Natural extensions and entropy of $\alpha$-continued fractions

TL;DR

This paper constructs natural extensions for Nakada's alpha-continued fractions, demonstrating the continuity of entropy and measure with respect to alpha, and establishing a specific product relation involving entropy and measure.

Contribution

It introduces a natural extension for each alpha-continued fraction and proves the continuity of entropy and measure as functions of alpha, including an explicit relationship between alpha-expansions.

Findings

Entropy and measure are continuous functions of alpha.

The product of entropy and measure equals π^2/6 for all 0<alpha≤1.

Explicit relationship between alpha-expansions of alpha-1 and alpha.

Abstract

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$.