Summary
Using periodic points, we study a notion of entropy with values in the p-adic numbers. This is done for actions of countable discrete residually finite groups \(\Gamma\). For suitable \(\Gamma = \mathbb{Z}^d\)-actions we obtain p-adic analogues of multivariable Mahler measures. For certain actions of more general groups, the p-adic entropy can be expressed in terms of a p-adic analogue of the Fuglede–Kadison determinant from the theory of von Neumann algebras. Many basic questions remain open.
2000 Mathematics Subject Classifications: 11D88, 16S34, 19B28, 37A35, 37C35, 37C85
Dedicated to Yuri Ivanovich Manin
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Deninger, C. (2009). p-adic Entropy and a p-adic Fuglede–Kadison Determinant. In: Tschinkel, Y., Zarhin, Y. (eds) Algebra, Arithmetic, and Geometry. Progress in Mathematics, vol 269. Birkhäuser Boston. https://doi.org/10.1007/978-0-8176-4745-2_10
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DOI: https://doi.org/10.1007/978-0-8176-4745-2_10
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