Stable-norm boundary geometry on once-punctured tori

Abstract

We use normal-turn estimates to study the local and global geometry of the boundaries of stable-norm balls $B_X$ for complete finite-area hyperbolic once-punctured tori $X$. We show that the boundary $\partial B_X$ is a convex-geometric detector of exponential Diophantine approximation: rational directions give genuine corners with exponentially small exterior angles, while at an irrational direction $\beta$ the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach $\beta$ and the ${\ell }^{\infty }$-radius of $B_X$ in the projective direction $\beta$. Thus irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets. The computation of the rational corners unexpectedly yields a new McShane-type identity for the area of the dual body $B_X^{\ast }$: $$\mathrm{Area}(B_X^{\ast })=4\sum _{\gamma }{\ell }_X(\gamma )\log \coth \left(\frac{{\ell }_X(\gamma )}{2}\right),$$ where the sum is over unoriented simple closed geodesics on $X$. As another global application, the normal-turn estimates also give a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly $L\ge 2$ is at most $C_X(\log L{)}^2$. For the modular torus this yields $$\#{\lambda }_M^{-1}(m)\le C(\log \log (3m){)}^2$$ for every Markoff number $m$, improving the previous logarithmic bounds for Markoff fibers.

arXiv commentTitle changed following referee comments. Introduction revised to clarify the history of the stable norm. Paper reorganized. Added a McShane-type identity expressing the area of the dual norm ball as a sum over simple closed geodesics. 34 pages, 7 figures. Comments welcome!