Question 1.Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example Hausdorff) dimensions of $(X,a)$ and $(X,b)$ are equal?

Question 2.Is there a notion of a dimension for topological (as opposed to metric) spaces which can assume non-integral values?

## My motivation

Let $G$ be a finitely generated group and let $p$ be a prime number. Consider the compact topological space $X:=\prod_G \mathbb Z/p$, infinite product of copies of the cyclic group of order $p$, indexed by elements of $G$. Let $T$ be an element of the integral group ring of $G$. Note that $T$ gives a map $X\to X$ in a natural way (in this context $T$ is sometimes called *cellular automaton*). Define $Y$ to be the subset of those points $x$ of $X$ such that $T(x)=0$.

I want to measure "how big" $Y$ is.

If we choose a generating set for $G$ then we get a metric $d$ on $G$, and we also get a metric on $X$: two sequences $x_i$ and $y_i$ of $X$ are $p^{-|B(1,k)|}$ apart if they agree on the ball $B(1,k)$ of radius $k$ around the neutral element $1$ of $G$, but they don't agree on any larger ball. It's straightforward to see that box counting dimension of $X$ is $1$.

Unfortunately the metric we just defined depends on the generators chosen for $G$, so I'd like to know if I have a chance to get any kind of "intrinsic size of $Y$" this way.