TY - JOUR TI - Cantor minimal systems from a descriptive perspective DO - https://doi.org/doi:10.7282/T3WD42Q2 PY - 2016 AB - In recent years, the study of the Borel complexity of naturally occurring classification problems has been a major focus in descriptive set theory. This thesis is a contribution to the project of analyzing the Borel complexity of the topological conjugacy relation on various Cantor minimal systems. We prove that the topological conjugacy relation on pointed Cantor minimal systems is Borel bireducible with the Borel equivalence relation $Delta_{mathbb{R}}^+$. As a byproduct of our analysis, we also show that $Delta_{mathbb{R}}^+$ is a lower bound for the Borel complexity of the topological conjugacy relation on Cantor minimal systems. The other main result of this thesis concerns the topological conjugacy relation on Toeplitz subshifts. We prove that the topological conjugacy relation on Toeplitz subshifts with separated holes is a hyperfinite Borel equivalence relation. This result provides a partial affirmative answer to a question asked by Sabok and Tsankov. As pointed Cantor minimal systems are represented by properly ordered Bratteli diagrams, we also establish that the Borel complexity of equivalence of properly ordered Bratteli diagrams is $Delta_{mathbb{R}}^+$. KW - Mathematics KW - Descriptive set theory LA - eng ER -