The Structure of a Non-Isometric Non-Contractive Bijection of a Hereditarily Precompact Quasi-Metric Space
On a hereditarily precompact quasi-metric space, every bijective non-contractive self-map has the following structure: the quasi-metric splits as $d = λ^- + D_0$. The invariant limit register $λ^-$ is a quasi-pseudo-metric below $d$ for which the map is an isometry, with totally bounded symmetrization on every orbit, so that the map's rigid shadow is a rotation of a compact monothetic group; the asymmetric limit structure on it, a priori only paratopological, carries the compact topology. The defect $D_0$ lies in a band between a past fee difference and the fee of its source, the fee nondecreasing along orbits and vanishing in the deep past. The defect is zero precisely when the map is an isometry. Deep sources are thus almost rigid: in the deep past the map is an isometry to within any $\varepsilon$, against every target. Every such map that is not an isometry is a toll over its rigid shadow, levied on a forward-invariant zone that sheds into the past. Two further facts are forced: the orbit of the source of any strict pair is infinite and already strict, and the source pays its whole defect on its own backward returns, uniformly on a chain-closed set of displacements. The band is attained at both ends, and beyond it the shape of the defect is not forced: three spirals over the circle show that the toll can depend on the target, that a source carrying a fee can descend for free, and that an ascent can carry defect; no rate of shedding is forced either, and for zone tolls (two-valued descent tolls that are shift-monotone and subadditive along their own routes) the fee reaches zero already at a finite depth, by the ascending chain condition on submonoids of $\mathbb{N}$. The results are machine-checked in Lean, apart from a few statements listed at the end.