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As a natural counterpart to Nakada's $\alpha$-continued fraction maps, westudy a one-parameter family of continued fraction transformations with anindifferent fixed point. We prove that matching holds for Lebesgue almost everyparameter in this family and that the exceptional set has Hausdorff dimension1. Due to this matching property, we can construct a planar version of thenatural extension for a large part of the parameter space. We use this toobtain an explicit expression for the density of the unique infinite$\sigma$-finite absolutely continuous invariant measure and to compute theKrengel entropy, return sequence and wandering rate of the corresponding maps.

Matching for a family of infinite measure continued fraction transformations