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.

翻译：Schellekens 引入的复杂性拟度量为计算比较的非对称本质——即声明一个算法比另一个更快，与声明第二个比第一个更慢所携带的信息不同——提供了精确的数学表达框架。本文在复杂性拟度量空间上建立了关于扩张同胚的完整理论。我们的核心结果表明，缩放变换 $ψ_α(f)(n)=αf(n)$ 在复杂性空间 $(\C,d_\C)$ 上是扩张的，当且仅当 $α\neq 1$。由此动力学产生的 $δ$-稳定集恰好对应于渐近复杂性类，从而为复杂性理论中的基本对象提供了动力学刻画。我们证明了与 $ψ_α$ 相关的规范坐标是双曲的，其收缩率为 $λ=1/α$，并建立了该动力系统中轨道分离与 Hartmanis 和 Stearns 的经典时间分层定理之间的精确联系。我们进一步研究了缩放映射的不稳定集、共轭动力学以及拓扑熵估计。贯穿全文，具体的算法和 Python 实现伴随证明过程，确保每个结果在计算上可复现。SageMath 验证代码片段内嵌于示例旁，完整代码可在配套代码库中获取。