Bounded deviations in higher genus II: minimal laminations

TL;DR

This work completes the bounded deviations program on closed hyperbolic surfaces of genus $g\ge 2$ by treating deviations relative to minimal nonclosed geodesic laminations, extending the torus irrational-direction results to higher genus. It develops a tracking-laminations framework, proving that such laminations are orientable and that bounded deviations hold both transversally to the lamination and along its direction, via a combination of forcing theory, tracking geodesics, and Atkinson-type arguments. The main contributions include (i) orientability of laminations for minimal nonclosed classes, (ii) a transverse bounded-deviation principle guaranteeing that orbits cannot drift unboundedly in directions orthogonal to tracking geodesics, and (iii) a directional bound along the lamination established through two controlled approximations and syndetic intersection arguments. These results constrain the shape of ergodic rotation sets on high-genus surfaces and lay groundwork for pseudo-foliations with irrational direction in the non-wandering, measure-preserving, and $C^r$ settings.

Abstract

This article follows and completes [GT25], where we study the problem of bounded deviations for homeomorphisms of closed surfaces of genus $\ge 2$. This second part deals with bounded deviations relative to geodesic minimal laminations that are not reduced to a closed geodesic. The combination of both articles generalises to the higher genus case most of the bounded deviations results already known for the torus.

Figures (12)

• Figure 1: The statement of Theorem \ref{['ThmBndDevIrrat']}. The sets $\widetilde{X}^-$ and $\widetilde{X}^+$ are complement of the lift of $X$. Theorem \ref{['ThmBndDevIrrat']} prevents from having a trajectory such as the one of $y_0$.
• Figure 2: Theorem \ref{['TheoBndedDirLam']} asserts that such a grey orbit cannot exist: an orbit cannot meet first the top red domain and then the bottom red domain (delimited by geodesics that are orthogonal to $\widetilde{\gamma}_{\widetilde{z}}$ and at a distance $C$).
• Figure 3: Proof of Theorem \ref{['TheoLaminMinim']}: such a red path cannot be the image of the transverse trajectory of $\tilde{z}$ by some deck transformation, otherwise it would have to meet either $\widetilde{B}$ or $T\widetilde{B}$.
• Figure 4: Proof of Lemma \ref{['LemIfFarThenNotEquiv']}, first case: when the geodesics $\widetilde{\gamma}_{\widetilde{z}}$ and $T'T_0\widetilde{\gamma}_{\widetilde{z}}$ do not have the same orientation.
• Figure 5: Proof of Lemma \ref{['LemIfFarThenNotEquiv']}: the leaves $\widetilde{\phi}_{I^{t_0}_{\widetilde{\mathcal{F}}}(T'\widetilde{z})}$ and $\widetilde{\phi}_{I^{t_1}_{\widetilde{\mathcal{F}}}(T'\widetilde{z})}$ are included in $R(T'T_0\widetilde{B})$ and hence, by ordering of the bands, in $R(\widetilde{B})$.

Theorems & Definitions (76)

• Theorem A
• Theorem B
• Corollary C
• Theorem D
• Remark 1.1
• Theorem E
• Corollary F
• Theorem 2.1
• Lemma 2.2
• Theorem 2.3