Multifractal analysis of homological growth rates for hyperbolic surfaces

TL;DR

This work analyzes the multifractal structure of homological growth rates along oriented geodesics on hyperbolic surfaces via a Bowen–Series coding of a finitely generated Fuchsian group $G$. By combining symbolic dynamics, finite and induced Markov partitions, and thermodynamic formalism, it derives a formula for the Hausdorff dimension of level sets $\ ext{$\mathscr{H}(\alpha)$}$ of growth rates in terms of the generalized Poincaré exponent $P(\beta)$, with the Legendre transform $P^*(-\alpha)$ giving the spectrum $b(\alpha)=P^*(-\alpha)/\alpha$ on its domain, and proves analytic dependence of the spectrum. The approach handles groups with and without parabolic elements, using an induced uniformly expanding system to manage neutral points and establishing the analyticity of the spectrum. The results extend Prior work for freely generated Kleinian groups to a broad class of Fuchsian groups and reveal how the Lyapunov and homological growth spectra align, yielding a unified thermodynamic picture of the conical limit set. Overall, the paper provides a rigorous framework linking geometric growth, symbolic coding, and multifractal spectra on hyperbolic surfaces.

Abstract

We perform a multifractal analysis of homological growth rates of oriented geodesics on hyperbolic surfaces. Our main result provides a formula for the Hausdorff dimension of level sets of prescribed growth rates in terms of a generalized Poincaré exponent of the Fuchsian group. We employ symbolic dynamics developed by Bowen and Series, ergodic theory and thermodynamic formalism to prove the analyticity of the dimension spectrum.

Figures (4)

• Figure 1: An oriented geodesic $\gamma$ crossing copies of the fundamental domain $R$.
• Figure 2: The graphs of $\beta\in\mathbb R\mapsto P(\beta)$ and $\alpha\in[\alpha_-,\alpha_+]\mapsto b(\alpha)$: $G$ has no parabolic element (upper); $G$ has a parabolic element (lower). We have $\delta_G=\min\{\beta\geq0\colon P(\beta)=0\}$. The constant $\alpha_G$ is the unique maximum point of the $\mathscr{H}$-spectrum, see \ref{['ag']} for the definition.
• Figure 3: An oriented geodesic $\gamma$ with the finite cutting sequence $g_0,g_1$ for a free Fuchsian group with two generators.
• Figure 4: A fundamental domain $R$ of a finitely generated Fuchsian group of the first kind with eight sides: $e_1$ and $e_8$, $e_2$ and $e_6$, $e_3$ and $e_7$, $e_4$ and $e_5$ are identified in pairs, which yields a hyperbolic surface of genus $2$. The bidirectional arrows indicate elements of the finite Markov partition constructed in Section \ref{['markov-sec']}.

Theorems & Definitions (79)

• Proposition 2.1: Ser86, Theorem 3.1(ii)
• Lemma 2.2
• proof
• Proposition 2.3: Ser86, Theorem 4.2
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof