Sparse matrices describing iterations of integer-valued functions

TL;DR

This paper studies the iterative behavior of integer-valued functions using sparse matrices, providing criteria for cycle detection and methods to compute matrix inverses, with applications to famous problems like the 3x+1 conjecture.

Contribution

It introduces a matrix-based framework for analyzing iterations of functions without fixed points, including cycle detection and inverse computation methods.

Findings

Determinant of matrix indicates cycle presence in iterations.

Inverse matrices can be explicitly computed for certain functions.

Framework connects matrix theory with classical problems like the 3x+1 problem.

Abstract

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.