Sparse matrices describing iterations of integer-valued functions

We consider iterations of integer-valued functions $\varphi$, which have no fixed points in the domain of positive integers. We define a local function ${\varphi }_n$, which is a sub-function of $\varphi$ being restricted to the subdomain $\{0,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. The iterations of ${\varphi }_n$ can be described by a certain $n \times n$sparsematrix$M_n$anditspowers\mathrm{.}Thedeterminantoftherelated$n \times n$matrix$\hat{M}_n = I - M_n$,where$I$ is the identity matrix, acts as an indicator, whether the iterations of the local function ${\varphi }_n$ enter a cycle or not. If ${\varphi }_n$ has no cycle, then $\det {\widehat{M}}_n=1$ and the structure of the inverse ${\widehat{M}}_n^{-1}$ can be characterized. Subsequently, we give applications to compute the inverse ${\widehat{M}}_n^{-1}$ for some special functions. At the end, we discuss the results in connection with the $3x+1$ and related problems.