset{i\in\mathcal{N}}{\sum}s_{i,2,r}$ , where

s_{i,2,r}=\begin{cases}s_{i},&\quad\text{if~{}}r=0,s_{i}\text{~{}is a non-negative constant and~{}}q_{i}\text{~{}is concave}\\ s_{i},&\quad\text{if~{}}r\neq 0,s_{i}\text{~{}is a non-negative constant and~{}}\\ &\quad q_{i}\text{~{}is concave with respect to the $r^{\text{th}}$ co-ordinate~{}}\\ 0,&\quad\text{~{}~{}otherwise;}\end{cases}

-

$\gamma_{3,r}=\underset{i\in\mathcal{N}}{\sum}s_{i,3,r}$ , where

s_{i,3,r}=\begin{cases}s_{i},&\quad\text{if~{}}r=0,s_{i}\text{~{}is a non-negative constant and~{}}q_{i}\text{~{}is convex}\\ s_{i},&\quad\text{if~{}}r\neq 0,s_{i}\text{~{}is a non-negative constant and~{}}\\ &\quad q_{i}\text{~{}is convex with respect to the $r^{\text{th}}$ co-ordinate~{}}\\ 0,&\quad\text{~{}~{}otherwise.}\end{cases}

Lemma 4.1.

Assume that there exist $y_{1},y_{2},y_{3}\in V$ and $\lambda\in(0,1)$ such that $y_{3}=(1-\lambda)y_{1}+\lambda y_{2}$ and the set $\{(y_{1},p(y_{1})),(y_{2},p(y_{2})),(y_{3},p(y_{3}))\}$ is not collinear i.e., $L:=p(y_{3})-((1-\lambda)p(y_{1})+\lambda p(y_{2}))\neq 0$ . Then for $\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ , we get

(i)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},1,0}s_{\omega_{2},1,0}\dots s_{\omega_{k},1,0}|L|$ ;
(ii)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},2,0}s_{\omega_{2},2,0}\dots s_{\omega_{k},2,0}L$ if $L>0$ ;
(iii)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},3,0}s_{\omega_{2},3,0}\dots s_{\omega_{k},3,0}|L|$ if $L<0$ .

Proof.

Let us suppose that $L>0$ .
Since $l_{i}$ ’s are similarities on $\mathbb{R}^{m}$ , they are affine maps.
Thus, for $\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ , we get

l_{\omega}(y_{3})=(1-\lambda)l_{\omega}(y_{1})+\lambda l_{\omega}(y_{2}).

(4.11)

For $i\in\mathcal{N}$ such that $s_{i}$ is a non-negative constant and $q_{i}$ is concave, we get

		$\displaystyle g_{i}(y_{3},p(y_{3}))-((1-\lambda)g_{i}(y_{1},p(y_{1}))+\lambda g_{i}(y_{2},p(y_{2})))$
		$\displaystyle~{}~{}~{}=s_{i}L+q_{i}((1-\lambda)y_{1}+\lambda y_{2})-((1-\lambda)q_{i}(y_{1})+\lambda q_{i}(y_{2}))\geq s_{i}L.$		(4.12)

For $i,j\in\mathcal{N}$ such that $s_{i},s_{j}$ are non-negative constants and $q_{i},q_{j}$ are concave, we get

	$\displaystyle g_{i}(f_{j}(y_{3},p(y_{3})))-((1-\lambda)g_{i}(f_{j}(y_{1},p(y_{1})))+\lambda g_{i}(f_{j}(y_{2},p(y_{2}))))$
	$\displaystyle~{}\overset{(\ref{1701e2})}{=}s_{i}(g_{j}(y_{3},p(y_{3}))-((1-\lambda)g_{j}(y_{1},p(y_{1}))+\lambda g_{j}(y_{2},p(y_{2}))))$
	$\displaystyle~{}\;\;+q_{i}((1-\lambda)l_{j}(y_{1})+\lambda l_{j}(y_{2}))-((1-\lambda)q_{i}(l_{j}(y_{1}))+\lambda q_{i}(l_{j}(y_{2})))$
	$\displaystyle~{}\;\overset{(\ref{81eq1})}{\geq}s_{i}s_{j}L.$

Via the mathematical induction method, for $i\in\mathcal{N}$ and $\omega\in\mathcal{N}^{k-1},k-1\in\mathbb{N}$ , we get

		$\displaystyle\|g_{i}(f_{\omega}(y_{3},p(y_{3})))-((1-\lambda)g_{i}(f_{\omega}(y_{1},p(y_{1})))+\lambda g_{i}(f_{\omega}(y_{2},p(y_{2}))))\|$
		$\displaystyle\;\geq s_{i,2,0}s_{\omega_{1},2,0}s_{\omega_{2},2,0}\dots s_{\omega_{k},2,0}L.$		(4.13)

Since $f^{*}$ is a continuous function passing through $f_{\omega}(y_{1},p(y_{1})),f_{\omega}(y_{2},p(y_{2}))$ and $f_{\omega}(y_{2},p(y_{2}))$ , and by using (4.11) and (4), we get $G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},2,0}s_{\omega_{2},2,0}\dots s_{\omega_{k},2,0}L$ , for $\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ .
In a similar way, we can prove the other cases also. ∎

Lemma 4.2.

Let us consider $l_{i}(x)=(l_{1i_{1}}(x_{1}),l_{2i_{2}}(x_{2}),\dots,l_{mi_{m}}(x_{m}))$ for $x\in\mathbb{R}^{m}$ and $i\in\mathcal{N}$ , where $l_{ui_{u}}:\mathbb{R}\to\mathbb{R}$ are affine maps for $u\in\{1,2,\dots,m\}$ . Assume that there exist $y_{1}=(t_{1},t_{2},\dots,t_{r-1},t_{r_{1}},t_{r+1},\dots,t_{m}),y_{2}=(t_{1},t_{2},\dots,t_{r-1},t_{r_{2}},t_{r+1},\dots,t_{m}),$ $y_{3}=(t_{1},t_{2},\dots,t_{r-1},t_{r_{3}},t_{r+1},\dots,t_{m})\in V$ for some $r\in\{1,2,\dots,m\}$ , and $\lambda\in(0,1)$ such that $y_{3}=(1-\lambda)y_{1}+\lambda y_{2}$ and $L=p(y_{3})-((1-\lambda)p(y_{1})+\lambda p(y_{2}))\neq 0$ . Then for $\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ , we get

(i)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},1,r}s_{\omega_{2},1,r}\dots s_{\omega_{k},1,r}|L|$ ;
(ii)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},2,r}s_{\omega_{2},2,r}\dots s_{\omega_{k},2,r}L$ if $L>0$ ;
(iii)

$G_{f^{*},\omega}$ must cover a set of height $s_{\omega_{1},3,r}s_{\omega_{2},3,r}\dots s_{\omega_{k},3,r}|L|$ if $L<0$ .

Proof.

Using similar arguments of Lemma 4.1, we can prove this result. ∎

Theorem 4.2.

Let $f^{*}$ be a multivariate FIF derived from Case 2. Assume that the interpolation points are not collinear, i.e., there exist $r\in\{1,2,\dots,m\},y_{1},y_{2},y_{3}\in V$ and $L\neq 0$ as in the framework of Lemma 4.2. If:

(i)

$\gamma_{1,r}\neq 0$ , then

$1+\log_{\Lambda_{0}}(\gamma_{1,r})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right);$
(ii)

$L>0$ and $\gamma_{2,r}\neq 0$ , then

$1+\log_{\Lambda_{0}}(\gamma_{2,r})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right);$
(iii)

$L<0$ and $\gamma_{3,r}\neq 0$ , then

$1+\log_{\Lambda_{0}}(\gamma_{3,r})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right).$

Proof.

Let ${N}_{0}(k)$ denote the minimum number of cubes of side length $\frac{|K|_{0}}{\Lambda_{0}^{k}}$ that covers $G_{f^{*}}$ , where $|K|_{0}$ is the minimum of the side lengths of $K$ .
Let us suppose that $\gamma_{1,r}\neq 0$ .
From Lemma 4.2, we get $G_{f^{*},\omega}$ must cover a cuboid of size $\frac{|K|_{0}}{\Lambda_{0}^{k}}\times\dots\frac{|K|_{0}}{\Lambda_{0}^{k}}\times s_{\omega_{1},1,r}s_{\omega_{2},1,r}\dots s_{\omega_{k},1,r}|L|$ , for $\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ .
Thus

{N}_{0}(k)\geq\underset{\omega\in\mathcal{N}^{k}}{\sum}\frac{\Lambda_{0}^{k}}{|K|_{0}}s_{\omega_{1},1,r}s_{\omega_{2},1,r}\dots s_{\omega_{k},1,r}|L|=\frac{\Lambda_{0}^{k}}{|K|_{0}}\gamma_{1,r}^{k}|L|~{}~{}~{}\text{for}~{}k\in\mathbb{N}.

(4.14)

Therefore

\displaystyle\underline{\dim}_{B}(G_{f^{*}})=\liminf_{k\to\infty}\frac{\log{N}_{0}(k)}{\log\left(\frac{\Lambda_{0}^{k}}{|K|_{0}}\right)}\geq 1+\frac{\log\gamma_{1,r}}{\log\Lambda_{0}}.

A similar argument works for other cases as well. ∎

Corollary 4.1.

Let $f^{*}$ be a multivariate FIF derived from Case 2, $q_{i}\in\mathcal{C}^{\eta}(K)$ and $s_{i}$ be a constant for $i\in\mathcal{N}$ . Assume that the interpolation points are not collinear (i.e., there exists $r\in\{1,2,\dots,m\}$ as in Lemma 4.2)

$-$

and either $q_{i}$ is affine with respect to the $r^{\text{th}}$ co-ordinate for $i\in\mathcal{N}$
or
$-$

$L>0$ , $q_{i}$ is concave with respect to the $r^{\text{th}}$ co-ordinate and $s_{i}\geq 0$ for $i\in\mathcal{N}$
or
$-$

$L<0$ , $q_{i}$ is convex with respect to the $r^{\text{th}}$ co-ordinate and $s_{i}\geq 0$ for $i\in\mathcal{N}$ .

If:

(i)

$0<\gamma\leq\frac{N}{\Lambda^{\eta^{\prime}}},$ then

1+\log_{\Lambda_{0}}\gamma\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1-\eta^{\prime}+\log_{\Lambda}N;

(ii)

$\gamma>\frac{N}{\Lambda^{\eta^{\prime}}},$ then

1+\log_{\Lambda_{0}}\gamma\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1+\log_{\Lambda}\gamma;

(iii)

$n_{u}=n$ and $\{x_{ui_{u}}\}_{i_{u}=0}^{n_{u}}$ are equally spaced points for $u\in\{1,2,\dots,m\}$ , then

\dim_{B}\left(G_{f^{*}}\right)=\begin{cases}1+\frac{\log\gamma}{\log n},&\quad\text{if~{}}\gamma>n^{m-\eta^{\prime}},\\ m,&\quad\text{if}~{}\gamma\leq n^{m-1}~{}\text{and}~{}\eta^{\prime}=1.\end{cases}

Theorem 4.3.

Let $n\in\mathbb{N}$ and $f^{*}$ be a FIF on SG with respect to the data $p$ on $V=V_{n}$ as in Remark 3.2. Assume that there exist $y_{1},y_{2},y_{3}\in V$ and $L\neq 0$ as in the framework of Lemma 4.1. If:

(i)

$\gamma_{1,0}\neq 0$ , then

$1+\log_{2^{n}}\gamma_{1,0}\leq\underline{\dim}_{B}\left(G_{f^{*}}\right);$
(ii)

$L>0$ and $\gamma_{2,0}\neq 0$ , then

$1+\log_{2^{n}}\gamma_{2,0}\leq\underline{\dim}_{B}\left(G_{f^{*}}\right);$
(iii)

$L<0$ and $\gamma_{3,0}\neq 0$ , then

$1+\log_{2^{n}}\gamma_{3,0}\leq\underline{\dim}_{B}\left(G_{f^{*}}\right);$

(iv)

$s_{i},q_{i}\in\mathcal{C}^{\eta}(\text{SG})$ for $i\in\mathcal{N}$ and either $\gamma_{1,0}=\gamma$ (or) $L>0$ and $\gamma_{2,0}=\gamma$ (or) $L<0$ and $\gamma_{3,0}=\gamma$ , then

\dim_{B}\left(G_{f^{*}}\right)=\begin{cases}1+\frac{\log\gamma}{\log 2^{n}},&\quad\text{if~{}}\gamma>\left(\frac{3}{2^{\eta^{\prime}}}\right)^{n},\\ \frac{\log 3}{\log 2},&\quad\text{if~{}}\gamma\leq\left(\frac{3}{2}\right)^{n}\text{~{}and~{}}\eta^{\prime}=1.\end{cases}

Proof.

From the assumption, we get $N=3^{n},r_{\nu}=\frac{1}{2^{n}}$ for $\nu\in\mathcal{N}=\{1,2,3\}^{n}$ and $\Lambda_{0}=\Lambda=2^{n}$ .
Let ${N}_{\text{S}}(k)$ and ${N}_{\text{S}}(k,\omega)$ denote the minimum number of sets belonging to the family

\left\{T\times\left[z,z+\frac{|K|}{2^{nk}}\right]:T~{}\text{is an equilateral triangle of side length~{}}\frac{|K|}{2^{nk}}\right\},

which covers $G_{f^{*}}$ and $G_{f^{*},\omega}$ respectively.
Let us suppose that $\gamma_{1,0}\neq 0$ .
From Lemma 4.1, for $k\in\mathbb{N},$ we get

{N}_{\text{S}}(k)=\underset{\omega\in\mathcal{N}^{k}}{\sum}{N}_{\text{S}}(k,\omega)\geq\underset{\omega\in\mathcal{N}^{k}}{\sum}\frac{2^{nk}}{|K|}s_{\omega_{1},1,0}s_{\omega_{2},1,0}\dots s_{\omega_{k},1,0}|L|=\frac{2^{nk}\gamma_{1,0}^{k}|L|}{|K|}.

(4.15)

Since

{N}_{\frac{|K|}{2^{nk}}}(G_{f^{*}})\leq{N}_{\text{S}}(k)\leq 3{N}_{\frac{|K|}{2^{nk}}}(G_{f^{*}})~{}~{}\text{for}~{}k\in\mathbb{N},

we have

\displaystyle\underline{\dim}_{B}(G_{f^{*}})=\liminf_{k\to\infty}\frac{\log{N}_{\text{S}}(k)}{\log\left(\frac{2^{nk}}{|K|}\right)}\overset{(\ref{7324e1})}{\geq}1+\log_{2^{n}}\gamma_{1,0}.

In a similar way, we can prove the other cases. ∎

Remark 4.2.

By using Lemma 4.1 and 4.2, in a similar way, we can estimate non-trivial lower bounds of the box dimension of FIFs on many different domains such as triangle, Sierpiński sickle etc.

Theorem 4.4.

Let $f^{*}$ be a FIF on an interval derived from Case 2, $s_{i},q_{i}\in\mathcal{C}^{\eta}(K)$ for $i\in\mathcal{N}$ and $\{x_{1i_{1}}\}_{i_{1}=0}^{n_{1}}$ be a equally spaced points collection. If $\gamma_{0}>N^{1-\eta}$ and $\underset{r\to\infty}{\lim}\frac{N(r)}{\left(N^{2-\eta}\right)^{r}}=\infty$ , then

1+\log_{N}(\gamma_{0})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1+\log_{N}(\gamma),

where $\gamma_{0}:=\sum_{i\in\mathcal{N}}\|s_{i}\|_{0}$ and $\|s_{i}\|_{0}=\inf\{|s_{i}(x)|:x\in K\}$ .

Proof.

From assumption, we have $m=1,r_{i}=\frac{1}{N}$ for $i\in\mathcal{N}$ and $\Lambda_{0}=\Lambda=N$ . From Theorem 4.1, we get the upper bound of $\overline{\dim}_{B}\left(G_{f^{*}}\right)$ .
For $i\in\mathcal{N},\omega\in\mathcal{N}^{k},k\in\mathbb{N}$ and $x,x^{\prime}\in l_{\omega}(K)$ , we have

	$\displaystyle\|f^{}(l_{i}(x))-f^{}(l_{i}(x^{\prime}))\|$
	$\displaystyle\overset{(\ref{2101e1})}{\geq}\\|s_{i}\\|_{0}\|f^{}(x)-f^{}(x^{\prime})\|-\\|f^{*}\\|_{\infty}\|s_{i}(x)-s_{i}(x^{\prime})\|-\|q_{i}(x)-q_{i}(x^{\prime})\|.$

For $i\in\mathcal{N},\omega\in\mathcal{N}^{k}$ and $k\in\mathbb{N}$ , we get $G_{f^{*},(i,\omega)}$ must cover at least a rectangle of size $\frac{|K|}{N^{k+1}}\times\frac{\|s_{i}\|_{0}({N}(k,\omega)-2)|K|}{N^{k}}-\|f^{*}\|_{\infty}\text{Osc}_{\omega}(s_{i})-\text{Osc}_{\omega}(q_{i})$ .
From (4.10), we have

	$\displaystyle{N}(k+1)=\underset{i\in\mathcal{N}}{\sum}\underset{\omega\in\mathcal{N}^{k}}{\sum}{N}(k+1,(i,\omega))$
	$\displaystyle~{}~{}\geq\underset{i\in\mathcal{N}}{\sum}\underset{\omega\in\mathcal{N}^{k}}{\sum}\frac{N^{k+1}}{\|K\|}\left(\frac{\\|s_{i}\\|_{0}({N}(k,\omega)-2)\|K\|}{N^{k}}-\\|f^{*}\\|_{\infty}\text{Osc}_{\omega}(s_{i})-\text{Osc}_{\omega}(q_{i})\right)$
	$\displaystyle~{}~{}\geq N\gamma_{0}({N}(k)-2N^{k})-\frac{N^{k+1}}{\|K\|}\underset{i\in\mathcal{N}}{\sum}\left(\\|f^{*}\\|_{\infty}[s_{i}]_{\eta}+[q_{i}]_{\eta}\right)N^{k\left(1-\eta\right)}$
	$\displaystyle~{}~{}\geq N\gamma_{0}{N}(k)-N^{k(2-\eta)}C^{\prime},$

for $k\in\mathbb{N}$ , where $C^{\prime}=\frac{N}{|K|}\underset{i\in\mathcal{N}}{\sum}\left(\|f^{*}\|_{\infty}[s_{i}]_{\eta}+[q_{i}]_{\eta}\right)+2N\gamma_{0}$ .
Via the mathematical induction method, we get

	$\displaystyle N(k)\geq\left(N\gamma_{0}\right)^{k-r}N(r)-C^{\prime}\left(\left(N\gamma_{0}\right)^{k-r-1}\left(N^{2-\eta}\right)^{r}+\left(N\gamma_{0}\right)^{k-r-2}\left(N^{2-\eta}\right)^{r+1}\right.$
	$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\quad\left.+\cdots+\left(N^{2-\eta}\right)^{k-1}\right)$
	$\displaystyle~{}~{}=\left(N\gamma_{0}\right)^{k-r}N(r)-C^{\prime}\left(N\gamma_{0}\right)^{k-r-1}\left(N^{2-\eta}\right)^{r}\left(1+\frac{N^{1-\eta}}{\gamma_{0}}+\cdots+\left(\frac{N^{1-\eta}}{\gamma_{0}}\right)^{k-r-1}\right)$
	$\displaystyle~{}~{}\geq\left(N\gamma_{0}\right)^{k-r}N(r)-C^{\prime}\left(N\gamma_{0}\right)^{k-r-1}\left(N^{2-\eta}\right)^{r}\frac{1}{1-\frac{N^{1-\eta}}{\gamma_{0}}}$
	$\displaystyle~{}~{}=\left(N\gamma_{0}\right)^{k}\left(\frac{N^{2-\eta}}{N\gamma_{0}}\right)^{r}\left(\frac{N(r)}{\left(N^{2-\eta}\right)^{r}}-\frac{C^{\prime}}{N\gamma_{0}\left(1-\frac{N^{1-\eta}}{\gamma_{0}}\right)}\right),$

for $k>r$ .
By assumption, there exists $r^{\prime}\in\mathbb{N}$ such that

\frac{N(r^{\prime})}{\left(N^{2-\eta}\right)^{r^{\prime}}}-\frac{C^{\prime}}{N\gamma_{0}\left(1-\frac{N^{1-\eta}}{\gamma_{0}}\right)}>0.

Thus

N(k)\geq\left(N\gamma_{0}\right)^{k}C^{\prime\prime},

for $k>r^{\prime}$ , where $C^{\prime\prime}=\left(\frac{N^{1-\eta}}{\gamma_{0}}\right)^{r^{\prime}}\left(\frac{N(r^{\prime})}{\left(N^{2-\eta}\right)^{r^{\prime}}}-\frac{C^{\prime}}{N\gamma_{0}\left(1-\frac{N^{1-\eta}}{\gamma_{0}}\right)}\right)$ .
Hence

\displaystyle\underline{\dim}_{B}(G_{f^{*}})=\liminf_{k\to\infty}\frac{\log{N}(k)}{\log\left(\frac{N^{k}}{|K|}\right)}\geq\lim_{k\to\infty}\frac{\log\left(N\gamma_{0}\right)^{k}}{\log N^{k}}=1+\frac{\log\gamma_{0}}{\log N}.

∎

Corollary 4.2.

Let $f^{*}$ be a FIF on an interval, $s_{i},q_{i}$ be continuous bounded variation maps for $i\in\mathcal{N}$ , $\{x_{1i_{1}}\}_{i_{1}=0}^{n_{1}}$ be a equally spaced points collection. If the interpolation points are not collinear (i.e., there exists $L\neq 0$ as in Lemma 4.1), and either $\gamma_{1,0}>1$ (or) $L>0$ and $\gamma_{2,0}>1$ (or) $L<0$ and $\gamma_{3,0}>1$ , then

1+\log_{N}(\gamma_{0})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1+\log_{N}(\gamma).

Proof.

From assumption, we get $s_{i},q_{i}\in\mathcal{C}^{\eta}(K)$ with $\eta=1$ for $i\in\mathcal{N}$ .
Let us suppose that $\gamma_{1,0}>1$ .
From Lemma 4.1, for $k\in\mathbb{N},$ we get

{N}(k)=\underset{\omega\in\mathcal{N}^{k}}{\sum}{N}(k,\omega)\geq\underset{\omega\in\mathcal{N}^{k}}{\sum}\frac{N^{k}}{|K|}s_{\omega_{1},1,0}s_{\omega_{2},1,0}\dots s_{\omega_{k},1,0}|L|=\frac{N^{k}\gamma_{1,0}^{k}|L|}{|K|}.

Since $\gamma_{1,0}>1$ , we get

\underset{k\to\infty}{\lim}\frac{N(k)}{N^{k}}=\infty.

Since $\gamma_{0}\geq\gamma_{1,0}>1$ , by using Theorem 4.4, we conclude

1+\log_{N}(\gamma_{0})\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1+\log_{N}(\gamma).

In a similar way, we can prove the cases. ∎

Example 4.1.

Let us consider the data set $\left\{(x_{0},0),(x_{1},1/2),(x_{2},1/3),(x_{3},0)\right\}$ and the signature $\epsilon_{1}=(0,0,0)$ , where $0=x_{0}<x_{1}<x_{2}<x_{3}=1$ .
Let $g_{i}:[0,1]\times\mathbb{R}\to\mathbb{R},i\in\{1,2,3\}$ given by

\displaystyle g_{1}(x,z)=\frac{x^{\eta_{1}}}{2}+\frac{f(x)z}{4},g_{2}(x,z)=\frac{-x^{\eta_{2}}}{6}+\frac{z}{2}+\frac{1}{2},g_{3}(x,z)=\frac{-x^{\eta_{3}}}{3}+\frac{3z}{4}+\frac{1}{3},

for $(x,z)\in[0,1]\times\mathbb{R}$ , where $f(x)=\sin(x)$ for $x\in[0,1]$ or $f(x)=1$ for $x\in[0,1],\eta_{1}\in(0,1]$ and $\eta_{2},\eta_{3}\in[1,\infty)$ .
We have:
$-$ $s_{i},q_{i}\in\mathcal{C}^{\eta}([0,1])$ for $i\in\{1,2,3\}$ , where $\eta=\min\{1,\eta_{1}\}$ ;
$-$ $\gamma=\frac{3}{2},\gamma_{2,1}=\frac{5}{4}$ if $f\equiv\sin$ and $\gamma_{2,1}=\gamma$ if $f\equiv 1$ ;
$-$ $\gamma\leq\frac{3}{\Lambda^{\eta}}$ if $\eta\leq\log_{\Lambda}2$ and $\gamma>\frac{3}{\Lambda^{\eta}}$ if $\eta>\log_{\Lambda}2$ .
$\mathit{Case}~{}(i).$ Consider $x_{1}=4/15$ and $x_{2}=3/5$ , then we get $\Lambda_{0}=\frac{15}{4}$ , $\Lambda=\frac{5}{2}$ , $x_{1}=\left(1-\frac{4}{9}\right)x_{0}+\frac{4}{9}x_{2}$ and $L=p(x_{1})-\left(\left(1-\frac{4}{9}\right)p(x_{0})+\frac{4}{9}p(x_{2})\right)>0$ .
Based on Theorem 4.1 and 4.2, we get

\displaystyle\underline{\dim}_{B}\left(G_{f^{*}}\right)\geq\begin{cases}1+\log_{\frac{15}{4}}\left(\frac{5}{4}\right),&\text{if~{}}f\equiv\sin,\\ 1+\log_{\frac{15}{4}}\left(\frac{3}{2}\right),&\text{if~{}}f\equiv 1\end{cases}

and

\displaystyle\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq\begin{cases}1-\eta+\log_{\frac{5}{2}}3,&\text{if~{}}\eta\leq\log_{2.5}2,\\ 1+\log_{\frac{5}{2}}\left(\frac{3}{2}\right),&\text{otherwise}.\end{cases}

$\mathit{Case}~{}(ii).$ Consider $x_{1}=1/3,x_{2}=2/3,f\equiv 1$ and $\eta_{1}>\log_{3}2$ , then we get $\Lambda_{0}=\Lambda=3,\gamma_{2,1}=\gamma$ , $x_{1}=\left(1-\frac{1}{2}\right)x_{0}+\frac{1}{2}x_{2}$ and $L=p(x_{1})-\left(\left(1-\frac{1}{2}\right)p(x_{0})+\frac{1}{2}p(x_{2})\right)>0$ .
From Corollary 4.1, we get

\dim_{B}(G_{f^{*}})=1+\frac{\log\left(1.5\right)}{\log 3}.

Refer to caption — Figure 1: Graph of FIF for $f\equiv\sin$ with $(\eta_{1},\eta_{2},\eta_{3})=(0.8,2,1)$ derived from the
Case $(i)$ of Example 4.1 and we get $1.1688\leq\underline{\dim}_{B}\left(G_{f^{*}}\right)\leq\overline{\dim}_{B}\left(G_{f^{*}}\right)\leq 1.44251$ .

5 Conclusion

According to the research done so far on the dimensional analysis of FIFs, FIFs on specific domains have been taken and their box dimensions have been analyzed. It is important to note that each domain so far considered can be written as the attractor of some suitable IFS. In this paper, we have analyzed the box dimension of FIF with its domain considered as an attractor of an arbitrary IFS. Hence, most of the FIFs constructed so far will become particular cases of our theory. Also, studies have shown that the domain of any new FIF can be likely be obtained as an attractor of a suitable IFS. Hence, in the future, if any new FIF is considered in a different domain, our dimensional results can be applied to such FIFs as well. Hence, the theory on which we have worked will act as a single platform for the study of the dimensional analysis of a large class of FIFs.

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	$\displaystyle d(f_{i}(x,z),f_{i}(x^{\prime},z^{\prime}))$
	$\displaystyle=\\|l_{i}(x)-l_{i}(x^{\prime})\\|_{2}^{\eta}+\theta\|s_{i}(x)z+q_{i}(x)-f^{}(l_{i}(x)))-(s_{i}(x^{\prime})z^{\prime}+q_{i}(x^{\prime})-f^{}(l_{i}(x^{\prime})))\|$
	$\displaystyle\overset{(\ref{2101e1})}{=}r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\|s_{i}(x)(z-f^{}(x))-s_{i}(x^{\prime})(z^{\prime}-f^{}(x^{\prime})))\|$
	$\displaystyle\leq r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\\|s_{i}\\|_{\infty}\|(z-f^{}(x))-(z^{\prime}-f^{}(x^{\prime}))\|$
	$\displaystyle~{}~{}~{}+\theta\|(z^{\prime}-f^{*}(x^{\prime}))\|\|s_{i}(x)-s_{i}(x^{\prime})\|$
	$\displaystyle\overset{(\ref{0702e2})\&(\ref{0702e11})}{\leq}r_{i}^{\eta}\\|x-x^{\prime}\\|_{2}^{\eta}+\theta\\|s\\|_{\infty}\|(z-f^{}(x))-(z^{\prime}-f^{}(x^{\prime}))\|+\theta 2MH_{i}\\|x-x^{\prime}\\|_{2}^{\eta}$
	$\displaystyle\leq c_{i}d((x,z),(x^{\prime},z^{\prime})),$

	$\displaystyle\text{Osc}(k,f)$	$\displaystyle=\underset{\omega\in\mathcal{N}^{k}}{\sum}\underset{x,x^{\prime}\in l_{\omega}(K)}{\sup}\|f(x)-f(x^{\prime})\|\leq H\underset{\omega\in\mathcal{N}^{k}}{\sum}\underset{x,x^{\prime}\in K}{\sup}\\|l_{\omega}(x)-l_{\omega}(x^{\prime})\\|_{2}^{\eta}$
		$\displaystyle\leq H\underset{\omega\in\mathcal{N}^{k}}{\sum}\left(\Lambda^{-k}\right)^{\eta}\underset{x,x^{\prime}\in K}{\sup}\\|x-x^{\prime}\\|_{2}^{\eta}=HN^{k}\Lambda^{-k\eta}\|K\|^{\eta}$
		$\displaystyle=H\|K\|^{\eta}\Lambda^{k\left(\log_{\Lambda}N-\eta\right)},$