Sparse matrices describing iterations of integer-valued functions

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