On $C^0$-centralizers of Anosov diffeomorphisms on the torus: algebraic and topological aspects
Volume 258 / 2022
Fundamenta Mathematicae 258 (2022), 1-24
MSC: Primary 37D20, 37C20, 37C15; Secondary 37F15, 37C05.
DOI: 10.4064/fm969-10-2020
Published online: 25 April 2022
Abstract
Anosov diffeomorphisms on $\mathbb T^d$ are well known to be topologically conjugate to hyperbolic automorphisms induced by integer valued matrices. We show that in general the internal algebraic structures of the centralizers of such hyperbolic matrices, in the multiplicative group of all integer matrices with determinant $\pm 1$, are not sufficient to determine the matrix centralizers of the automorphisms and, ultimately, the $C^0$-centralizers of the original Anosov diffeomorphisms. We also obtain necessary and sufficient conditions, relying on entropy sequences, for the centralizer to be elementary.
