The Manhattan curve and the correlation of length spectra on hyperbolic surfaces
Let \(\Gamma_1\) and \(\Gamma_2\) be two isomorphic co-compact Fuchsian groups, and consider the conjugacy classes for which the corresponding closed geodesics on \(H^2/\Gamma_1\) and \(H^2/\Gamma_2\) have approximately the same length. It is known that if \(\Gamma_1\) and \(\Gamma_2\) are not conjugate in \(\text{PSL}(\alpha,\mathbb{R})\), then there exists constants \(C=C(\varepsilon)\) and \(\alpha\in(0,1)\) such that \[ \text{Card}\{[\gamma]:\ell_1[\gamma],\ell_2[\gamma]\in (T,T+\varepsilon)\}\sim C\;\frac{e^{\alpha T}}{T^{3/2}},\;\text{as} T\to+\infty, \] where \(\ell_i[\gamma]\) denotes the length of the closed geodesic on \(H^2/\Gamma_i\) lying in the conjugacy class \([\gamma]\). Note that \(\alpha\in (0,1)\) is called the correlation number and is independent of \(\varepsilon\). The main goal of this note is to identify the exponent \(\alpha=\alpha(\Gamma_1,\Gamma_2)\) as the value \(\alpha=a+b\) where \((a,b)\) is the point on the curve \(C(\Gamma_1,\Gamma_2)\) for which the normal has slope 1 and where \(C(\Gamma_1,\Gamma_2)\) is the Manhattan curve associated to \((\Gamma_1,\Gamma_2)\).
scientific article; zbMATH DE number 1229593