p-adic entropy and a p-adic Fuglede-Kadison determinant

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.