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 $ψ_α(f)(n)=αf(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $α\neq 1$. The $δ$-stable sets 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 $ψ_α$ are hyperbolic with contraction rate $λ=1/α$, 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}.

翻译：舍勒肯斯复杂性拟度量是一个拓扑框架，其中计算比较的不对称性——"A至多与B一样快"和"B至多与A一样慢"所携带的信息不同——被直接构建在距离函数中。本文发展了由此得到的空间上的扩张同胚理论。核心结论是：在复杂性空间(C, d_C)上，缩放变换ψ_α(f)(n)=αf(n)是扩张的当且仅当α≠1。该动力学的δ-稳定集恰好对应于渐近复杂度类，从而给出了复杂度理论中熟悉对象的动力学刻画。我们进一步证明ψ_α的规范坐标是双曲的，其收缩率为λ=1/α，并将动力系统中的轨道分离与哈特马尼斯和斯特恩斯的经典时间层级定理联系起来。本文还推导了缩放映射的不稳定集、共轭动力学和拓扑熵估计。每个证明均配以具体算法和Python实现，所有结果均可通过计算验证；SageMath代码片段与示例并存，完整代码见配套代码仓库。