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Cantor sets, Fat cantor sets and homeo and diffeo

  1. Oct 16, 2011 #1
    1. The problem statement, all variables and given/known data

    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

    3. 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!
     
  2. jcsd
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