How to prove that two dilations of $\mathbb R^n$ are conjugate? Let $\lambda_1,\lambda_2>1$. How to prove that the dilations $f_i:\mathbb R^n\to \mathbb R^n$, $f_i(x)=\lambda_ix$ for $i=1,2$ are conjugate? That is, how to prove there exists an homeomorphism $h:\mathbb R^n\to \mathbb R^n$ such that $h\circ f_1=f_2\circ h$?
It seems natural to set $h$ to be the identity on the unit sphere and $h(z)=\frac{\lambda_2}{\lambda_1}z$ for the points of norm $\lambda_1$. Then we get $h\circ f_1=f_2\circ h$ on the unit sphere. How to use this idea to define a good $h$ that works for all of $\mathbb R^n$?
 A: One of the most straightforward way to construct conjugacy is the following. You take fundamental domains for both dilations, 
for example $H_1 = \lbrace 1 < \vert\vert x \vert \vert_2 \leqslant \lambda_1 \rbrace$ 
and 
$H_2 = \lbrace 1 < \vert\vert x \vert \vert_2 \leqslant \lambda_2 \rbrace$ for $f_1$ and $f_2$ respectively. You define homeomorphism $h^{*}$ from $H_1$ to $H_2$ as 
$x \mapsto \left (\frac{\lambda_1 - \lambda_2}{\lambda_1 -1} + \frac{\lambda_2-1}{\lambda_1 -1} \cdot \vert \vert x \vert \vert \right) \frac{x}{\vert\vert x \vert \vert}$. Then you extend it in following way: for each point $p \in \mathbb{R}^n\setminus{O}$ exists $k \in \mathbb{Z}$ such that $f_1^k(p) \in H_1$ (this is because $H_1$ is a fundamental domain); you map $p$ to a point $f_2^{-k}(h^{*}(f_1^k(p)))$. Then you extend this homeomorphism  by continuity to a homeomorphism of $\mathbb{R}^n$ (by setting $O \mapsto O$). It's easy to show that constructed homeomorphism conjugates two dilations.
A: First take logs to reduce the problem to two translations $g_i:R\to R$, where $g_i(t)=t+\mu_i$.  to show that these are conjugate, show that each one is conjugate to $g(t)=t+1$.  This is done by conjugating by a scaling map with factor $\mu_i$.
