Let $(\Sigma_k, f)$ be an adding machine on $k$ symbols. That is, $\Sigma_k$ is the set of all sequences of numbers $0, 1, \ldots, k - 1$ and $f: x = (x_0, x_1, \ldots) \mapsto (x_0, x_1, x_2, \ldots) + (1, 0, 0, \ldots) = (z_0, z_1, z_2, \ldots)$, where $z_0 = x_0 + 1 \mod k$ and $z_1 = x_1 + t_1 \mod k$. Here $t_1 = 0$ if $x_0 + 1 < k$ and $t_1 = 1$ if $x_1 + 1 = k$. So, we carry a one in the second case. Continue adding and carrying in this way for the whole sequence.
What are all the possible topological conjugacies between $(\Sigma_k, f)$ and itself?
Obviously, the group of maps generated by $f$ (i.e. $f^n, n \in \mathbb{Z}$ and limit points of this set) consists of topological conjugacies. But are there any more?