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razmtaz
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Homework Statement
let f(x) = x3 and g(x) = x - 2x3. Show there is no homeomorphism h such that h(g(x)) = f(h(x))
Homework Equations
Def let J and K be intervals. the function f:J->K is a homeomorphism of J onto K if it is one to one, onto, and both f and its inverse are continuous
Def let f:J->J and g:K->K, then f is conjugate to g if there is a homeomorphism h:J->K such that h(f(x)) = g(h(x)). (Then use the inverse of h as the homeomorphism that makes g conjugate to f, as in the given question)
Thm. let f conjugate to g by h. then:
1. h(fn) = gn(h) for n >=1
2. if x* is a periodic point of f, then h(x*) is a periodic point of g
3. if f has a dense set of periodic points, so does g
Another question in the book goes: let f(x) = x3 and g[itex]\mu[/itex](x) = x - [itex]\mu[/itex]x3. Show there is no nontrivial polynomial h such that f is conjugate to g by h.
The Attempt at a Solution
I know nothing of topology so this question is difficult for me to start.
I want to show there is no h(x) such that h(x)^3 = h(x-2x3). this is from the definition/question statement, but doesn't really seem useful in helping me solve the problem.
My next intuition is that certain properties are preserved by conjugacy, so if I could show that one of them is violated then I would have a solution.
for example, periodic points are preserved by conjugacy, so if I can find a periodic point x* of g(x) for which h(x*) is not a periodic point of f(x), then i'd be in business. I used maple to determine that g(x) has period 2 points (1 and -1) and no period 3 points. I also know that 1 and -1 are periodic points of f(x). Can I use this in any way? The trouble I have here is that I don't know what h is going to "do", so I can't really use the theorem I quoted to claim that h(x*) won't be a periodic point for f(x).
Any ideas would be excellent. I found this other thread on proving no homeomorphism exists (https://www.physicsforums.com/showthread.php?t=423411&highlight=no+homeomorphism) but it seems a little different in that theyre showing two spaces arent homeomorphic to one another whereas I am showing that two functions arent conjugate... also much of the language of topology (compactness, connectedness..) is supposedly outside the scope of the course I am taking and not needed to solve this problem at all. so any hints/help?
thanks alot