The complexity of the Structure and Classification of Dynamical Systems

TL;DR

This paper explores the Borel complexity of structure and classification theorems in dynamical systems, categorizing classical problems by their computational complexity and showing some are infeasible with countable data.

Contribution

It provides a systematic classification of dynamical systems problems based on Borel complexity, revealing limitations in classical classification approaches.

Findings

Classical problems are sorted by known complexity benchmarks.

Some problems are shown to be infeasible with countable information.

The paper offers a framework for understanding the complexity of dynamical systems classification.

Abstract

This is an expository paper about the Borel complexity of structure and classification theorems. It sorts several classical problems relative to known benchmarks of complexity. As a corollary various problems proposed by people such as von Neumann and Smale are shown to be infeasible using inherently countable information.