The bundle over dynamical system $(X,\varphi)$ and associated $C^*$-algebra

The pair $(X,\varphi)$, which consists of the countable set $X$ and the preset self-mapping $\varphi$, can be regarded as a dynamical system. Here we consider the dynamical systems with a bounded in common cardinality of preimage. In this case the system $(X,\varphi)$ generates the $C^*$-algebra $C^*\varphi(X)$. We introduce the notions of covering and bundle of the system and consider the extension of $C^*\varphi(X)$ in the terms of bundle over $(X,\varphi)$.