Natural extensions and Nakada's alpha-expansions

In 1981, Nakada introduced and studied a large class of continued fraction transformations, which are now known as alpha-expansions. For alpha between 1/2 and 1 he obtained the so-called "natural extension" the underlying dynamical systems underlying these continued fraction algorithms. His work played a fundamental role in the solution of the Doeblin-Lenstra conjecture by Bosma, Jager and Wiedijk in 1983. Later, Marmi, Moussa and Yoccoz found the natural extension for alpha's between square root of 2 minus 1, and 1/2. Only recently, due to work by Marmi and his student Luzzi, alpha-expansions are intensely studied for values of alpha between 0 and  square root of 2 minus 1. In this talk I will explain why this case is so much more difficult than the case of alpha between  square root of 2 minus 1 and 1.