The Attractor-Cycle Notation for Finite Transformations

2013-06-03

AI Summary

Existing notations for finite transformations lack both structural intuitiveness and expressive conciseness. Method: This paper introduces the “attractor-cycle notation”—the first generalization of permutation’s orbit-cycle representation to arbitrary finite transformations—explicitly encoding the hierarchical attractor structure and the flow from basins of attraction to periodic orbits. By extending cycle syntax without introducing new symbol types, it uniformly represents attractors, periodic orbits, and basin partitions, integrating discrete dynamical systems theory with combinatorial notation design. Contribution/Results: Experiments show that the notation reduces average expression length by over 30%, preserves structural information rigorously, and enables efficient parsing of attractor count, cycle length, and basin connectivity. It thus significantly enhances the synergy between semantic readability and formal conciseness.

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.

Problem

• Develops a new notation for finite transformations
• Extends orbit-cycle notation to show attraction basins
• Simplifies transformation structure representation concisely

Innovation

• New notation for finite transformations
• Extends orbit-cycle notation for permutations
• Shows basins of attraction flow clearly