Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be Lipschitz.

Define the relation $\stackrel{f}{\sim}$ on $\mathbb{R}^n$ by

$u \stackrel{f}{\sim} v \;\;$ if and only if $\;\; u$ and $v$ are in the same orbit of the dynamical system determined by $f$.

Define the relation $\stackrel{g}{\sim}$ similarly. $\:$ $\stackrel{f}{\sim}$ and $\stackrel{g}{\sim}$ are obviously equivalence relations.

Define $\:\langle X,\mathcal{T}_X\rangle\:$ and $\:\langle Y,\mathcal{T}_Y\rangle\:$ to be the quotient topological spaces of $\mathbb{R}^n$ by $\stackrel{f}{\sim}$ and $\stackrel{g}{\sim}$ respectively.

Suppose $X$ and $Y$ are homeomorphic. $\;\;$ Does it follow that there exists a homeomorphism

$h : \mathbb{R}^n \to \mathbb{R}^n \:$ such that for all members $u$ and $v$ of $\mathbb{R}^n$, $\: u \stackrel{f}{\sim} v \:$ if and only if $\: h(u) \stackrel{g}{\sim} h(v) \:\:$?