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arXiv · 2604.00257

Smoothness of Markov Partitions for Expanding Toral Endomorphisms

Chayce Hughes·Huub de Jong
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Abstract

We show that an expanding toral endomorphism in dimension 2 admits a smooth (in fact linear) Markov partition if and only if some power of the corresponding integer matrix is diagonalizable with integer eigenvalues. We exhibit examples of qualitatively different smoothness behavior, and highlight the existence of a hybrid type of smoothness in dimension 2. For dimension d, we show that expanding toral endomorphisms satisfying the eigenvalue condition above admit a linear Markov partition. Finally, we provide an estimate on the Hausdorff dimension of the boundary of a Markov partition using techniques from symbolic dynamics.

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Chayce Hughes, Huub de Jong. 2026-03-31. Smoothness of Markov Partitions for Expanding Toral Endomorphisms. https://arxiv.org/abs/2604.00257

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