Polynomial convergence rate at infinity for the cusp winding spectrum of generalized Schottky groups
Yuya Arima
TL;DR
This work studies cusp winding spectra for a generalized Schottky group with a single parabolic generator acting on the Poincaré disk. It develops a multifractal framework using thermodynamic formalism on a countable Markov shift to relate the spectrum $b(\alpha)$ to equilibrium measures, proving that $b(\alpha)$ converges to $\dim_H(\Lambda_c(G))$ at a polynomial rate governed by a sharp exponent $x_0=\frac{1}{2-2\dim_H(\Lambda_c(G))}-1$. The main analytical machinery combines pressure theory, Gibbs states, and Dirichlet-series techniques via Mellin transforms and the Gamma and zeta functions to connect large $\alpha$ behavior with spectral data of the induced system. The results provide a new geometric interpretation of the cusp winding in relation to the geodesic flow on $\mathbb{D}/G$ and highlight a novel phenomenon where the Hausdorff dimension of the limit set shapes multifractal convergence rates.
Abstract
We show that the convergence rate of the cusp winding spectrum to the Hausdorff dimension of the limit set of a generalized Schottky group with one parabolic generator is polynomial. Our main theorem provides the new phenomenon in which differences in the Hausdorff dimension of the limit set generated by a Markov system cause essentially different results on multifractal analysis. This paper also provides a new characterization of the geodesic flow on the Poincaŕe disc model of two-dimensional hyperbolic space and the limit set of a generalized Schottky group. To prove our main theorem we use thermodynamic formalism on a countable Markov shift, gamma function, and zeta function.