Cantor set

Under the Cantor set, Cantor set, also cantor MOORISH Diskontinuum, Cantor dust or wipe amount called, is meant in mathematics a certain subset of the amount of real numbers with special topological , measure-theoretic , geometric and set-theoretic properties: it is

• compact , perfect , totally incoherent (a "discontinuum") and nowhere close ;
• a Lebesgue - null set ;
• self-similar and has a non-integer Hausdorff dimension (so it is a fractal );
• equal to the continuum (the set of all real numbers), so in particular uncountable .

The Cantor set is named after the mathematician Georg Cantor .

More generally, one also calls certain sets or topological spaces Cantor sets if they have some of these properties. Which of these properties are required depends on the mathematical field and often also on the context. A topological space that is homeomorphic to the Cantor set is called a Cantor space .

In addition to many multi-dimensional variants of the Cantor set, the main example of this article, the middle third Cantor set , is one-dimensional.

construction

The first five iteration steps for the construction of the Cantor set

Cuts from intervals

The Cantor set can be constructed using the following iteration:

You start with the closed interval of real numbers from 0 to 1. The open middle third is removed from this interval (wiped away), i.e. all numbers that are strictly between 1/3 and 2/3. The two intervals and remain . The open middle third is from these two intervals each turn away and you will now receive four intervals: , , and . The open middle thirds are in turn removed from these intervals. This step is repeated an infinite number of times. ${\displaystyle [0,1]}$ ${\displaystyle \left[0,{\tfrac {1}{3}}\right]}$ ${\displaystyle \left[{\tfrac {2}{3}},1\right]}$ ${\displaystyle \left[0,{\tfrac {1}{9}}\right]}$ ${\displaystyle \left[{\tfrac {2}{9}},{\tfrac {1}{3}}\right]}$ ${\displaystyle \left[{\tfrac {2}{3}},{\tfrac {7}{9}}\right]}$ ${\displaystyle \left[{\tfrac {8}{9}},1\right]}$

Using the function

${\displaystyle {\begin{array}{llll}T\;\colon \;&{\mathcal {P}}([0,1])&\to &{\mathcal {P}}([0,1])\\&A&\mapsto &{\frac {1}{3}}{\bigl (}A\cup (2+A){\bigr )},\end{array}}}$

which maps any subsets of the interval back into subsets of , the wiping away of the middle third can be formalized. Here are translational and scaling a set element by element made. ${\displaystyle [0,1]}$ ${\displaystyle [0,1]}$

You start from the crowd and bet ${\displaystyle A_{0}:=[0,1]}$

${\displaystyle A_{n}:=T(A_{n-1})}$   for   . ${\displaystyle n\in \mathbb {N} ^{+}}$

The intersection of all these sets is then the Cantor set

${\displaystyle {\mathcal {C}}:=\bigcap _{n=0}^{\infty }A_{n}=\lim _{n\to \infty }A_{n}}$,

and it is . The Cantor set thus consists of all points that have survived each wiping away. In the borderline case (intersection over all -th wiped quantities, ) the proportion of the original interval is zero, although there are still an uncountable number of elements. This construction method is related to that for the Koch curve . ${\displaystyle T({\mathcal {C}})={\mathcal {C}}}$ ${\displaystyle n}$ ${\displaystyle n\to \infty }$

Explicit formulas for the Cantor set are

${\displaystyle {\mathcal {C}}=[0,1]\smallsetminus \bigcup _{n=1}^{\infty }\;\bigcup _{k=0}^{3^{n-1}-1}\;\;\left]{\frac {3k+1}{3^{n}}},{\frac {3k+2}{3^{n}}}\right[}$,

where every middle third   is deleted from the closed interval as the open interval   by quantity subtraction , or ${\displaystyle \textstyle \left]{\frac {3k+1}{3^{n}}},{\frac {3k+2}{3^{n}}}\right[}$ ${\displaystyle \textstyle \left[{\frac {3k+0}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]=\left[{\frac {k+0}{3^{n-1}}},{\frac {k+1}{3^{n-1}}}\right]}$

${\displaystyle {\begin{array}{lll}{\mathcal {C}}&=\lim _{n\to \infty }&\bigcup _{k=0}^{3^{n-1}-1}\;\;\left[{\frac {3k+0}{3^{n}}},{\frac {3k+1}{3^{n}}}\right]\cup \left[{\frac {3k+2}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]\\&=\bigcap _{n=1}^{\infty }&\bigcup _{k=0}^{3^{n-1}-1}\;\;\left[{\frac {3k+0}{3^{n}}},{\frac {3k+1}{3^{n}}}\right]\cup \left[{\frac {3k+2}{3^{n}}},{\frac {3k+3}{3^{n}}}\right],\end{array}}}$

where the middle third is     removed from the completed previous interval by averaging with the union set . ${\displaystyle \textstyle \left]{\frac {3k+1}{3^{n}}},{\frac {3k+2}{3^{n}}}\right[}$ ${\displaystyle \textstyle \left[{\frac {k+0}{3^{n-1}}},{\frac {k+1}{3^{n-1}}}\right]=\left[{\frac {3k+0}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]}$ ${\displaystyle \textstyle \left[{\frac {3k+0}{3^{n}}},{\frac {3k+1}{3^{n}}}\right]\cup \left[{\frac {3k+2}{3^{n}}},{\frac {3k+3}{3^{n}}}\right]}$

As a ternary development

The Cantor set can also be described as the set of all numbers in the interval that are represented as a decimal point on base 3, in which only the digits 0 and 2 occur. The representation for base 3 is also called "ternary development". Each number from the interval can be represented as ${\displaystyle [0,1]}$ ${\displaystyle [0,1]}$

${\displaystyle x=\sum _{i=1}^{\infty }{\frac {x_{i}}{3^{i}}}}$,

where is. For example, with and for . ${\displaystyle x_{i}\in \{0,1,2\}}$ ${\displaystyle {\frac {23}{27}}={\frac {2}{3}}+{\frac {1}{9}}+{\frac {2}{27}}}$ ${\displaystyle x_{1}=2,\,x_{2}=1,\,x_{3}=2}$ ${\displaystyle x_{i}=0}$ ${\displaystyle i>3}$

Incidentally, the set constructed above is equal to the set of numbers whose ternary expansion does not include the digit 1 up to and including the -th place: ${\displaystyle A_{n}}$ ${\displaystyle n}$

${\displaystyle A_{n}={\bigl \{}x\in [0,1]\,{\big |}\,x_{i}\in \{0,2\}{\text{ f}}\mathrm {\ddot {u}} {\text{r alle }}i\leq n{\bigr \}}}$

The intersection of all these sets is again the Cantor set and thus contains all numbers whose 3-adic expansion does not contain a 1. In particular, the Cantor set contains more than just the edge points of the removed intervals; these edge points are the abbreviated fractions in , the denominator of which is a power of 3, which can be written with a period 0 end but also with a period 2 end, for example ${\displaystyle p/q}$ ${\displaystyle [0,1]}$ ${\displaystyle q}$

${\displaystyle 0{,}1{\overline {0}}_{3}=1\cdot 3^{-1}+0\cdot 3^{-2}+0\cdot 3^{-3}+\dotsb =1/3=2\cdot 3^{-2}+2\cdot 3^{-3}+\dotsb =0{,}0{\overline {2}}_{3}}$

the left edge point of the interval removed in the first step. The use of the digit 1 is circumvented by the period 2 end, which can be used to represent the same number. (This is only possible for a 1 directly before the end of period 0. However, no 1 can appear elsewhere, since the number would otherwise be in the middle of one of the deleted intervals.) A right edge point of an interval that has been deleted has the number in its 1 avoiding ternary representation a period 0 end, such as B. . If these numbers have neither a period-0-end nor a period-2-end, then they belong to the Cantor set, e.g. B. also 1/4: ${\displaystyle 2/3=0{,}2{\overline {0}}_{3}=0{,}1{\overline {2}}_{3}}$

${\displaystyle 1/4=2\cdot 3^{-2}+2\cdot 3^{-4}+2\cdot 3^{-6}+\cdots =0{,}{\overline {02}}_{3}}$

properties

• The Cantor set is closed in : In each iteration step open sets are removed, the union of these sets is then open and the complement of the Cantor set. Thus the Cantor set is complete. ${\displaystyle \mathbb {R} }$
• With the boundedness of the Cantor set and the Heine-Borel Theorem , it follows that the Cantor set is compact .
• The uncountability of the Cantor set can be shown with a diagonalisability argument and the ternary expansion of the numbers in the Cantor set. The numbers in the Cantor set are represented in their ternary expansion of all elements of , that is, sequences that contain only the zero and the two. If one assumes that this set can be counted, this can lead to a contradiction by constructing a number with a ternary expansion that is not included in the count. ${\displaystyle \{0,2\}^{\mathbb {N} }}$
• The interior of the Cantor crowd is empty. The Cantor set consists only of edge points , all of which are accumulation points .
• No point in the Cantor set is isolated . The Cantor set is thus insecure and, since it is closed, also perfect.
• Since the set of edge points of the removed intervals can be counted, the difference set remains uncountable after their removal. This is no longer complete as a subset in , but it is also not open. ${\displaystyle \{x\in [0,1]\mid \exists k\in \mathbb {N} _{0}:x\,3^{k}\in \mathbb {Z} \}\;\subset \mathbb {N} _{0}\,3^{-\mathbb {N} _{0}}}$ ${\displaystyle \mathbb {R} }$
• The subspace topology (relative topology) of the Cantor set is both open and closed. Equipped with this topology, it is homeomorphic to ( see below ) as well as to the whole -adic numbers . ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle p}$
• The Hausdorff dimension and the Minkowski dimension of the Cantor set are . This follows from the fact that in each construction step two copies of the quantity are created, which are scaled by the factor . ${\displaystyle D=\ln(2)/\ln(3)=0{,}6309\ldots }$ ${\displaystyle 1/3}$
• The one-dimensional Lebesgue-Borel measure of the Cantor set is zero, so it is a - zero set . At first it is closed, i.e. contained in Borel's σ-algebra and therefore Borel can be measured . A measure can therefore be assigned to the Cantor set. When the function is iterated, the number of intervals doubles as a result of the translation in each step, with the length of each interval doubling in each step. Since all intervals are disjoint, the Lebesgue-Borel measure then applies due to the σ-additivity ${\displaystyle \lambda ^{1}}$ ${\displaystyle \lambda ^{1}}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f}$

${\displaystyle \lambda ^{1}({\mathcal {C}})=\lim _{n\to \infty }\lambda ^{1}(A_{n})=\lim _{n\to \infty }{\frac {2^{n}}{3^{n}}}=0}$.

Thus, the Lebesgue measure of the Cantor set is also zero, since the Borel σ-algebra is contained in the Lebesgue σ-algebra and the measures there match.

0-1 episodes

The Cartesian product of countable infinite number of copies of the two-element set is the set of all infinite sequences which only take the values ​​0 and 1, i.e. H. the set of all functions . This amount is also referred to as or . The natural bijection is a homeomorphism between the Cantor set and the topological space if this is equipped with its natural topology (namely with the product topology induced on the set by the discrete topology ). The topological space is therefore called Cantor space . ${\displaystyle \{0,1\}}$ ${\displaystyle x\colon \mathbb {N} \to \{0,1\}}$ ${\displaystyle \{0,1\}^{\mathbb {N} }}$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle {\mathcal {C}}\to 2^{\mathbb {N} }}$ ${\displaystyle 2^{\mathbb {N} }}$ ${\displaystyle \{0,1\}}$ ${\displaystyle 2^{\mathbb {N} }}$

The mentioned bijection can be extended to surjection

${\displaystyle {\begin{array}{rlll}f\;\colon &{\mathcal {C}}&\to &[0,1]\\&\sum _{i\in \mathbb {N} }x_{i}3^{-i}&=&\sum _{i\in \mathbb {N} }{\frac {x_{i}}{2}}2^{-i}&&{\bigl (}\,\forall i\in \mathbb {N} :x_{i}\in \{0,2\}\,{\bigr )}\end{array}}}$

between the Cantor set and the interval . This surjection is not injective because the left and right edge points of a wiped-away interval are mapped onto the same point, e.g. is at the interval ${\displaystyle [0,1]}$ ${\displaystyle \left]{\tfrac {1}{3}},{\tfrac {2}{3}}\right[}$

${\displaystyle {\begin{array}{lcl}f{\bigl (}{}^{1}\!\!/\!_{3}{\bigr )}=f(0{,}0{\overline {2}}_{3})=0{,}0{\overline {1}}_{2}=\!\!&\!\!0{,}1_{2}\!\!&\!\!=0{,}1{\overline {0}}_{2}=f(0{,}2{\overline {0}}_{3})=f{\bigl (}{}^{2}\!\!/\!_{3}{\bigr )}.\\&\parallel \\&{}^{1}\!\!/\!_{2}\end{array}}}$

On the other hand is . ${\displaystyle f(1/4)=f(0{,}{\overline {02}}_{3})=0{,}{\overline {01}}_{2}=1/3}$

Cantor distribution and Cantor function

Closely related to the Cantor set is the Cantor distribution . It is constructed similarly to the Cantor set. Their distribution function is also called the Cantor function .

The Cantor distribution often serves as an example for the existence of continuously singular distributions that are singular with respect to the Lebesgue measure , but still have a continuous distribution function (functions with so-called singular-continuous behavior).

Other Cantor sets

The Cantor set (also middle thirds Cantor set) was described above. Under a Cantor set is defined as a set of real numbers that you get with a variant of the above wiping process, which can now vary the lengths and numbers of weggewischten intervals:

You start with any closed interval of real numbers. In the first step one removes finitely many open and (including their edge ) disjoint subintervals (but at least one) and thus obtains finitely many closed intervals (at least two) of non-vanishing length. ${\displaystyle [a,b]}$

In the second step, you remove a finite number of sub-intervals (at least one in each case) from each of the included intervals.

Again, this process - infinitely repeated steps - defines a set of real numbers, namely those points that have never fallen into one of the wiped out intervals.

If all interval lengths become arbitrarily small in this process, then all Cantor sets constructed in this way are homeomorphic to one another and are equal to the set of all real numbers. By suitably varying the proportion “lengths of the wiped-away intervals: lengths of the remaining intervals”, one can generate a Cantor set whose Hausdorff dimension is any given number in the interval [0,1].

A two-dimensional analogue of the Cantor set is the Sierpinski carpet , a three-dimensional one the Menger sponge .

literature

• Jürgen Appell: Analysis in examples and counterexamples. An introduction to the theory of real functions. Springer, Berlin a. a. 2009, ISBN 978-3-540-88902-1 , pp. 233-237, ( excerpt (Google) ).
• Steven G. Krantz : A Guide to Topology (= The Dolciani Mathematical Expositions. Vol. 40 = MAA Guides. Vol. 4). Mathematical Association of America, Washington DC 2009, ISBN 978-0-88385-346-7 , pp. 33-36, ( excerpt (Google) ).

Web links

Commons : Cantor Granules  - collection of images, videos and audio files

• Cantor Sets and Cantor Set and Function on cut-the-knot.org (English)
• Eric W. Weisstein : Cantor Set . In: MathWorld (English).
• Cantor set . In: Encyclopaedia of Mathematics (English)

Individual evidence

1. ↑ Mohsen Soltanifar: A Different Description of A Family of Middle-a Cantor Sets . In: American Journal of Undergraduate Research . 5, No. 2, 2006, pp. 9-12.