Non-Trivial Minimal Horocyclic Orbit Closures

• Non-trivial minimal horocyclic orbit closures are closed, invariant sets where every orbit is dense, yet they are neither periodic nor the full nonwandering set.
• They are explicitly constructed via loom surfaces using geodesic slack functions and Busemann-type invariants to precisely control fractal dimensions within (1,2).
• These constructions challenge classical rigidity in homogeneous dynamics by introducing new infinite, ergodic, and singular invariant measures in complex infinite-volume settings.

Non-trivial minimal horocyclic orbit closures are closed invariant subsets for the horocycle flow that are neither a single periodic orbit nor the entire non-wandering set, typically manifesting in spaces where classical rigidity and homogeneity break down. Their existence, structure, and dimension reflect deep facets of the geometric, dynamical, and ergodic landscape of negative curvature and homogeneous flows.

1. Conceptual Definition and Historical Context

A minimal set for a flow is a closed, invariant subset such that every orbit is dense in it and no strictly smaller closed invariant subset supports this property. For the horocycle flow on homogeneous spaces—such as on unit tangent bundles of finite-volume hyperbolic manifolds, lattice quotients, or translation surface moduli spaces—minimal sets are classically either a single closed orbit (periodic horocycle) or the whole nonwandering set. On infinite volume surfaces or non-compact spaces, rigidity fails, and the existence of "non-trivial" minimal closures (i.e., strictly intermediate, nonperiodic nonwandering sets) becomes both possible and nontrivial to construct.

Prior to 2025, no explicit geometrically infinite hyperbolic surface was known to support a minimal closed invariant subset for the horocycle flow which was neither a single orbit nor the full nonwandering set. Recent work—see "Weaving Geodesics and New Phenomena in Horocyclic Dynamics" (Dal'Bo et al., 28 Oct 2025)—provides the first concrete constructions and a systematic understanding of their structure and invariance.

2. Structures and Explicit Constructions

The occurrence of non-trivial minimal horocyclic orbit closures depends on the ambient geometry, and the dynamical system in play:

• Classical homogeneous hyperbolic surfaces (finite volume): Only trivial minimal sets: periodic or dense orbits (Matsumoto, 2014, Clotet, 2023, Knopp et al., 2011).
• Laminations and foliations with trivial holonomy-free loops: All orbits are dense (fully minimal dynamics) (Alcalde et al., 2014).
• Non-compact or infinite-type hyperbolic spaces: Potential for complicated orbit closure behavior, including minimal sets that are not the whole nonwandering set.

The explicit construction in (Dal'Bo et al., 28 Oct 2025) is via loom surfaces—double covers of bands in the hyperbolic plane minus infinitely many disjoint half-planes arranged to control injectivity radius and geodesic/hocycle recurrence. Key features:

• The "slack" function $S(\alpha)$ for a geodesic segment $\alpha$ measures inefficiency: $$S(\alpha) = \text{length}(\alpha) - (\tau(\alpha(b))-\tau(\alpha(a)))$$, where $\tau$ is a height function.
• The Busemann-type function $\beta(y) = \tau(y) - S(A_+ y)$ is $N$-invariant (horocycle invariant).
• Minimal sets arise as level sets of $\beta$ for certain "weaving" patterns—geodesic paths alternating sides in a way that the total slack is summable.

Such level sets constitute closed $N$-invariant subsets ($F = \{\beta = 0\}$, for instance) which support a horocycle flow whose orbits are all dense in $F$ but $F$ is neither the whole nonwandering set nor a periodic horocycle. This realizes a genuinely non-trivial minimal horocyclic orbit closure.

3. Dynamical and Measure-Theoretic Properties

These sets $F$ produced in (Dal'Bo et al., 28 Oct 2025) possess notable features:

• Minimality: $F$ is closed, $N$-invariant, all $N$-orbits are dense in $F$, $F \neq \mathcal{E}$ (full nonwandering set), $F \neq Nx$ (single orbit).
• Invariant Measure: $F$ supports an infinite, locally finite, ergodic, conservative $N$-invariant measure $\mu$ that is singular with respect to geodesic flow; i.e., for any $t \neq 0$, $a_t \mu \perp \mu$ (no mutual absolute continuity under the flow, reflecting new measure rigidity phenomena).
• Fractal Structure: Control over the sequence of slacks allows the orbit closure to have prescribed Hausdorff dimension $\alpha \in (1,2)$; unlike previous known examples (e.g., $\mathbb{Z}$-covers (Farre et al., 16 Sep 2024, Farre et al., 2023)) where dimension is always integer.

4. Classification and Comparison with Other Settings

The landscape of horocyclic orbit closures depends strongly on geometry and dynamical context. The following table summarizes the state of minimal sets for the horocycle flow in several settings:

Geometry / Surface Class	Possible Minimal Sets for Horocycle Flow	Structure of Non-trivial Closures
Compact hyperbolic / lattice surfaces	Periodic or entire space only (Knopp et al., 2011, Chaika et al., 2015)	None (except periodic)
Open tight hyperbolic surfaces	No minimal sets; almost all orbits are dense (Matsumoto, 2014)	No non-trivial minimal closure
Laminations with holonomy-free loops	Horocycle flow is minimal (Alcalde et al., 2014)	All orbits dense (no proper minimal set)
Geometrically finite (cusped / expanding)	Periodic orbits; possibly union with large nonwandering set (Clotet, 2023)	No non-trivial minimal closure
$\mathbb{Z}$-covers of compact surfaces	No non-trivial minimal set; closures are fractal, but never minimal (Farre et al., 16 Sep 2024, Farre et al., 2023)	Closures never minimal
Loom surface / tailored infinite surface	Non-trivial minimal sets exist: $F = \overline{Nx_0} \neq Nx_0,\mathcal{E}$ (Dal'Bo et al., 28 Oct 2025)	Explicitly constructed, flexible dimension

Notably, in regular covers ($\mathbb{Z}$-covers), non-trivial non-maximal closures abound (e.g., horoballs, fractal sets (Farre et al., 2023, Farre et al., 16 Sep 2024)), but none are minimal for the horocycle flow: their closure is always strictly larger, reflecting chain-proximality or recurrence semigroups of finite/combinatorial depth. By contrast, the loom surface construction yields compact (in the appropriate topology), proper minimal sets supporting new invariant measures and flexible fractal dimension.

5. Geometric and Fractal Dimension Control

The new loom surface examples (Dal'Bo et al., 28 Oct 2025) demonstrate that the Hausdorff dimension of the minimal orbit closure can be prescribed arbitrarily in $(1,2)$. Specifically, for a tailored sequence controlling the slack sizes and weaving pattern, the accumulation set $E$ of slacks yields orbit closures whose dimension is computed via sumset techniques from fractal geometry (see formula $\Delta_0 = \bigcup_{k=1}^\infty 2k E$), in contrast to the rigidity of $\mathbb{Z}$-cover closures, which only allow dimension $1$ or $2$ (see (Farre et al., 16 Sep 2024)).

6. Impact and Open Problems

The explicit existence and structure theory for non-trivial minimal horocyclic orbit closures break longstanding paradigms in infinite-volume homogeneous dynamics. The new invariant measures—conservative, infinite, ergodic, and singular—introduce unanticipated flexibility into ergodic theory for horocycle flows. These constructions provoke revisiting classical rigidity conjectures (Ratner-type theorems) in infinite volume, inform understanding of orbit closure fractal geometry, and suggest new connections to recurrence semigroups and proximality. A plausible implication is that similar phenomena may appear in even broader settings beyond hyperbolic surfaces, such as higher-rank laminations or regular covers with tailored metric and topological features.

7. Summary Table: Features Across Constructions

Feature / Citation	Classical Finite Volume	$\mathbb{Z}$-Cover (Farre et al., 16 Sep 2024)	Loom Surface (Dal'Bo et al., 28 Oct 2025)
Non-trivial minimal horocyclic closure	Never	Never	Yes, explicit construction
Hausdorff dimension of minimal closure	1 or 2	1 or 2 (integer only)	Any value in $(1,2)$
Invariant ergodic measures (infinite vol.)	Supported on orbits / Lebesgue	Non-minimal, always non-ergodic	Infinite, conservative, singular
Orbit closure structure	Homogeneous (manifold/periodic)	Fractal, non-minimal, chain-proximal	Highly irregular, engineered fractal
Recurrence semigroup structure	Trivial / full group	Countable or contains ray	Can be prescribed / designed

8. Future Directions

Further investigation into the flexible control of orbit closure topology, measure-theoretic properties, and higher-rank analogues may yield additional classes of non-trivial minimal sets. Understanding the full spectrum of horocycle invariant measures—especially those singular to geodesic flow—in settings with variable negative curvature, higher genus, or more intricate limit sets (e.g., Sierpiński carpets (Kim et al., 23 Jan 2025)) remains a key area for ongoing research.