Genericity of sublinearly Morse directions in general metric spaces

TL;DR

The paper addresses the genericity of sublinearly Morse directions in general metric spaces under a proper, non-elementary SCC action with a contracting element. It develops a unified conformal-density framework on convergence boundaries, builds Patterson–Sullivan measures on the horofunction boundary, and leverages shadow lemmas and barrier analysis to show that regularly contracting (and hence sublinearly Morse) directions occupy full measure with respect to the conformal density. This yields a broad, quasi-isometry-invariant understanding of sublinear Morse behavior that extends prior CAT(0) and Teichmüller-space results to general proper metric spaces. The findings highlight the robustness of sublinear Morse directions as a measure-theoretic generic phenomenon and illuminate connections to divergence-type dynamics and conical points, with implications for geometric group theory and boundary dynamics in diverse spaces.

Abstract

In this paper, we show that for a proper statistically convex-cocompact action on a proper geodesic metric space, the sublinearly Morse boundary has full Patterson-Sullivan measure in the horofunction boundary.

Figures (7)

• Figure 1: A $\kappa$--neighbourhood of a geodesic ray $\tau$ with multiplicative constant ${\sf n}$.
• Figure 2: A sublinearly contracting geodesic ray
• Figure 3: Illustrate Assumptions (A)(B)(C) in Definition \ref{['ConvBdryDefn']}
• Figure 4: The proof of Lemma \ref{['RegContrRayClass']}. The dotted area indicates the $r$-neighborhood of $p_n$, and the double lines $\beta_1$ and $\beta_2$ are $\theta$-segments of $[o,u]_\gamma$ and $[o,v]_\alpha$ accordingly.
• Figure 5: Illustration of $\mathcal{O}_{M_1,M_2}$.

Theorems & Definitions (76)

• Conjecture 1.1
• Definition 2.1: Contracting subset
• Lemma 2.2
• proof
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof