- #1
mkkrnfoo85
- 50
- 0
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
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