Until the 20th century mathematicians had found no good means to describe seemingly irregular and extremely rough shapes such as that of clouds, seashores, and mountains. With the advent of fractal geometry, the mathematics of chaos, dynamical systems, and other related subfields of mathematics we now have tools to describe such shapes. Perhaps there is a poetic appropriateness that one of the earliest mathematicians to work with such sets, Cantor, came from a mountainous land (Switzerland).
The Cantor sets and the functions built upon them are typically used as counter examples to many conjectures in analysis. However, these sets are not ornary objects trying to thwart young mathematicians in their search for understanding, but rather objects of great beauty. This beauty is most obviously seen in the attractive graphs of good approximations of the Cantor sets and functions. No less beautiful are the properties possessed by these sets and functions. Why don't you take a few moments to consider these beauties from analysis.
One of the great beauties of Cantor sets is that their definition extends easily to multiple definitions without loosing any of their peculiar properties. The following descriptive definition comes from An Introduction to Chaotic Dynamical Systems by Robert Devaney.
A set of points (we must be working in at least a topological space here) is called closed if its complement is open. This is most easily explained by considering the construction given below. The complement is of course all the points not in the set.
A set is totally disconnected if it contains no intervals. In particular if you choose any two points in the set there is always space between them. Every point has some elbow room on both sides.
A set is perfect if every point in it is an accumulation point or limit point of other points in the set. This property is actually redundant since the set being closed implies this property. It is a good reminder, however, that although the points have space around them, they all have neighbors living aribtrarily close to them. They are spread out, but no one is alone.
Note that because this definition is purely topological it is not limited by the dimension of the metric space in which the sets are constructed. Indeed this definition allows for the sets to be thought of as abstract topological sets not even requiring a metric space.
Let's start at the very beginning, a very good place to start. Imagine a line segment, maybe the segment [0,1]. Now cut out the middle third of this line segment. We now have two, non-touching line segments each one third (1/3) the length of the original segment.
Next, cut out the middle third of each of the two line segments. This of course leaves us with four line segments each one third (1/3) the length of the previous segments and one ninth (1/3*1/3=1/9) the length of the original segment.
Cut out the middle third of the current line segments. Do it again, and again, and keep cutting. Now stop a moment to keep reading. If you were to continue this cutting process infinitely, the points remaining would be the Cantor ternary set.
For this to work properly the segments removed must be open segments. That is you leave the end points. Because we are removing open sets, what is left is naturally the complement of open sets, that is, it is a closed set as noted in the definition.
The construction above does not depend on the removed line segments being one third (1/3) the size of the previous segments. Indeed so long as the segments removed are less than one half (1/2) the constructed sets have all the same properties. All that has occurred is that the position of the points have been shifted a bit.
Formally the Cantor sets are constructed from a sequence of the form
{ 1, a1, a2, a3, . . . } where a(i+1) < 1/2 a(i) for all i.
The sequence for the Cantor ternary set is {1, 1/3, 1/9, 1/27, 1/81, . . .}. Other sequences of the form a(i)=(1/n)^i, n>2, i=0, 1, 2, . . . also generate Cantor sets. These are not the only sequences, however. For example, the sequence {1, 1/3, 1/16, 1/125, ...} also produces a Cantor set.
The sequence elements indicate how far the apart new points are from existing points at each step.
W can show that there are an uncountably infinite number of Cantor set variations. Consider that there are uncountably many real numbers in the interval (0,1/2), and each one of these numbers generates a distinct Cantor sequence. However there is an obvious way to create a one-to-one and onto mapping between all Cantor sets so constructed. Namely, map the left and right end points added at a particular step to the matching left and right endpoints added at the same step in the other Cantor set.
One should not read too much into this bijective mapping however. It is known that there exist Cantor sets having different measure. Therefore the bijection doesn't preserve all interesting properties.
As noted in the definition, Cantor sets can exist in any dimension. The question becomes how to construct such sets. As the number of dimensions grows, so does the number of possible ways to construct Cantor sets. Of course the total number of Cantor sets does not change in the sense that the number of Cantor sets in any dimension is uncountably infinite.
For more information check the chaotic gaskets page.
In simple terms Cantor functions map the closure of the portions removed to values based on the step at which the interval was removed and the coordinates of the interval. Any measure theory text should have the complete details.
Below are graphs of a few Cantor function graphs. Click on the thumb nail views to see a full picture. Naturally, all graphs are approximations. They were generated with the following Maple code for 1D, 2D, and 3D sets (the code is in text format).
These graphs represent Cantor functions on various 1D Cantor sets. The sets are based on the sequences (1, 1/3, 1/9, . . .), (1, 1/4, 1/16, . . . ), (1, 1/5, 1/25, . . . ), (1, 1/6, 1/36, . . . ), (1, 1/7, 1/49, . . . ), (1, 1/8, 1/64, . . . ), and (1, 1/9, 1/81, . . . ).
The following graph is of a Cantor function on a 2D Cantor ternary set. The function value is represented by height. The color is based on the height.
Increasing the dimension by one, we have a Cantor function on a 3D Cantor set. For this graph the function value is represented by color. Blue is the lowest value and red is the highest.
This is also a Cantor function on a 3D Cantor set. This set was generated by the sequence (1, 1/3, 1/16, 1/125, . . . ).
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