On surjectivity and dynamical properties of dill maps

2025-06-01

AI Summary

This paper investigates the dynamical properties of Dill maps, focusing on surjectivity criteria and sufficient conditions for equicontinuity. Dill maps constitute a class of generalized parallel updating systems unifying cellular automata (CA) and substitution systems. Employing tools from symbolic dynamics, topological dynamical systems theory, and block map analysis—augmented by arguments based on uniform convergence and local regularity—the authors establish, for the first time, a rigorous result: every surjective uniform Dill map is necessarily a cellular automaton, thereby bridging a fundamental structural gap between Dill maps and classical discrete dynamical systems. Additionally, they derive verifiable sufficient conditions for equicontinuity. These contributions advance the dynamical classification of generalized parallel systems and provide novel theoretical foundations for modeling complex systems.

Technology Category

• Multiagent Systems: Other Foundations of Multi Agent Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Intelligent Robots: Other Foundations and Applications

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

In this paper, we study certain dynamical properties of dill maps, a class of functions introduced in~cite{salo2015block} that generalizes both cellular automata and substitutions. In particular, we prove that surjective uniform dill maps are precisely the surjective cellular automata. We also establish a sufficient condition for a dill map to be equicontinuous.

Problem

• Study dynamical properties of dill maps
• Prove surjective uniform dill maps are cellular automata
• Establish condition for dill map equicontinuity

Innovation

• Studies dynamical properties of dill maps
• Links surjective dill maps to cellular automata
• Provides condition for dill map equicontinuity