Expansive homeomorphisms on complexity quasi-metric spaces

The complexity quasi-metric, introduced by Schellekens, provides a topological framework where the asymmetric nature of computational comparisons – stating that one algorithm is faster than another carries different information than stating the second is slower than the first – finds precise mathematical expression. In this paper we develop a comprehensive theory of expansive homeomorphisms on complexity quasi-metric spaces. Our central result establishes that the scaling transformation $ψ_α(f)(n)=αf(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $α\neq 1$. The $δ$-stable sets arising from this dynamics correspond exactly to asymptotic complexity classes, providing a dynamical characterisation of fundamental objects in complexity theory. We prove that the canonical coordinates associated with $ψ_α$ are hyperbolic with contraction rate $λ=1/α$ and establish a precise connection between orbit separation in the dynamical system and the classical time hierarchy theorem of Hartmanis and Stearns. We further investigate unstable sets, conjugate dynamics, and topological entropy estimates for the scaling map. Throughout, concrete algorithms and Python implementations accompany the proofs, making every result computationally reproducible. SageMath verification snippets are inlined alongside the examples, and the full code is available in the companion repository.

💡 Research Summary

The paper develops a systematic theory of expansive homeomorphisms on the “complexity quasi‑metric” space introduced by Schellekens, thereby forging a precise bridge between dynamical systems and computational complexity. The authors begin by recalling the definition of a quasi‑metric, a distance function that drops the symmetry axiom, and illustrate how this asymmetry naturally captures the directional cost of moving from a faster algorithm to a slower one (zero cost) versus the opposite direction (positive cost). The specific quasi‑metric (d_{\mathcal C}) on the set (\mathcal C) of positive‑valued functions (f:\mathbb N\to(0,\infty)) is defined by

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