Natural extensions and entropy of $\alpha$-continued fraction expansions with odd partial quotients

TL;DR

This paper investigates the natural extensions and entropy behavior of a class of odd partial quotient continued fractions parameterized by lpha, demonstrating isomorphisms across parameter ranges and analyzing entropy variations using matching phenomena.

Contribution

It extends the understanding of natural extensions and entropy for lpha-continued fractions, showing isomorphisms and detailed entropy behavior across a broader parameter range.

Findings

Natural extensions are isomorphic for lpha in [g,G]

Entropy remains constant across these isomorphisms

Entropy varies (increases, decreases, or stays constant) near zero due to matching phenomena

Abstract

In an article (which we will refer to as [BM]) of Boca and the fourth author of this paper, a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $\alpha\in [g,G]$, where $g=\tfrac{1}{2}(\sqrt{5}-1)$ and $G=g+1=1/g$ are the two golden mean numbers is introduced. In this article, by using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [BM] are obtained, and it is shown that for each $\alpha,\alpha^*\in [g,G]$ the natural extensions from [BM] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $\alpha\in [g,G]$, a fact already observed in [BM]. Furthermore, it is shown that this approach can be extended to values of $\alpha$ smaller than $g$, and that for values of $\alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g]$ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $\alpha\in [0,G]$. It is shown that in any neighborhood of $0$ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. In order to prove this we use a phenomena called matching.