A Ternary Cantor Set proof

In summary, the map x |--> f(x) = (x+2)/3 on [0,1] is a contraction and maps the ternary Cantor set into itself. The fixed point of the map is 1. To show that it maps the Cantor set into itself, it must be shown that if x is in the Cantor set, then f(x) is also in the Cantor set. This can be proven by using the fact that x is in the Cantor set if and only if its base 3 expansion contains no 1s. Since dividing by 3 in base 3 just shifts the decimal point, it is a trivial result.
  • #1
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Hey all,

I would really like help on this probably simple proof:

That the map x |--> f(x) = (x+2)/3 on [0,1] is a contraction,
and maps the ternary Cantor set into itself. Also, find it's fixed point.


(1) I can easily show the fixed point (where f(x) = x) is 1.
(2) I can also pretty easily show it is a contraction:
where |f(x) - xo| <= q*|x - xo|, where q < 1, and xo is the fixed point.

(3) However, I can't seem to find a way to tell whether it maps the ternary Cantor set into itself. I kno the definition of the ternary Cantor set is taking the interval [0,1] and deleting the middle-third of the interval, and then repeating the process on each remaining interval, infinitely.

What does it mean by mapping the ternary Cantor set into itself? Does x have to start out being in the ternary Cantor set? If so, how is it possible if x can vary from [0,1]? What am I interpreting wrong?

Thanks,

Mark
 
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  • #2
Look at how the function maps certain intervals. What sort of intervals must x be contained in for it to be in the Cantor set? What then happens to the intervals under function f(x)?
 
  • #3
ah thx. just the boost i needed =)
 
  • #4
What does it mean by mapping the ternary Cantor set into itself?

It means that if x is in the subset of [0,1] called the Cantor set, then f(x) is also in the Cantor set. You might use the fact that x is in the Cantor set if and only if its base 3 expansion contains no 1s.

(I've edited to change "contains no 0s" to "contains no 1s"!)

I've also given a little more thought to this. Since dividing by 3 base 3 just shifts the "decimal" point, this is trivial!
 
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1. What is a Ternary Cantor Set?

A Ternary Cantor Set is a fractal set constructed by removing the middle third of each line segment in an initial line segment, and repeating this process infinitely.

2. What is the proof for the Ternary Cantor Set?

The proof for the Ternary Cantor Set involves using mathematical induction to show that the set is uncountable, meaning it has an infinite number of points and cannot be put into a one-to-one correspondence with the natural numbers.

3. How does the proof for the Ternary Cantor Set relate to the Cantor's Diagonal Argument?

The proof for the Ternary Cantor Set is based on Cantor's Diagonal Argument, which states that for any given infinite set, there exists at least one element that is not in the set. In the case of the Ternary Cantor Set, this element is the infinite number of points that are removed during the construction process.

4. Why is the Ternary Cantor Set important in mathematics?

The Ternary Cantor Set is important in mathematics because it is one of the earliest examples of a fractal set, which has self-similar patterns at different scales. It also has important applications in number theory and topology.

5. Can the Ternary Cantor Set be generalized to other bases?

Yes, the Ternary Cantor Set can be generalized to any base, not just base 3. This results in different types of Cantor Sets, such as the Binary Cantor Set (base 2) and the Quaternary Cantor Set (base 4).

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