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New 5adic Cantor sets and fractal string
SpringerPlus volume 2, Article number: 654 (2013)
Abstract
In the year (1879–1884), George Cantor coined few problems and consequences in the field of set theory. One of them was the Cantor ternary set as a classical example of fractals. In this paper, 5adic Cantor onefifth set as an example of fractal string have been introduced. Moreover, the applications of 5adic Cantor onefifth set in string theory have also been studied.
Introduction
During the late eighteenth century, mathematicians delighted in producing sets with ever more weird properties, many of them now recognized to be fractal in nature (Crilly et al.). George Cantor (1879–1884) wrote a series of papers entitled “Uber unendliche lineare punktmannichfaltigkeiten” (Cantor 1879; 1880; 1882; 1883a; 1883b; 1884) that contained the first systematic treatment of the point set topology of real line, in which he triggered some problems and consequences in the field of set theory. One of these is the classical Cantor set problem devised by Cantor in the footnote to a statement saying that perfect sets do not need to be everywhere dense (Fleron 1994). In last two decades, Devil’s and other researchers established the graphical representation of Cantor sets in the form of staircases (Horiguchi and Morita 1984a; 1984b; Rani and Prasad 2010).
Middle onethird, a classical Cantor set found a celebrated place in the mathematical analysis and in its applications (Hutchinson 1981; Mendes 1999; Shaver 2010). For a fundamental work on Cantor set and its applications, one may refer to (Peitgen et al. 2004), (Devaney 1992), (Beardon 1965), (Falconer 1985), (Lapidus and van Frankenhuijsen 2006), (Gutfraind et al. 1990) and (Lee 1998). In recent years, padic analysis has been used in various areas of mathematics as well as in aspects of quantum physics and string theory (Lapidus and van Frankenhuijsen 2006). For a detailed analysis of fractal string and padic integers, one may refer to (Chistyakov 1996; Hung 2007; Koblitz 1984; Robert 2000; Schikhof 1984; Vladimirov et al. 1994).
Lapidus and van Frankenhuijsen (2000; 2006) introduced the concept of fractal string and established the geometric zeta function, zeros of zeta function, spectra of fractal string and the complex dimension of the fractal string. In 2008, (Lapidus 2008) suggested that fractal string and their quantization may be related to aspects of string theory. In last few decades, M. L. Lapidus, jointly with other researchers generalized and introduced the various properties of fractal string (see (Edgar 2008; Lapidus 1992; Lapidus and Maier 1995; Lapidus and Pearse 2006; 2008; Lapidus and Pomerance 1993)).
In 2008, (Lapidus and Hung 2008; 2009) provided a framework for unifying the archimedean and padic (nonarchimedean) fractal string with their geometric zeta functions and complex dimensions for 3adic Cantor sets and also the general case for padic Cantor sets respectively. Recently, (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted and Ashish, Mamta Rani and Renu Chugh, Study of Variants of Cantor sets., submitted) studied the variants of Cantor sets and established their mathematical analysis using mathematical feedback system and iterated function system respectively.
Our goal in this paper is to study the Cantor onefifth set as a new classical example of fractal string. Moreover, the nonarchimedean (5adic) Cantor onefifth set with their applications in string theory has also been established. In the third section, the main results of our study have been presented, followed by the “Concluding remarks” section.
Preliminaries
In this section, we recall some basic definitions pertaining to the notion of (ordinary) fractal string and introduce several new ones such as the most important of which are quinary expansion and Cantor onefifth set:
Definition 2.1. Cantor onefifth set
The Cantor onefifth set for unequal intervals is defined as the F = ∩ F_{n+1}, where F_{n+1} is constructed by dividing F_{ n } in five unequal line segments and removing second and fourth onefifth line segment, F_{0} being the closed interval 0 ≤ x ≤ 1 (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted).
Definition 2.2. Quinary expansion
The sequence 0.x_{1}x_{2}x_{3}x_{4}x_{5}…, where each x_{ i } is either 0, 1, 2, 3, or 4 is called quinary expansion of x if x = x_{1}/5 + x_{2}/5^{2} + x_{3}/5^{3} + ....
For example, the sequence 0.04444… is the quinary expansion of 1/5 since we have
Lapidus and van Frankenhuijsen (2000) and (2006), introduced the concept of fractal strings as follows:
Definition 2.3. Fractal string
A fractal string Ω is a bounded open subset of the real line R. The collection of lengths ℓ_{ j } of the disjoint intervals is denoted by L.
For example, the complement of the Cantor set in the closed unit interval [0, 1] is a Cantor string. Moreover, the topological boundary of Cantor string is the Cantor set C itself.
Definition 2.4. Geometric zeta function
The geometric zeta function of a fractal string Ω with lengths L is
where ℓ_{1}, ℓ_{2}, …, ℓ_{ k } are the lengths of open intervals and m_{ k } be the corresponding multiplicity of open intervals (Lapidus and van Frankenhuijsen 2000).
For example, Cantor string consists of intervals of lengths ℓ_{1} = (l_{1} = 1/3), ℓ_{2} = (l_{2} = l_{3} = 1/9), ℓ_{3} = (l_{4} = l_{5} = l_{6} = l_{7} = 1/27), and so on, that is, the lengths are the numbers 3^{k1} with multiplicity {m}_{{3}^{k1}}={2}^{k} for k = 0, 1, 2, 3, …. . So, the geometric zeta function is:
where D = log2/log3 is the dimension of usual Cantor set.
Recently, (Ashish, Mamta Rani and Renu Chugh, Variants of Cantor Sets Using IFS, submitted), established the selfsimilarity of the Cantor onefifth set using the iteration function system as follows:
Theorem 2.1
Let f_{1}, f_{2} and f_{3} be the similarity contraction mappings on ℝ defined by
where all the mappings have the ratio 1/5. Then, the Cantor onefifth set F satisfies the selfreferential equation
for the iterated function system (f_{1}, f_{2}, f_{3}).
Main results
5adic (nonarchimedean) Cantor onefifth set
A sequence (s_{ i })_{i ∈ ℕ} of natural numbers between 0 and p1 (inclusive) is a padic integer. We write this conventionally as .....s_{ i }.....s_{2} s_{1} s_{0}. If ‘n’ is any natural number, and
is its padic representation (in other words, n={\displaystyle {\sum}_{i=0}^{k1}{s}_{i}}{p}^{i} with each s_{ i } is a padic digit), then we identify ‘n’ with the padic integer (s_{ i }) with s_{ i } = 0 if i ≥ k (Madore 2000). Further, the set of padic integers, which we call ℤ_{ p } with two binary operations on it (addition and multiplication) is a ring. The relation between the set (ring) ℤ_{ p } of padic integers and the set (field) ℚ_{ p } of padic numbers is the same as between the set (ring) ℤ of integers and the set (field) ℚ of rationals (Madore 2000). Since, ℤ_{ p } is an important subspace of ℚ_{ p }, it can be represented as follows:
For this padic expansion, we can also write
where c + pℤ_{ p } = {y ∈ ℚ_{ p } : y  c_{ p } ≤ 1/p} (Lapidus and van Frankenhuijsen 2006) It is also known that there are topological models of ℤ_{ p } in the Euclidean space ℝ^{d} as fractal spaces such as the Cantor set and the Sierpinsky gasket (Robert 2000), where ℤ_{ p } is homeomorphic to the ternary Cantor set. Now, we consider the ring of 5adic integers ℤ_{5}, that is, homeomorphic to Cantor onefifth set.
Figure 1 below shows the representation of 5adic Cantor onefifth set ‘N’. To start the construction, initiator N_{0} = ℤ_{5} is subdivided into five equal subintervals 0 + 5ℤ_{5}, 1 + 5ℤ_{5}, 2 + 5ℤ_{5}, 3 + 5ℤ_{5} and 4 + 5ℤ_{5}. Drop the subintervals 1 + 5ℤ_{5} and 3 + 5ℤ_{5} and repeat the same process for the remaining subintervals. Further, repeating the same process over and over again, by removing the open subintervals of second and fourth position at each step from each closed interval, we obtain a sequence N_{ k } for k = 1, 2, . . . The 5adic Cantor onefifth set (see Figure 1) N_{ k } consists of 3^{k} disjoint closed intervals. Thus, the 5adic Cantor onefifth set would be the limit ‘N’ of the sequence N_{ k } of sets. So, we define limit ‘N’ as the intersection of the sets N_{ k } i.e.
Theorem 3.1
Let f_{1}, f_{2} and f_{3} be the similarity contraction mappings on 5adic integer ℤ_{5} defined by
with scaling ratio 1/5. Then, the 5adic Cantor onefifth set N satisfies the selfreferential equation
Proof: Using above construction of 5adic Cantor onefifth set, we can say that
for all k ≥ 1. Since, the mapping f_{ j } for j = 1, 2, 3 is onetoone and N = ∩ N_{ k }, then it implies that
f_{ j }[N] = f_{ j }[ ∩ N_{ k }] = ∩ f_{ j }[N_{ k }], for k = 1, 2, ….
so that, we can write f_{1}[N] = ∩ f_{1}[N_{ k }], f_{2}[N] = ∩ f_{2}[N_{ k }]and f_{3}[N] = ∩ f_{3}[N_{ k }],
therefore, f_{1}[N] ∪ f_{2}[N] ∪ f_{3}[N] = ( ∩ f_{1}[N_{ k }]) ∪ ( ∩ f_{2}[N_{ k }]) ∪ ( ∩ f_{3}[N_{ k }])
which gives the proof of the theorem.
Figure 2 shows the graphical representation of 5adic Cantor onefifth set using iterated function system (f_{1}, f_{2}, f_{3}).
Quinary expansion of 5adic Cantor onefifth set
Theorem 3.2
The 5adic Cantor onefifth set is represented by the quinary expansion of its elements in the form
for all j = 0, 1, 2, .....
Proof: Let us define the inverse of similarity contraction mappings f_{1}, f_{2} and f_{3}, on ℤ_{5} as follows:
Now, for x_{ j } ∊ {0, 1, 2, 3, 4}, for all j ≥ 0, either
if and only if either x_{0} = 1 or x_{0} = 3, respectively. Let η, μ ∈ ℕ be the fixed subscript numbers such that x_{ η } = 1and x_{ μ } = 3. Thus, x_{ j } = 0, 2 or 4, for all j > η and all j > μ. Since, we have divided the real line into five equal line segments denoted by 0, 1, 2, 3, and 4 respectively. Thus, if x_{0} = 0, then we use the function f_{1}^{1} for all x ∊ N, if x_{0} = 2, then use the function f_{2}^{1} for all x ∊ N and if x_{0} = 4, then use the function f_{3}^{1} for all x ∊ N. Thus, from these three cases, we obtain
again repeating the process in this manner, we obtain the general case
which lie in the intervals 1 + 5ℤ_{5} and 3 + 5ℤ_{5} respectively. Thus, we found that
Hence either x ∈ 1 + 5ℤ_{5} or x ∈ 3 + 5ℤ_{5} which deduce that x ∉ N. Hence we proved that for x_{ j } ∊ {0, 2, 4}, x ∊ N.
Conversely, let all the variables x = x_{0} + 5^{1}x_{1} + 5^{2}x_{2} + …, belong to ℤ_{5} for all x_{ j } ∊ {0, 2, 4}, and j = 0, 1, 2, …. Then, from Eq. (3) and (5), we can say that neither x ∈ 1 + 5ℤ_{5} nor x ∈ 3 + 5ℤ_{5} which implies that x ∉ f_{ j }(1 + 5ℤ_{5}) and also x ∉ f_{ j }(3 + 5ℤ_{5}), for j ∊ W_{ l } = {1, 2, 3}^{l}, l = 0, 1, 2, ..... Thus,
Thus, N ∪ Y = ℤ_{5} and hence x ∊ N, which completes the proof of the theorem.
Cantor onefifth set as fractal string
It is well known from the definition of fractal string that such a set consists of countably many disjoint open intervals. The lengths of which form a sequence L = ℓ_{1}, ℓ_{2}, ℓ_{3}, …, called the lengths of the string. We can assume without loss of generality that
where each length is counted according to its multiplicity. An ordinary fractal string can be thought of as a onedimensional drum with fractal boundary. In the literature of fractal geometry, we found a classical example of the fractal string as Cantor string. It is the set, complement of the interval [0, 1] of the usual ternary Cantor set. It is one of the simplest and most important example in the research of fractal string by (Lapidus and van Frankenhuijsen 2006). Information about the geometry of Cantor string like Minkowski dimension and the Minkowski measurability is obtained from its geometric zeta function. Motivated by the research of Lapidus with other researcher’s (Lapidus and Hung 2008) on the Cantor string, we introduce a new Cantor onefifth set as an example of fractal string.
The Cantor onefifth string ℵ, is the complement of [0, 1] of the usual Cantor onefifth set F. The Figure 3 shows the geometrical representation of Cantor onefifth string.
Thus, we obtain
where, ℓ_{1} = (l_{1} = l_{2} = 1/5), ℓ_{2} = (l_{3} = l_{4} = l_{5} = l_{6} = l_{7} = l_{8} = 1/25) and so on. Continuing in this way, we find that the lengths of open intervals is consist of ℓ_{ k } = 5^{k1} with multiplicity {m}_{{5}^{k1}}={2.3}^{k} for k = 0, 1, 2, ....
Thus, the geometric zeta function of the Cantor onefifth string is determined by the sequence ℵ:
The poles of the such function are the set of complex numbers (see (Lapidus and Hung 2008), pp. 7) and given by
where D = log 3/log 5 = 0.6826 is the dimension of Cantor onefifth set and p = 2π/log 5 oscillatory period of Cantor onefifth string ℵ, is called complex dimension of Cantor onefifth string.
Further, representation of Cantor onefifth string may be seen in Figure 4 using fractal harp.
5adic Cantor onefifth set as fractal string
Since, the construction of 5adic Cantor onefifth string (ξ) is analogue to the usual Cantor onefifth set. We start, by subdividing the interval ℤ_{5} into closed subintervals
since, fractal string is complement of the usual Cantor onefifth set in the closed interval [0, 1], the remaining open subintervals after this step are given by
then, the G_{1} ∪ G_{2} is the first subring of self similar 5adic Cantor onefifth string. The lengths of G_{1} and G_{2} are given by using the Haar measure (Gupta and Jain 1986) as follows:
Again repeating the same process, by subdividing the closed intervals of first step (see Figure 1), we get
Thus, the remaining open subintervals are given by
The subring G_{3} ∪ G_{4} ∪ G_{5} ∪ G_{6} ∪ G_{7} ∪ G_{8}is the second set of selfsimilar 5adic Cantor onefifth string. Thus, the length is given by
Repeating the same process over and over again, we obtain a sequence ℓ_{1} = ℓ_{2} = ℓ_{3} = ℓ_{4} = ℓ_{5} = ..... which consists of lengths 5^{k1} with multiplicity 2.3^{k}. Using Figure 5 the 5adic Cantor onefifth string can also be written as follows:
From Definition 2.3 (Lapidus and Hung 2009), the geometric zeta function of ξ is given by
the poles of the such function are the set of complex numbers
where D = log 3/log 5 = 0.6826 is the dimension of 5adic Cantor onefifth string and p = 2π/log 5 oscillatory period is the volume of the inner tubular neighborhood of ξ.
Concluding remarks
Based on the results, our conclusions are following:

1.
In Subsection “5adic (nonarchimedean) Cantor onefifth set”, using 5adic integer it has been concluded that Cantor onefifth set satisfies the nonarchimedean properties of a set and also studied that nonarchimedean Cantor onefifth set satisfies selfsimilarity property using selfreferential equation.

2.
Further, it has been concluded that quinary Cantor onefifth set is homeomorphic to 5adic Cantor onefifth set N in subsection “Quinary expansion of 5adic Cantor onefifth set”.

3.
In Subsection “Cantor onefifth set as fractal string” and “5adic Cantor onefifth set as fractal string”, it has been analyzed that Cantor onefifth set and 5adic Cantor onefifth set both satisfy the properties of fractal string. Moreover, we found that the geometric zeta function and the complex dimension of both the sets are perfectly same.
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This Research is supported by the University Grant Commission of India (Grant No. 3929/2010(SR)).
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Kumar, A., Rani, M. & Chugh, R. New 5adic Cantor sets and fractal string. SpringerPlus 2, 654 (2013). https://doi.org/10.1186/219318012654
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DOI: https://doi.org/10.1186/219318012654
 Cantor onefifth set
 padic integers
 5adic Cantor onefifth set
 Iterated function system (IFS)
 Fractal string
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 28A80
 11E20
 26E30
 28E30
 26A80
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