Event Type:

Number Theory Seminar

Date/Time:

Tuesday, November 8, 2016 - 16:00 to 16:45

Location:

BAT 250

Guest Speaker:

Institution:

TU Delft

Abstract:

In 1981, Hitoshi Nakada introduced his now famous class of continued fraction expansions, now known as Nakada's $alpha$-expansions. Here $\alpha$ is a parameter which runs between 0 and 1. In case $\alpha =1$, the continued fraction expansion at hand is the well-known regular continued fraction expansion. Other well-known cases are $\alpha = 1/2$ (the nearest integer continued fraction expansion) and $\\alpha =0$ (the "by-access expansion"). For $\alpha$ between 1/2 and 1, Nakada was able to find the natural extension of the underlying ergodic system. Later Marmi, Moussa and Yoccoz extended his results to $\alpha$ between $\sqrt{2}-1$ and 1/2.

Using these systems, various authors have obtained a large number of new results in the metric theory of continued fraction, but also arithmetic results were obtained, e.g. a Legendre-type theorem for these $\alpha$-expansions (but only for $\alpha$'s between $\sqrt{2}-1$ and 1).

In recent years, $\alpha$-expansions have attracted a lot of attention due to the erratic behavior of the entropy-function (as a function of $\alpha$). As a spin-off I will show in this talk how one can obtain the underlying natural extension for values of $\alpha$ slightly smaller than $\sqrt{2}-1$. Immediately it will be clear that serious problems occur.