We give natural extensions for the α-Rosen continued fractions of Dajani et al. for a set of small α values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions. It follows that the underlying maps have equal entropy.
Cornelis KRAAIKAMP. Thomas A. SCHMIDT. Ionica SMEETS. "Natural extensions for α-Rosen continued fractions." J. Math. Soc. Japan 62 (2) 649 - 671, April, 2010. https://doi.org/10.2969/jmsj/06220649