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arXiv · 2609.12031

The Borel complexity of conjugacy for Cantor minimal systems

Xinan Dai·Wenhao Deng·Yingdong Shi·Tailin Wu·Yuchen Yang
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Abstract

We prove that conjugacy of minimal homeomorphisms of the Cantor space is Borel bireducible with isomorphism of countable graphs, answering the Cantor minimal case of a question of Foreman. We obtain the lower bound by encoding countably based profinite groups. Finite quotient homomorphisms are represented by factor maps between a common family of minimal subshifts. Amalgamation makes the resulting inverse limit independent, up to conjugacy, of the quotient presentation. Conversely, finite-stage factorization recovers the group from any conjugacy of these inverse limits.

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Xinan Dai, Wenhao Deng, Yingdong Shi, Tailin Wu, Yuchen Yang. 2026-09-10. The Borel complexity of conjugacy for Cantor minimal systems. https://arxiv.org/abs/2609.12031

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