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An exercise in Katok and Hasselblat's Introduction to the Modern Theory of Dynamical Systems (Section 2.1, exercise 2) goes as follows:

Let $f$, $g$ be $C^1$ maps defined in a neighborhood of the origin of the real line with $f(0) = g(0) = 0$ and $0 < f'(0), g'(0) < 1$. There then exists an homeomorphic embedding $h$ from an open interval $I$ containing the origin to $\mathbb R$ such that $h(0) = 0$ and $f(h(x)) = h(g(x))$ for all $x$ in $I$. Prove that $h$ can be chosen Lipschitz continuous if and only if $f'(0) \leq g'(0)$.

A given hint is I should compare the speed of convergence of orbits to the origin. Supposing $h$ is $K$-Lipschitz, I have found some things none of which seem to lead to the answer, and perhaps more gravely none of which seem to relate to the hint:

  • $f^n(h(x)) = h(g^n(x)) \leq Kg^n(x)$

  • $(f^n)'(0) = (f'(0))^n$, and similarly for $g$ (since $f(0)=g(0)=0$).

The homeomorphic embedding $h$ was constructed in a previous exercise; I did it using fundamental domains and have a feeling this construction should tie in with the hint but can't wrap my head around it.

Any help is appreciated, thanks in advance.

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