Compact Quasi-Metric Spaces Are Expand-Contract Plastic

Abstract

On a totally bounded metric space every non-contractive self-map is an isometry. For quasi-metric spaces, taken T_1 throughout, this fails: Zavarzina gave a hereditarily precompact counterexample and asked whether bijectivity restores rigidity; the toll spiral shows that it does not. We prove that compactness does: every compact quasi-metric space is expand-contract plastic, that is, each of its bijective non-contractive self-maps is an isometry. Countable compactness suffices, as does compactness of the conjugate topology. Bijectivity cannot be dropped: a compact, hereditarily precompact quasi-metric on the natural numbers makes the shift non-contractive and not an isometry. For arbitrary non-contractive self-maps we prove rigidity when the space is hereditarily precompact and its conjugate topology is sequentially compact, in particular when it is hereditarily precompact and Yoneda complete; neither hypothesis can be dropped. In both theorems the T_1 axiom can be weakened to R_0, with hereditary precompactness read sequentially, but not to T_0: we construct a T_0 space with all the remaining hypotheses of both theorems, carrying a bijective non-contractive self-map that is not an isometry. Under R_0 the two-sided case reduces to the metric theorem; the one-sided theorems do not, and plasticity does not pass to the symmetrization even for compact spaces. In finite dimension the unit ball of a T_1 asymmetric normed space, forward or symmetric, and its sphere are rigid for every non-contractive self-map; without T_1 the forward ball need not be; in infinite dimension the symmetric ball and sphere are not plastic in general, and the forward versions carry the quasi-metric form of the unit-ball question of Cascales, Kadets, Orihuela and Wingler, which is posed. The two theorems are machine-checked in Lean, and so are the counterexamples, apart from a few statements listed at the end.