1. The problem statement, all variables and given/known data
let f(x) = x3
and g(x) = x - 2x3
. Show there is no homeomorphism h such that h(g(x)) = f(h(x))
2. Relevant equations
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
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)
let f conjugate to g by h. then:
) = 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
(x) = x - [itex]\mu[/itex]x3
. Show there is no nontrivial polynomial h such that f is conjugate to g by h.
3. 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 doesnt 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 dont know what h is going to "do", so I cant really use the theorem I quoted to claim that h(x*) wont be a periodic point for f(x).
Any ideas would be excellent. I found this other thread on proving no homeomorphism exists (http://physicsforums.com/showthread....+homeomorphism
) but it seems a little different in that theyre showing two spaces arent homeomorphic to one another whereas Im showing that two functions arent conjugate... also much of the language of topology (compactness, connectedness..) is supposedly outside the scope of the course Im taking and not needed to solve this problem at all. so any hints/help?