Let $G$ be  a  topological group. We define  an  equivalent  relation on $G$ as follows:

For $a,b\in G$ we set $a\sim b$ if the  following two maps are topological conjugate:

$$x\mapsto ax,\qquad x\mapsto bx$$

By topological conjugacy of two maps $f, g$ on a  topological space $G$ we mean, as usual, existence of a homeomorphism $H:G\to G$  with $Hf=gH$.

Obviousely the usual group theoretical conjugate elements are equivalent in this new sense. But possibly the  equivalent classes are larger than algebraic conjugacy class.

In the  finite  (discrete ) group case  the equivalent relation is the following: two elements are equivalents if the  corresponding permutations are  conjugate in $P(G)$, the  permutation group of $G$.

**My first question:** 

>In the  finite group case, the  cardinality of the conjugacy class of an element $a$ is  $\frac{|G|}{|C(a)|}$  where $C(a)$ is  a commutator of $a$. Now  what is a  formula for  the  cardinality of an equivalent class containig $a$. I mean a formula in terms of $G$ itself  not in terms of $P(G)$. More precisely can  we canonically associate a  subgroup $\tilde{C}(a)\subset G$   to  every element $a\in G$ such that the  equivalent class $[a]$ is in bijection correspondance to $\tilde{C}(a)$ in a canonical way. Another question: From undergraduate group theory I remember that various  equivalents relations on finite groups enable us to obtain useful  counting theorems. Even Sylows theorem were direct or undirect  consequence of  some equivalent relations. So I wonder is this  equivalent relation useful to produce some  new counting results?

**My second  question:**

> For the  obvious case $G=\mathbb{R}$ we get triviality. There are only two equivalent class. On the  other hand for  circle case every equivalent class is the  (singleton) conjugacy class since the rotation number is a topological invariant. Are there examples for which the measure of  an equivalent class approach to zero  or  approach to 1 (the  Haar measure)? Is every equivalent class necessarilly measurable?
I mean can one produce a sequence of example such that an equivalent class has  measure $1/n$ or $1-1/n$? This possibly prevent us to give finite group example