Expansive homeomorphisms on complexity quasi-metric spaces

TL;DR

This work builds a rigorous bridge between dynamical systems and computational complexity by endowing the space of running-time profiles with the complexity quasi-metric $(\mathcal{C},d_\mathcal{C})$. Its central result shows that the scaling map $\psi_α(f)(n)=αf(n)$ is expansive if and only if $α\neq 1$, with stable sets corresponding to complexity-neighborhoods and unstable sets capturing pointwise faster functions; the dynamics exhibit exact hyperbolicity with canonical coordinates contracting by $λ=1/α$ in forward time and expanding by $α$ in backward time. Moreover, the time-hierarchy phenomenon of Hartmanis–Stearns has a dynamical counterpart as orbit separation in the symmetrized metric, linking classical complexity theory to geometric/dynamical properties. The paper also develops practical tools, including numerical checks and SageMath/Python implementations, and provides entropy estimates that illuminate the growth of dynamical information under scaling. Overall, it offers a principled framework to reinterpret complexity classes through the lens of expansive dynamics, with clear avenues for future exploration such as non-linear transforms, space-bcomplexity analogues, and shadowing phenomena.

Abstract

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.

Figures (9)

• Figure 1: Two worlds connected by quasi-metrics. Dynamical systems study orbit divergence; complexity theory studies resource growth. The asymmetry inherent in both settings is encoded by the quasi-metric distance.
• Figure 2: From metrics to quasi-metrics. Dropping symmetry allows the distance to encode directional information.
• Figure 3: The standard quasi-metric on $\mathbb{R}$: going from $x=2$ up to $y=7$ costs $u(2,7)=5$, but going from $y=7$ down to $x=2$ is free: $u(7,2)=0$.
• Figure 4: The complexity landscape. Moving from a faster class to a slower one is free ($d_\mathcal{C}=0$); the reverse direction is costly ($d_\mathcal{C}>0$).
• Figure 5: An expansive homeomorphism: the orbits of any two distinct points are eventually separated by more than $\delta$.

Theorems & Definitions (84)

• Definition 2.1: Quasi-metric
• Definition 2.2: Conjugate and symmetrization
• Example 2.3: Standard quasi-metric on $\mathbb{R}$
• Example 2.4: Weighted quasi-metric on $\mathbb{R}$
• Example 2.5: Discrete quasi-metric
• Example 2.6: Non-example: failing the triangle inequality
• Example 2.7: Asymmetric topologies on $\{a,b,c\}$
• Remark 2.8: Topological considerations
• Definition 3.1: Complexity quasi-metric schellekens1995
• Theorem 3.2: Basic properties of $d_\mathcal{C}$