  In this note we describe instances where values of the $K$-theoretical
regulator map evaluated on topological cycles equal entropies of topological
actions by a group $\Gamma$. These entropies can also be described by
determinants on the von Neumann algebra of $\Gamma$. The relations were first
observed for real regulators. The latter have $p$-adic analogues and both
$p$-adic entropy and $p$-adic determinants were then defined so that similar
relations hold as in the real case. We describe this $p$-adic theory in the
second part of the paper. This note is almost entirely a survey of known
results with the exception of some results in section 3.1. However the
different aspects of the theory have not been discussed together before. Along
the way we point out several open questions and possible directions for further
research.
