5-adic (nonarchimedean) Cantor one-fifth set
A sequence (s
i
)i ∈ ℕ of natural numbers between 0 and p-1 (inclusive) is a p-adic integer. We write this conventionally as .....s
i
.....s2 s1 s0. If ‘n’ is any natural number, and
is its p-adic representation (in other words, with each s
i
is a p-adic digit), then we identify ‘n’ with the p-adic integer (s
i
) with s
i
= 0 if i ≥ k (Madore 2000). Further, the set of p-adic 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 p-adic integers and the set (field) ℚ
p
of p-adic 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 p-adic 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 ℝd 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 5-adic integers ℤ5, that is, homeomorphic to Cantor one-fifth set.
Figure 1 below shows the representation of 5-adic Cantor one-fifth set ‘N’. To start the construction, initiator N0 = ℤ5 is subdivided into five equal subintervals 0 + 5ℤ5, 1 + 5ℤ5, 2 + 5ℤ5, 3 + 5ℤ5 and 4 + 5ℤ5. Drop the subintervals 1 + 5ℤ5 and 3 + 5ℤ5 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 5-adic Cantor one-fifth set (see Figure 1) N
k
consists of 3k disjoint closed intervals. Thus, the 5-adic Cantor one-fifth 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 f1, f2 and f3 be the similarity contraction mappings on 5-adic integer ℤ5 defined by
(1)
with scaling ratio 1/5. Then, the 5-adic Cantor one-fifth set N satisfies the self-referential equation
(2)
Proof: Using above construction of 5-adic Cantor one-fifth set, we can say that
for all k ≥ 1. Since, the mapping f
j
for j = 1, 2, 3 is one-to-one 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 f1[N] = ∩ f1[N
k
], f2[N] = ∩ f2[N
k
]and f3[N] = ∩ f3[N
k
],
therefore, f1[N] ∪ f2[N] ∪ f3[N] = ( ∩ f1[N
k
]) ∪ ( ∩ f2[N
k
]) ∪ ( ∩ f3[N
k
])
which gives the proof of the theorem.
Figure 2 shows the graphical representation of 5-adic Cantor one-fifth set using iterated function system (f1, f2, f3).
Quinary expansion of 5-adic Cantor one-fifth set
Theorem 3.2
The 5-adic Cantor one-fifth set is represented by the quinary expansion of its elements in the form
(3)
for all j = 0, 1, 2, .....
Proof: Let us define the inverse of similarity contraction mappings f1, f2 and f3, on ℤ5 as follows:
(4)
Now, for x
j
∊ {0, 1, 2, 3, 4}, for all j ≥ 0, either
(5)
if and only if either x0 = 1 or x0 = 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 x0 = 0, then we use the function f1-1 for all x ∊ N, if x0 = 2, then use the function f2-1 for all x ∊ N and if x0 = 4, then use the function f3-1 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 + 5ℤ5 and 3 + 5ℤ5 respectively. Thus, we found that
Hence either x ∈ 1 + 5ℤ5 or x ∈ 3 + 5ℤ5 which deduce that x ∉ N. Hence we proved that for x
j
∊ {0, 2, 4}, x ∊ N.
Conversely, let all the variables x = x0 + 51x1 + 52x2 + …, belong to ℤ5 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 + 5ℤ5 nor x ∈ 3 + 5ℤ5 which implies that x ∉ f
j
(1 + 5ℤ5) and also x ∉ f
j
(3 + 5ℤ5), for j ∊ W
l
= {1, 2, 3}l, l = 0, 1, 2, ..... Thus,
Thus, N ∪ Y = ℤ5 and hence x ∊ N, which completes the proof of the theorem.
Cantor one-fifth 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 = ℓ1, ℓ2, ℓ3, …, 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 one-dimensional 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 one-fifth set as an example of fractal string.
The Cantor one-fifth string ℵ, is the complement of [0, 1] of the usual Cantor one-fifth set F. The Figure 3 shows the geometrical representation of Cantor one-fifth string.
Thus, we obtain
where, ℓ1 = (l1 = l2 = 1/5), ℓ2 = (l3 = l4 = l5 = l6 = l7 = l8 = 1/25) and so on. Continuing in this way, we find that the lengths of open intervals is consist of ℓ
k
= 5-k-1 with multiplicity for k = 0, 1, 2, ....
Thus, the geometric zeta function of the Cantor one-fifth string is determined by the sequence ℵ:
(6)
The poles of the such function are the set of complex numbers (see (Lapidus and Hung 2008), pp. 7) and given by
(7)
where D = log 3/log 5 = 0.6826 is the dimension of Cantor one-fifth set and p = 2π/log 5 oscillatory period of Cantor one-fifth string ℵ, is called complex dimension of Cantor one-fifth string.
Further, representation of Cantor one-fifth string may be seen in Figure 4 using fractal harp.
5-adic Cantor one-fifth set as fractal string
Since, the construction of 5-adic Cantor one-fifth string (ξ) is analogue to the usual Cantor one-fifth set. We start, by subdividing the interval ℤ5 into closed subintervals
since, fractal string is complement of the usual Cantor one-fifth set in the closed interval [0, 1], the remaining open subintervals after this step are given by
then, the G1 ∪ G2 is the first sub-ring of self similar 5-adic Cantor one-fifth string. The lengths of G1 and G2 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 G3 ∪ G4 ∪ G5 ∪ G6 ∪ G7 ∪ G8is the second set of self-similar 5-adic Cantor one-fifth string. Thus, the length is given by
Repeating the same process over and over again, we obtain a sequence ℓ1 = ℓ2 = ℓ3 = ℓ4 = ℓ5 = ..... which consists of lengths 5-k-1 with multiplicity 2.3k. Using Figure 5 the 5-adic Cantor one-fifth string can also be written as follows:
From Definition 2.3 (Lapidus and Hung 2009), the geometric zeta function of ξ is given by
(8)
the poles of the such function are the set of complex numbers
(9)
where D = log 3/log 5 = 0.6826 is the dimension of 5-adic Cantor one-fifth string and p = 2π/log 5 oscillatory period is the volume of the inner tubular neighborhood of ξ.