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arXiv · 2610.13973

Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries

Thomas Duzer
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Abstract

Zavarzina asked whether every bijective non-contractive self-map of a precompact quasi-metric space is an isometry. We answer the question in the negative, in a stronger form: on a hereditarily precompact quasi-metric space (a tolled extension of the irrational rotation over Z), the shift is a bijective non-contractive map that is not an isometry. The construction generalizes: for every metric space Y, every bijective isometry $σ$ of Y, and every point whose $σ$-orbit is aperiodic and totally bounded, the tolled Z-extension is such a counterexample; instances include the irrational rotation and the p-adic odometer. The conjugate space is not precompact, so the two-sided hypothesis of Zavarzina's positive result is sharp, and the space is not expand-contract plastic. The core results, including the general scheme and the circle instance, are machine-checked in Lean 4 over mathlib, sorry-free (no unproved steps), under a pinned toolchain. A structural section proves a monochromy theorem: for every strict pair on a hereditarily precompact space, one endpoint's backward self-returns have self-defect at least a quarter of the pair's defect, uniformly on a chain-closed displacement set. Boundary results delimit what remains open: one-sided compactness together with separation.

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Thomas Duzer. 2026-09-17. Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries. https://arxiv.org/abs/2610.13973

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