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=> Apollonian Gasket
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=> Box Fractal
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=> Cactus Fractal
=> Cantor Dust
=> Cantor Function
=> Cantor Set
=> Cantor Square Fractal
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Ziyaretçi defteri
 

Cantor Function

 
CantorFunction

The Cantor function F(x) is defined as the function on [0,1] such that for values of x on the Cantor set, i.e.,

 x=sum_(i)2·3^(-n_i),
(1)

then

 F(x)=sum_(i)2^(-n_i),
(2)

which is then extended to other values by noting that F is monotone and has the same values on each removed endpoint (Chalice 1991).

The Cantor function is a particular case of a devil's staircase (Devaney 1987, p. 110), and can be extended to a function F_q for q>2, with q=3 corresponding to the usual Cantor function (Gorin and Kukushkin 2004).

Chalice (1991) showed that any real-valued function F(x) on [0,1] which is monotone increasing and satisfies

1. F(0)=0,

2. F(x/3)=F(x)/2,

3. F(1-x)=1-F(x)

is the Cantor function (Chalice 1991; Wagon 2000, p. 132).

Gorin and Kukushkin (2004) give the remarkable identity

 I_q(n)=int_0^1[F_q(t)]^ndt
            =1/(n+1)-(q-2)sum_(k=1)^(|_n/2_|)(n; 2k)(2^(2k-1)-1)/(q·2^(2k-1)-1)(B_(2k))/(n-2k+1)
(3)

for integer n. For q=3 and n=1, 2, ..., this gives the first few values as 1/2, 3/10, 1/5, 33/230, 5/46, 75/874, ... (Sloane's A095844 and A095845).

M. Trott (pers. comm., June 8, 2004) has noted that

 int_0^1[F(t)]^(F(t))dt approx 0.750387...
(4)

(Sloane's A113223), which seems to be just slightly greater than 3/4.

 

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