Bad Wadge-like reducibilities on the Baire space

TL;DR

This paper investigates various reducibility hierarchies on the Baire space induced by different classes of functions, revealing they are significantly more complex than the classical Wadge hierarchy, with large antichains and chains.

Contribution

It introduces and analyzes new degree-structures based on computable, contraction, nonexpansive, and Lipschitz functions, showing their complexity surpasses classical hierarchies.

Findings

Hierarchies contain large infinite antichains.

Most hierarchies also contain infinite descending chains.

Complexity exceeds that of the classical Wadge hierarchy.

Abstract

We consider various collections of functions from the Baire space X into itself naturally arising in (effective) descriptive set theory and general topology, including computable (equivalently, recursive) functions, contraction mappings, and functions which are nonexpansive or Lipschitz with respect to suitable complete ultrametrics on X (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of X, and we show that the resulting hierarchies of degrees are much more complicated than the classical Wadge hierarchy; in particular, they always contain large infinite antichains, and in most cases also infinite descending chains.