Cantor sets, Fat cantor sets and homeo and diffeo

[LBloom]

Homework Statement

Ok so the first part of the problem was constructing fat cantor sets, cantor sets with some positive lesbegue measure. The second part involved proving that any fat cantor set is homeomorphic to the regular cantor set. The third part asks whether there is a diffeomorphism from [0,1] to itself such that a fat cantor set is mapped to the ternary cantor set

The Attempt at a Solution

I was able to get the first two parts. For constructing the homeomorphism it seemed simple when you look at the formation of the cantor set in an algorithmic way. At each stage lines are removed from the middle and the number of intervals left grows like 2^n (eventually the intervals shrink to points of course).

What I did was for each stage was define a homeomorphism from the cantor set to the fat cantor set. This is obviously possible since we're just mapping lines segments to line segments. This defines a sequence of functions and as n->∞ we'll have a homeomorphism from a fat cantor set to the regular one

For the diffeomorphism it appears more difficult. On the one hand it seems simple because both spaces are totally disconnected and are uncountably infinite so we could just map points to points and have diffeomorphisms fitting the missing intervals in between them correctly. On the other hand, the ternary cantor set shrinks much quicker than the fat cantor sets and it would appear that as n->∞ the map won't be diffeomorphic when we're mapping a point from one cantor set to another. The mapping of the intervals around these points seems to be too violent to be a diffeomorphism (derivative wouldn't be defined).

Thanks for any help!

 

[mighty2000]

Thank you for your post and for sharing your solution for the first two parts of the problem. It seems like you have a good understanding of the construction of fat cantor sets and homeomorphisms. I agree with your approach of defining a sequence of homeomorphisms from the fat cantor set to the regular cantor set as n->∞, which would then give us a homeomorphism between the two sets.

As for the third part of the problem, it is indeed more challenging to find a diffeomorphism between a fat cantor set and the ternary cantor set. As you mentioned, the shrinking of the intervals in the ternary cantor set is much quicker than in a fat cantor set, which poses a difficulty in finding a smooth mapping between the two sets. However, I believe this problem can be solved by considering a different approach.

Instead of trying to map points to points, we can consider a mapping between the two sets that preserves the structure of the sets. For example, we can define a mapping that takes the middle third of each interval in the ternary cantor set and maps it to the middle third of each interval in the fat cantor set. This way, we are still preserving the structure of the sets, while also taking into consideration the different rates of shrinking. This mapping may not be smooth at every point, but it is a continuous mapping and can be extended to a diffeomorphism by using a smoothing technique.

I hope this helps and provides some guidance for finding a solution to the third part of the problem. Keep up the good work!