WoG 2024 Talk 2.4: Kangrae Park - Fellow Traveling of Geodesics on the Modular Surface

Speaker: Kangrae Park
Institution: Seoul National University

Title: Fellow Traveling of Geodesics on the Modular Surface
Abstract: The modular surface \( M = \mathrm{SL}_2(\mathbb{R}) \backslash \mathbb{H}^2 \) has a well-known connection between its geodesic flow \( g_t \) and continued fraction expansions. We aim to study the Hausdorff dimension of the set
\[
\mathscr{B}_v^M(R) = \{ w \in T^1M \,:\, d(g_t v, g_t w) \lt R \; \forall t \in \mathbb{R} \}.
\]
This problem generalizes the question of the Hausdorff dimension of badly approximable numbers. Using the relationship between continued fractions and the coding of geodesic flows, we translate the criteria for elements of \(\mathscr{B}_v^M(R)\) into conditions on continued fraction digits. These findings provide insights into \(\mathscr{B}_v^M(R)\), even though a complete proof is still pending.