  It is known that the classical Banach--Stone theorem does not extend to the
class of $AC(\sigma)$ spaces of absolutely continuous functions defined on
compact subsets of the complex plane. On the other hand, if $\sigma$ is
restricted to the set of compact polygons, then all the corresponding
$AC(\sigma)$ spaces are isomorphic. In this paper we examine the case where
$\sigma$ is the spectrum of a compact operator, and show that in this case one
can obtain an infinite family of homeomorphic sets for which the corresponding
function spaces are not isomorphic.
