The Dynamics of One-Dimensional Quasi-Affine Maps

TL;DR

This paper provides a comprehensive analysis of the dynamics of one-dimensional quasi-affine maps, detailing fixed points, cycles, and limit sets across all parameter values, thus advancing understanding of their long-term behavior.

Contribution

It offers a complete characterization of the fixed points, cycles, and limit sets of quasi-affine maps, including explicit formulas and parameter regions, which was previously lacking.

Findings

Existence of parameter regions with any number of fixed points

Explicit formula for the number of 2-cycles

Classification of omega-limit sets for all parameters

Abstract

We study the dynamics of the one-dimensional quasi-affine map $x\mapsto \left\lfloor \lambda x +\mu \right\rfloor$, providing a complete description of the map's periodic points, and of the limit points of every $x\in\mathbb{R}$ under the map, for all real parameter values. Specifically, we establish the existence of regions of parameter values for which the map possesses $n$ fixed points for all $n\in\mathbb{N}_0\cup \{\infty\}$, an explicit formula for the number of 2-cycles possessed by the map, and the $\omega$-limit set of any $x\in\mathbb{R}$ under the map, which, depending on the parameter values, is either a singleton of a fixed point, a 2-cycle, $\{-\infty,\infty\}$, $\{\infty\}$, or $\{-\infty\}$.