Are [itex]f(x)=2x, x\in R[/itex] and [itex]g(x)=x^2, x>0[/itex] topologically conjugate?
i.e. does there exist an h(x) such that
The Attempt at a Solution
My professor gave one example in class about finding such a function h which was by guessing it to be equal to xn and subsequently solving and finding the value of n. However, when I tried to apply the same idea to this problem, I come off short.
[tex]h(g(x)) = h(x^2)[/tex]
[tex]f(h(x)) = 2h(x)[/tex]
If we let [itex]h(x)=x^n[/itex] then we want to solve for n in
[tex]x^n=2 = h(x)[/tex]
Hence I find h(x)=2 but this doesn't work. Are there other guesses I could make? Or better yet, is there a more systematic approach to these sorts of problems? Does there even exist an h?