Cantor Set as a Fractal and Its Application in Detecting Chaotic Nature of Piecewise Linear Maps

Abstract

We have investigated the Cantor set from the perspective of fractals and box-counting dimension. Cantor sets can be constructed geometrically by continuous removal of a portion of the closed unit interval [0, 1] infinitely. The set of points remained in the unit interval after this removal process is over is called the Cantor set. The dimension of such a set is not an integer value. In fact, it has a ‘fractional’ dimension, making it by definition a fractal. The Cantor set is an example of an uncountable set with measure zero and has potential applications in various branches of mathematics such as topology, measure theory, dynamical systems and fractal geometry. In this paper, we have provided three types of generalization of the Cantor set depending on the process of removal. Also, we have discussed some characteristics of the fractal dimensions of these generalized Cantor sets. Further, we have shown its application in detecting chaotic nature of the dynamics produced by iteration of piecewise linear maps.

Appendix

1.1 Proof of Theorem 2.1

Let $\lambda \,\,{\text{and}}\,\,\varLambda$, respectively, be the lub (least upper bound) and glb (greatest lower bound) of the subset S of R. Consider the complement $S^{c} = \left[ {\lambda ,\varLambda } \right] - S$ of S. Clearly, $S^{c}$ is union of countably many open intervals. Let G be the collection of all such open intervals. Also, $C^{c} = \left[ {0,\,\,1} \right] - C$ is a countable collection of open intervals. We denote this collection of open intervals by H. Now define $\phi :$ G $\to$ H as follows:

Let $F_{1} \in$ G be the interval of maximal length. Then, $\phi \left( {F_{1} } \right) = \left( {\frac{1}{3},\frac{2}{3}} \right) \in C^{c}$. Next, let $F_{21}$ and $F_{22}$ be the open intervals in G such that $F_{21}$ is left and $F_{22}$ is right to $F_{1}$ and of maximal length among the members of G other than $F_{1}$, then

$$ \phi \left( {F_{21} } \right) = \left( {\frac{1}{9},\frac{2}{9}} \right)\,\,\;{\text{and}}\,\,\;\,\,\,\phi \left( {F_{22} } \right) = \left( {\frac{7}{9},\frac{8}{9}} \right) $$

In this way one, can define $\phi$ on the whole set G, as G contains only finitely many sets of length greater than some fixed $\varepsilon > 0$ and since any two intervals in G or in H have different end points. Construction of $\phi$ ensures that $\phi$ is bijective and order preserving in the sense that if $F_{\alpha }$ is to the left of $F_{\beta }$, then $\phi \left( {F_{\alpha } } \right)$ is to the left of $\phi \left( {F_{\beta } } \right)$.

Now let us define $f:\left[ {\lambda ,\varLambda } \right] \to \left[ {0,1} \right]$ as follows:

For $F \in$ G, let $\left. f \right|_{F} :F \to \phi \left( F \right)$ be the unique linear increasing map, which maps $F$ bijectively to $\phi \left( F \right)$. Since S and C are totally disconnected, they are nowhere dense, and thus, there is a continuation $\xi :\left[ {\lambda ,\varLambda } \right] \to \left[ {0,1} \right]$ given by:

$$ \xi \left( x \right) = \sup \left\{ {f\left( y \right):y \notin S,y \le x} \right\} $$

Let $\psi = \left. \xi \right|_{S}$, then $\psi :S \to C$ is monotone increasing, continuous bijective map. Now, we want to show that $\zeta = \psi^{ - 1}$ is continuous. Clearly $\zeta$ is again monotone increasing. Let $x \in C\,\,{\text{and}}\;\,x_{n} \to x$. We need to show that $\zeta \left( {x_{n} } \right) \to \zeta \left( x \right)$. Since the sequence $\left\langle {x_{n} } \right\rangle$ contains a monotone subsequence, without loss of generality assume that $\left\langle {x_{n} } \right\rangle$ itself is monotone increasing.

Now, $y = \mathop {\lim }\limits_{n \to \infty } \zeta \left( {x_{n} } \right) = \mathop {\sup }\limits_{n \ge 1} \zeta \left( {x_{n} } \right) \le \zeta \left( x \right)$

If possible, suppose $y < \zeta \left( x \right)$. Since S is closed, we have $y \in S$ and $\zeta^{ - 1} \left( y \right) < x.$ This implies that $y < x_{n}$ for large n. Therefore, by monotonicity $y < \zeta \left( {x_{n} } \right)$ which contradicts the fact $y = \mathop {\lim }\limits_{n \to \infty } \zeta \left( {x_{n} } \right)$. Thus, we must have $y = \zeta \left( x \right)\,\,\,{\text{i}} . {\text{e}},\,\,\,\zeta \left( {x_{n} } \right) \to \zeta \left( x \right)$.□