A canonical thickening of Q and the dynamics of continued fractions

TL;DR

This paper constructs a measure-full set of quadratic surd endpoints within (0,1] and proves these are the monotonicity intervals of the entropy for alpha-continued fractions, confirming a conjecture.

Contribution

It establishes a precise link between quadratic surd endpoints and entropy monotonicity intervals in alpha-continued fractions, confirming Nakada and Natsui's conjecture.

Findings

Constructed a measure-full set of quadratic surd endpoints.

Identified these endpoints as the monotonicity intervals of entropy.

Proved the conjecture of Nakada and Natsui.

Abstract

We construct a countable family of open intervals contained in (0,1] whose endpoints are quadratic surds and such that their union is a full measure set. We then show that these intervals are precisely the monotonicity intervals of the entropy of alpha-continued fractions, thus proving a conjecture of Nakada and Natsui.