Title: On the complexity of topological conjugacy of compact metrizable $G$-ambits,
Abstract: In this note, we analyze the classification problem for compact metrizable
$G$-ambits for a countable discrete group $G$ from the point of view of
descriptive set theory. More precisely, we prove that the topological conjugacy
relation on the standard Borel space of compact metrizable $G$-ambits is Borel
for every countable discrete group $G$.