Martin's conjecture and strong ergodicity
type: Item
instance of: scholarly article
title: Martin's conjecture and strong ergodicity
zbMATH Open document ID: 1188.03032
author: Simon Thomas
published in: Archive for Mathematical Logic
publication date: 2009-12-14T00:00:00Z
review text: The author explores some consequences of Martin's Conjecture on Turing-degree-invariant Borel maps. Let \(E\) and \(F\) be countable Borel equivalence relations on standard Borel spaces \(X\) and \(Y\), respectively. Then \(f : X \rightarrow Y\) is a \textit{homomorphism} from \(E\) to \(F\) if \(x E y\) implies \(f (x) F f(y)\) for all \(x, y \in X\). If \(f\) is Borel and satisfies \(x E y \Leftrightarrow f (x) F f(y)\) for all \(x, y \in X\), then \(f\) is a \textit{Borel reduction} and we write \(E \leq_{B} F\). If there exists a countable-to-one Borel homomorphism \(f : X \rightarrow Y\) from \(E \) to \(F\), then \(E\) is \textit{weakly Borel reducible} to \(F\), denoted by \(E \leq^{w}_{B} F\). A countable Borel equivalence relation \(E\) is \textit{weakly universal} if \(F \leq^{w}_{B} E\) for every countable Borel equivalence relation \(F\). For \(r, s \in 2^{\mathbb N}\), we say \(r\) is \textit{Turing reducible} to \(s\), \(r \leq_{T} s\) in notation, if there exists an oracle Turing machine which computes \(r\) when its oracle tape contains \(s\). If both \(r \leq_{T} s\) and \(s \leq_{T} r\), then \(r\) and \(s\) are \textit{Turing equivalent}, what we denote by \(r \equiv_{T} s\). For each \(r \in 2^{\mathbb N}\), the cone corresponding to \(r\) is the set \(C \subseteq 2^{\mathbb N}\) defined by \(C = \{s \in 2^{\mathbb N} : r \leq_{T} s\}\). Martin's Conjecture, denoted by MC, states that if \(f : 2^{\mathbb N} \rightarrow 2^{\mathbb N}\) is a Borel homomorphism from \(\equiv_{T}\) to \(\equiv_{T}\) , then exactly one of the following holds: {\parindent=6mm \begin{itemize}\item[(i)] There exists a cone \(C \subseteq 2^{\mathbb N}\) such that \(f\) maps \(C\) into a single \(\equiv_{T}\)-class. \item[(ii)] There exists a cone \(C \subseteq 2^{\mathbb N}\) such that \(x \leq_{T} f (x)\) for all \(x \in C\). \end{itemize}} Let \(E\) be a countable Borel equivalence relation on a standard Borel space \(X\). Then \(\equiv_{T}\) is \textit{\(E\)-\(m\)-ergo…
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03D28
Mathematics Subject Classification ID: 37A20
zbMATH DE Number: 5646250
zbMATH Keywords: Borel equivalence relation
zbMATH Keywords: Turing reducibility
zbMATH Keywords: Martin's Conjecture
zbMATH Keywords: Borel homomorphism
zbMATH Keywords: weakly universal countable Borel equivalence relation
zbMATH Keywords: Borel boundedness
Wikidata QID: Q123026772
MaRDI profile type: Publication
full work available at URL: s00153-009-0148-0
OpenAlex ID: W1969482738
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DOI: 10.1007/S00153-009-0148-0
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description: scientific article; zbMATH DE number 5646250