In the book Dynamic Reported - The definition of the solenoidal sets is:
Let $I_{0} \supset I_{1}\supset I_{2}\dots$ be periodic intervals with periods $m_{0}$, $m_{1}$,$\dots$. If $m_{i} \to \infty$ the intervals $\{I_{i}\}_{i=0}^{\infty}$ are said generating and for any invariant closed set $S \subset Q=\bigcap\limits_{i\geq 0} orbI_{i}$ is called a solenoidal set.
One can use a transitive shift in an Abelian zero-dimensional infinite group as a model for the map on a solenoidal set.
What signifies a model for the map on a solenoidal set? Why a transitive shift in an Abelian zero-dime signal infinite groups in a good choice?