arXiv subjects

Claire Merriman

Publications and source records attributed to Claire Merriman.

5 recordsLinked to original sources

arXiv2023

Geodesic flows and slow downs of continued fraction maps

The connection between cutting sequences of geodesics on the modular surface $\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$ and regular continued fractions was established by Series, and Heersink expanded the cross-section of the geodesic flow on the unit tangent bundle to the modular surface to describe the Farey tent-map as a slowdown of the Gauss map for the regular continued fractions. Boca and the author expanded the connection between cutting sequences of geodesics on the modular surface $Θ\backslash\mathbb{H}$ and even continued fractions, which was previously established as a billiard flow by Bauer and Lopes. We will similarly expand the cross-section of the geodesic flow on this unit tangent bundle to describe the three-branch slowdown of the even Farey map.

math.DS↗

arXiv2021

Geodesic flows and the mother of all continued fractions

We extend the Series' connection between the modular surface $\mathcal{M}=\operatorname{PSL}(2,\mathbb{Z})\backslash\mathbb{H}$, cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits $(1,+1)$ and $(2,-1)$ in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that $\frac{dxdy}{(1+xy)^2}$ is invariant under the natural extension map.

math.DS↗

arXiv2021

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

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 $α\in [g,G]$, 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 $α,α^*\in [g,G]$ 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 $α\in [g,G]$, a fact already observed in [BM]. Furthermore, it is shown that this approach can be extended to values of $α$ smaller than $g$, and that for values of $α\in [\tfrac{1}{6}(\sqrt{13}-1), g]$ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $α\in [0,G]$. 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.

math.DS↗

arXiv2019

Coding of geodesics on some modular surfaces and applications to odd and even continued fractions

The connection between geodesics on the modular surface $\operatorname{PSL}(2,{\mathbb Z})\backslash {\mathbb H}$ and regular continued fractions, established by Series, is extended to a connection between geodesics on $Γ\backslash {\mathbb H}$ and odd and grotesque continued fractions, where $Γ\cong {\Bbb Z}_3 \ast {\Bbb Z}_3$ is the index two subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by the order three elements $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 1 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 0 & 1 \\ -1 & 1 \end{smallmatrix} \right)$, having an ideal quadrilateral as fundamental domain. A similar connection between geodesics on $Θ\backslash {\mathbb H}$ and even continued fractions is discussed in our framework, where $Θ$ denotes the Theta subgroup of $\operatorname{PSL}(2,{\mathbb Z})$ generated by $\left( \begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix} \right)$ and $\left( \begin{smallmatrix} 1 & 2 \\ 0 & 1 \end{smallmatrix} \right)$.

math.DS↗

arXiv2019

$α$-Expansions with odd partial quotients

We consider an analogue of Nakada's $α$-continued fraction transformation in the setting of continued fractions with odd partial quotients. More precisely, given $α\in [\frac{1}{2}(\sqrt{5}-1),\frac{1}{2}(\sqrt{5}+1)]$, we show that every irrational number $x\in I_α=[α-2,α)$ can be uniquely represented as $$ x= \cfrac{e_1 (x;α)}{d_1 (x;α) +\cfrac{e_2(x;α)}{d_2(x;α)+\cdots}} , $$ with $e_i(x;α) \in \{ \pm 1\}$ and $d_i(x;α) \in 2{\mathbb N} -1$ determined by the iterates of the transformation $$φ_α(x) := \frac{1}{| x|} - 2 \bigg[ \frac{1}{2| x|} +\frac{1-α}{2} \bigg]-1$$ of $I_α$. We also describe the natural extension of $φ_α$ and prove that the endomorphism $φ_α$ is exact.

math.DS↗