Bad Wadge-like reducibilities on the Baire space

Luca Motto Ros. "Bad Wadge-like reducibilities on the Baire space." Fundamenta Mathematicae 224.1 (2014): 67-95. <http://eudml.org/doc/283248>.

@article{LucaMottoRos2014,
abstract = {We consider various collections of functions from the Baire space $^\{ω\}ω$ 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 $^\{ω\}ω$ (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of $^\{ω\}ω$, 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.},
author = {Luca Motto Ros},
journal = {Fundamenta Mathematicae},
keywords = {Wadge reducibility; Lipschitz reducibility; computable function; recursive function; contraction mapping; nonexpansive function; Lipschitz function; (ultra)metric Polish space},
language = {eng},
number = {1},
pages = {67-95},
title = {Bad Wadge-like reducibilities on the Baire space},
url = {http://eudml.org/doc/283248},
volume = {224},
year = {2014},
}

TY - JOUR
AU - Luca Motto Ros
TI - Bad Wadge-like reducibilities on the Baire space
JO - Fundamenta Mathematicae
PY - 2014
VL - 224
IS - 1
SP - 67
EP - 95
AB - We consider various collections of functions from the Baire space $^{ω}ω$ 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 $^{ω}ω$ (compatible with its standard topology). We analyze the degree-structures induced by such sets of functions when used as reducibility notions between subsets of $^{ω}ω$, 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.
LA - eng
KW - Wadge reducibility; Lipschitz reducibility; computable function; recursive function; contraction mapping; nonexpansive function; Lipschitz function; (ultra)metric Polish space
UR - http://eudml.org/doc/283248
ER -