Expansive homeomorphisms on complexity quasi-metric spaces

TL;DR

This paper explores the dynamics of expansive homeomorphisms on a topological complexity space, linking dynamical systems concepts with computational complexity classes and providing computational tools and code for verification.

Contribution

It develops the theory of expansive homeomorphisms on the complexity quasi-metric space, connecting dynamical properties with complexity classes and classical theorems.

Findings

Scaling transformation is expansive iff alpha ≠ 1.

Stable sets correspond to asymptotic complexity classes.

Orbit separation relates to the time hierarchy theorem.

Abstract

The complexity quasi-metric of Schellekens is a topological framework in which the asymmetry of computational comparisons -- ``$A$ is at most as fast as $B$'' carrying different information than ``$B$ is at most as slow as $A$'' -- is built into the distance itself. This paper develops the theory of expansive homeomorphisms on the resulting space. The central result is that the scaling transformation $\psi_\alpha(f)(n)=\alpha f(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $\alpha\neq 1$. The $\delta$-stable sets of this dynamics turn out to coincide with asymptotic complexity classes, giving a dynamical characterisation of objects familiar from complexity theory. We then show that the canonical coordinates of $\psi_\alpha$ are hyperbolic with contraction rate $\lambda=1/\alpha$, and we connect orbit separation in the dynamical system to the classical time hierarchy theorem of Hartmanis and Stearns. Unstable sets, conjugate dynamics, and topological entropy estimates for the scaling map are also worked out. Concrete algorithms and Python implementations accompany every proof, so each result can be checked computationally; SageMath snippets sit alongside the examples, and the full code is in the \href{https://github.com/gabayae/expansive-homeomorphisms-complexity-qmetric}{companion repository}.