Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries
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.
Related
- The Structure of a Non-Isometric Non-Contractive Bijection of a Hereditarily Precompact Quasi-Metric Space
- Fixed Point Theorems for Paired Contractive Mappings in b-Metric Spaces
- Hereditarily Lindelöf and hereditarily separable subspaces of function spaces
- A Bijective Proof of an Unbalanced Wilf Equivalence
- Three Problems on Separable Quotients of Precompact Abelian Groups
BibTeX
@misc{duzer2026bijective,
title = {{Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries}},
author = {Thomas Duzer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13973},
eprint = {2610.13973},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13973},
}
RIS
TY - UNPB
TI - Bijective Non-Contractive Maps of Hereditarily Precompact Quasi-Metric Spaces Need Not Be Isometries
AU - Thomas Duzer
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.13973
UR - https://arxiv.org/abs/2610.13973
AB - 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.
ER -