The Attractor-Cycle Notation for Finite Transformations

TL;DR

This paper introduces the attractor-cycle notation, a new way to represent finite transformations that enhances understanding of their structure and simplifies expressions without adding complexity.

Contribution

It extends orbit-cycle notation for permutations, providing a clearer visualization of basins of attraction and flow into orbit cycles.

Findings

The notation makes the structure of transformations more transparent.

It reduces expression length without increasing symbol types.

Provides new insights into transformation dynamics.

Abstract

We describe a new notation for finite transformations. This attractor-cycle notation extends the orbit-cycle notation for permutations and builds upon existing transformation notations. How the basins of attraction of a finite transformation flow into permuted orbit cycles is visible from the notation. It gives insight into the structure of transformations and reduces the length of expressions without increasing the number of types of symbols.