Absoluteness of Fixed Points

TL;DR

This work investigates when the existence of fixed points for non-expanding maps on complete $W$-metric spaces is absolute across models of set theory. It develops a framework based on relativization, distance monoids, and Cauchy completion to transfer generalized dynamical systems between models and to analyze fixed-point existence. The authors prove upwards absoluteness in broad cases (notably when $W$ is not continuous at $0$, or when $\mathrm{coinit}(W)=\omega$ with suitable niceness conditions) and construct non-absolute scenarios for uncountable coinitiality using $\kappa$-trees and related trees-with-paths techniques. These results clarify the interaction between set-theoretic background and metric-dynamical properties, guiding when fixed-point results are genuinely model-dependent and informing theory for generalized metric spaces and their dynamics.

Abstract

We characterize those complete commutative positive linear ordered monoids $W$ such that whenever $f$ is a map from a Cauchy complete $W$-metric space to itself, the existence of a fixed point of $f$ is independent of the background model of set theory.

Theorems & Definitions (77)

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