  We study endomorphisms of complete Noetherian local rings in the context of
motivic integration. Using the notion of an auto-arc space, we introduce the
(reduced) auto-Igusa zeta series at a point, which appears to measure the
degree to which a variety is not smooth that point. We conjecture a closed
formula in the case of curves with one singular point, and we provide explicit
formulas for this series in the case of the cusp and the node. Using the work
of Denef and Loeser, one can show that this series will often be rational.
These ideas were obtained through extensive calculations in Sage. Thus, we
include a Sage script which was used in these calculations. It computes the
affine arc spaces $\nabla_{\mathfrak{n}}X$ provided that $X$ is affine,
$\mathfrak{n}$ is a fat point, and the ground field is of characteristic zero.
Finally, we show that the auto Poincar\'e series will often be rational as well
and connect this to questions concerning new types of motivic integrals.
