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In this note, we analyze the classification problem for compact metrizable$G$-ambits for a countable discrete group $G$ from the point of view ofdescriptive set theory. More precisely, we prove that the topological conjugacyrelation on the standard Borel space of compact metrizable $G$-ambits is Borelfor every countable discrete group $G$.

On the complexity of topological conjugacy of compact metrizable $G$-ambits