Can one visualise the dual groups to Cantor groups?

Question

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the dual groups of $C_p^{\mathbb{N}}$, where $C_p$ is the cyclic group of order $p$. Is there any illuminating description of the duality here like, for example, between $\mathbb{T}$ and $\mathbb{Z}$?

Answer 1

Well, the dual group is the direct sum $\bigoplus_{\mathbb N} C_p$, right? If we visualize $C_p^{\mathbb N}$ as a fractal -- so $C_2^{\mathbb N}$ is something like the middle-thirds Cantor set, and $C_3^{\mathbb N}$ is something like a disconnected Sierpinski triangle -- then an element of the dual group can be visualized as a locally constant $p$-coloring of $C_p^{\mathbb N}$ with lots of symmetry. Or, an element of the dual group can be visualized as a finite sequence of simultaneous rotations of the "components" of $C_p^{\mathbb N}$.

I'm not sure if that's what you were looking for... did it make any kind of sense?

Edit: Ah, we're trying to prove that $\widehat{C_p^{\mathbb{N}}} \cong \bigoplus_{\mathbb{N}} C_p$. Well, I think you can get that directly from the fact that $\widehat{C_p}\cong C_p$ and the categorical duality of products and coproducts. So if you have a visualization of $\widehat{C_p}\cong C_p$, then repeating it on multiple "scales" or "locations" gives you a visualization of $\widehat{C_p^{\mathbb{N}}} \cong \bigoplus_{\mathbb{N}} C_p$.

At this point I should admit that I haven't worked out the categorical duality myself. The topological part looks kind of daunting. So, don't trust me on that! If you prefer, you can find the dual manually. A character $f:C_p^{\mathbb{N}}\to\mathbb R/\mathbb Z$ must take values in the finite group $(\frac{1}{p}\mathbb Z)/\mathbb Z\cong C_p$. Since $f$ is continuous, the sequence $$f(1,0,0,\ldots), f(1,1,0,\ldots), f(1,1,1,0\ldots),\ldots$$ converges, so it is eventually constant, so only finitely many coordinates are nonzero; that's the definition of a direct sum.