After Hitoshi Nakada introduced his family of $\alpha$-continued fraction expansions in the eighties of the previous century, continued fraction expansions have seen a kind of renaissance. Nakada studied the ergodic properties of his continued fraction expansions for certain values of $\alpha$ (actually, for $\alpha$'s between $\frac{1}{2}$ and $1$). Later, Marmi, Moussa and Yoccoz were able to extend his results for $\alpha$'s between $\sqrt{2}$ and $\frac{1}{2}$. More recently, interest in these $\alpha$-expansions resurfaced, after Stefano Marmi and Laura Luzzi, and more later Hitoshi Nakda and Rie Natsui were able to show that the entropy as a function of $\alpha$ behaves rather wild for values of $\alpha$ between 0 and $\sqrt{2} - 1$. In this talk I want to explain to some extend why such a rich and strange behavior can be observed. This is joint work with my PhD-student Jaap de Jonge.