Two Dimensional Discrete Dynamics of Integral Value Transformations

TL;DR

This paper introduces and analyzes two-dimensional integral value transformations (IVTs) over natural numbers, exploring their algebraic dynamics, cycle structures, and dependence on Boolean functions, with implications for higher-dimensional systems.

Contribution

It systematically studies 2D IVTs using Boolean functions, identifies Collatz-like IVTs, and examines their dynamical properties and cycle structures, advancing understanding of their algebraic and dynamical behavior.

Findings

Dynamics depend on Boolean function state transitions

Identified 16 Collatz-like IVTs in 2D

Studied attractors with cycles of various lengths

Abstract

A notion of dimension preservative map, \textit{Integral Value Transformations} (IVTs) is defined over $\mathbb{N}^k$ using the set of $p$-adic functions. Thereafter, two dimensional \textit{Integral Value Transformations} (IVTs) is systematically analyzed over $\mathbb{N} \times \mathbb{N}$ using pair of two variable Boolean functions. The dynamics of IVTs over $\mathbb{N} \times \mathbb{N}=\mathbb{N}^2$ is studied from algebraic perspective. It is seen that the dynamics of the IVTs solely depends on the dynamics (state transition diagram) of the pair of two variable Boolean functions. A set of sixteen \textit{Collatz-like} IVTs are identified in two dimensions. Also, the dynamical system of IVTs having attractor with one, two, three and four cycles are studied. Additionally, some quantitative information of \textit{Integral Value Transformations} (IVTs) in different bases and dimensions are also discussed.