  As a natural counterpart to Nakada's $\alpha$-continued fraction maps, we
study a one-parameter family of continued fraction transformations with an
indifferent fixed point. We prove that matching holds for Lebesgue almost every
parameter in this family and that the exceptional set has Hausdorff dimension
1. Due to this matching property, we can construct a planar version of the
natural extension for a large part of the parameter space. We use this to
obtain an explicit expression for the density of the unique infinite
$\sigma$-finite absolutely continuous invariant measure and to compute the
Krengel entropy, return sequence and wandering rate of the corresponding maps.
