Ergodicity of asymmetric lemon Billiards

TL;DR

This work proves ergodicity for the asymmetric lemon billiard by establishing uniform hyperbolicity of a return map on a carefully chosen section. The authors introduce an invariant cone framework in the p-metric, control singularity curves (including slope-sign changes), and deploy a detailed case-by-case expansion analysis across four collision-pattern regimes (a0, a1, b, c), handling contraction regions with precise geometric estimates. The combination of invariant half-quadrant cones, controlled expansion through iterates, and novel management of singularities yields global ergodicity (with potential for K-property and higher mixing rates). The results advance the understanding of hyperbolic billiards beyond Wojtkowski-type criteria by exploiting the lemon table’s geometry and a perturbative link to the 1-petal case.

Abstract

We show ergodicity of (asymmetric) lemon billiards, billiard tables that are the intersection of two circles of which one contains the centers of both. These do not satisfy the Wojtkowski criteria for hyperbolicity, but we establish \emph{uniform} expansion of vectors in an invariant cone family and alignment of singularity curves. Both of these are difficult, and the approach to the latter seems new. Together, these properties imply ergodicity.

Figures (52)

• Figure 1: Lemon billiard Table
• Figure 2: The 1-petal billiard and its unfolding to the 2-petal billiard
• Figure 3: The region $M_r$. The blue parallelogram region is ${\color{blue}M^{\text{\upshape in}}_r}$, and the red one is ${\color{red}M^{\text{\upshape out}}_r}$. The small sectors are 'bad' regions (\ref{['REMBadRegions']}).
• Figure 4: The region $M_R$. The red region is ${\color{red}M^{\text{\upshape out}}_R}$, the blue region is ${\color{blue}M^{\text{\upshape in}}_R}$.
• Figure 5: \ref{['fig:Mr']} with $\phi_*<\pi/6$ and more detail. The two pink strips are $\mathcal{F}^{-1}({\color{red}M^{\text{\upshape out}}_r}\setminus{\color{blue}M^{\text{\upshape in}}_r})$ and have slope $-1/4$. The green strips are $\mathcal{F}^{-2}({\color{red}M^{\text{\upshape out}}_r}\setminus({\color{red}M}^{\text{\color{red}\upshape out}}_{r,0}\cup{\color{red}M}^{\text{\color{red}\upshape out}}_{r,1}\cup{\color{red}M}^{\text{\color{red}\upshape out}}_{r,2}))$ and have slope $-1/6$. The intersections ${\color{blue}M}^{\text{\color{blue}\upshape in}}_{r,0}$, $\textcolor{blue}{M^{\upshape in}_{r,1}}$ and $\textcolor{blue}{M^{\upshape in}_{r,2}}$ are marked. The leftmost point of the closure of $M^{\text{\upshape in}}_{r,0}$ is $(\pi-\phi_*,\frac{\pi}{2})$. The leftmost point of the closure of $M^{\text{\upshape in}}_{r,1}$ is $(\frac{2\pi}{3}-\phi_*,\frac{\pi}{3})$. The leftmost point of the closure of $M^{\text{\upshape in}}_{r,2}$ is $(\frac{\pi}{2}-\phi_*,\frac{\pi}{4})$$\hat{M}$ is the union of the two pink strips and the center rhombus ${\color{red}M}^{\text{\color{red}\upshape out}}_{r,0}={\color{blue}M}^{\text{\color{blue}\upshape in}}_{r,0}={\color{blue}M^{\text{\upshape in}}_r}\cap{\color{red}M^{\text{\upshape out}}_r}$. ${\color{blue}N^{\text{\upshape in}}}$ and ${\color{red}N^{\text{\upshape out}}}$ are neighborhoods of $x_*,y_*,Ix_*,Iy_*$ respectively.

Theorems & Definitions (228)

• Theorem A
• proof : Proof of uniform hyperbolicity in \ref{['RMASUH']}
• Remark 2.2
• Definition 2.3: Symmetries
• Remark 2.5
• Definition 2.6: Returns to $\hat{M}$
• Remark 2.7: "Bad regions"
• Proposition 2.8: Location of bad regions
• proof
• Remark 2.9