First return systems for some continued fraction maps

TL;DR

The paper proves that for all n≥3 and α∈(0,1), the first expansive power of the interval maps T_{3,n,α} is realized as the first return map of the geodesic flow to a cross-section of the unit tangent bundle on the hyperbolic surface associated with G_{3,n}, by reducing to the m=2 Hecke-group setting and employing Arnoux’s method with planar extensions. It establishes a precise entropy–volume correspondence h(T)μ(Ω) = vol T^1(G\H) for the relevant families, and provides a detailed analysis of the entropy function h(T_{2,n,α}) in the m=2 setting, including continuity, symmetry, maximal values, and asymptotic decay as n→∞. The work also corrects a Mass formula in the Rosen-fraction (m=2) case, develops the machinery of matching intervals and planar extensions, and constructs the necessary bridge from Ω_{2,n,1} to Ω_{3,n,1} to finish the conjecture’s proof. The results deepen the connection between continued-fraction-like dynamics and hyperbolic geometry, with explicit, computable entropy profiles and a robust geometric interpretation via geodesic flows. This yields a comprehensive framework for viewing a broad family of interval maps as cross-section return maps on hyperbolic surfaces, enriching both dynamical-systems and geometric-structures perspectives.

Abstract

We prove a conjecture of Calta, Kraaikamp and the author: For all $n\ge 3$, each member of their one-parameter family of interval maps, denoted $T_{3,n,α}$, has its `first expansive return map' of natural extension given by the first return map under the geodesic flow to a section of the unit tangent bundle of the hyperbolic surface uniformized by the underlying Fuchsian group $G_{3,n}$. To achieve the proof, we first prove the corresponding result for analogous one-parameter families related to the Hecke triangle Fuchsian group $G_{2,n}$. A direct comparison per $n$ of the $α=1$ planar domains allows the Hecke group setting to provide sufficient information to prove the conjecture. We also give details about the entropy functions for the Hecke triangle Fuchsian group maps, $α\mapsto h(T_{2,n,α})$. Each is continuous on $(0,1)$, increasing on $(0,1/2)$, decreasing on $(1/2,1)$, with a central interval of constancy. We give precise formulas for the end points of the central intervals and also give precise formulas for the maximal entropy per family. For fixed $α$, the entropy of $T_{2,n,α}$ goes to zero as $n$ tends to infinity.

Figures (7)

• Figure 1: Birkhoff sums approximation to entropy functions $\alpha \mapsto h(T_{2,n,\alpha})$, for $n\in \{3, 5,8,50\}$ and $0.1< \alpha \le 1/2$. Horizontal dashed lines at maximum value as per Theorem \ref{['t:mIsTwoSimpleEntropyBehavior']}; vertical dashed lines at $1- R_n/t_{2,n}$ per $n$.
• Figure 2: The $\mu$-mass of the Rosen planar extension $\Omega_{2s+3}$ of BKS (in red) is comparable to that of $\Omega_{2s+2}$ (in blue), with both going to infinity as $s\to \infty$. See § \ref{['ss:Correction']}.
• Figure 3: On the left: The planar domain $\Omega_{2,3,1/5}$, and its partitioning 'blocks'; each $\mathcal{B}_i$ fibers over the cylinder where $T_{2,3,1/5}$ is given by $x \mapsto A^iC\cdot x$. The $T_{2,3,1/5}$-orbit of $r_0(1/5) = 1/5$ is labeled by $r_0, r_1$; the $T_{2,3,1/5}$-orbit of $\ell_0(1/5) = -4/5$ is labeled by $\ell_0,\ell_1, \cdots$. One has $r_2(\alpha) = \ell_5(\alpha)$ for other $\alpha$ in the matching interval $J_{k,v} = J_{5,1}$, but here the special equality arises: $r_1 = \ell_4 = 0$. The $y$-values of the horizontal boundaries are derived from appropriate $T_{2,3,\eta_{5,1}}$- and $T_{2,3,\eta_{5,1}}$-orbits, see the discussion of § \ref{['sss:OneFifth']}. The blue rectangle indicates the region $\Omega^{+}_{\alpha, d}$ of Lemma \ref{['l:firstReturnTo2ndQuadrant']}. On the right: The image of $\Omega_{2,3,1/5}$ under $\mathcal{T}_{2,3,1/5}$ is $\Omega_{2,3,1/5}$ itself.
• Figure 4: Decreasing mass along $J_{1,5}$. With $m=2, n=3$, both $\alpha = 0.18$ and $\alpha' = 0.2$ lie in the matching interval $J_{1, 5}$. Along any matching interval, the coordinates of all tops and bottoms of the rectangles which comprise the various $\Omega_{2,n,\alpha}$ are constant (here $y_1, \dots y_5$ and $y_{-2}, y_{-1}$). The function $\mathcal{T}_A$ maps the rectangle $\mathcal{D} = [\ell_0, \ell'_0)\times [r_{-\overline{S}-1}, y_1]$ to $\mathcal{A} = [r_0, r'_0)\times [r_{-1}, y_{\overline{S}+1}]$, where here we use simplified notation of $\ell_0$ for $\ell_0(\alpha)$ and $\ell'_0$ for $\ell'_0(\alpha)$, and similarly for $r_0, r'_)$. To pass from $\Omega_{2,n,\alpha}$ to $\Omega_{2,n,\alpha'}$, we delete the initial $\mathcal{T}_{2,n,\alpha}$-orbit of $\mathcal{D}$ and add $\mathcal{A}$ and its initial $\mathcal{T}_{2,n,\alpha}$-orbit. Since each step preserves $\mu$-measure, the sign of the change in mass is that of $(\overline{S}(k) - \underline{S}(k,v))$. See Lemma \ref{['c:massIncreasesWhere']}.
• Figure 5: The domain $\Omega_{2,5,0}$ is the union of the gray and the hatched regions. These latter are excised to leave the planar domain of the two-dimensional system defined by the interval map given by by accelerating $T_{2,5,0}$ by applying appropriate powers of $W$ for $x$ in $(\ell_0, \epsilon_0) = (-t_{2,5}, \epsilon_0)$, where $\epsilon_0 = W^{-1}\cdot 0$. The negative sloped hatched rectangles form $\mathcal{D}_1$ of \ref{['e:deletionRects']}, the positive sloped hatched rectangles form $\mathcal{D}_2$.

Theorems & Definitions (35)

• Theorem 1
• Theorem 2
• Definition 2.1
• Theorem 2.2
• Theorem 2.3: Nakada 2010
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 3