Infinite Locally-Finite Horocyclic Invariant Measures

• Infinite locally-finite conservative horocyclic invariant measures are nontrivial, ergodic, and infinite Radon measures on the unit tangent bundle of hyperbolic surfaces, invariant under the horocycle flow.
• They are constructed via loom surfaces which break classical rigidity, producing exotic minimal sets with tailored recurrence and fractal supports of arbitrary Hausdorff dimension.
• Their singular behavior under the geodesic flow, achieved through engineered combinatorial and geometric data, opens a new paradigm in the study of horocyclic dynamics and ergodic theory.

An infinite locally-finite conservative horocyclic invariant measure is a nontrivial, locally finite (Radon), infinite, ergodic, and conservative measure on the unit tangent bundle of a hyperbolic surface that is invariant under the action of the horocycle flow. Such measures have traditionally been subject to strong rigidity: on finite and most infinite volume geometrically finite hyperbolic surfaces, all ergodic, locally-finite horocycle invariant measures are either supported on single orbits or are quasi-invariant under the geodesic flow. The new constructions in geometrically infinite settings depart dramatically from this paradigm, revealing a previously unknown diversity of horocyclic invariant measures and minimal sets.

1. Mathematical Context and Definitions

Let $\Sigma$ be an orientable hyperbolic surface (possibly of infinite type), $T^1\Sigma$ its unit tangent bundle, and $G = PSL_2(\mathbb{R})$. The horocycle flow is generated by the subgroup $$N = \left\{\begin{pmatrix} 1 & s \ 0 & 1 \end{pmatrix} : s \in \mathbb{R} \right\} \leq G$$, while the geodesic flow is generated by the diagonal subgroup $$A = \left\{ a_t = \mathrm{diag}(e^{t/2}, e^{-t/2}) : t \in \mathbb{R} \right\}$$.

A Borel measure $\mu$ on $T^1 \Sigma$ is:

• $N$-invariant if it is invariant under the horocycle flow,
• locally finite if it gives finite mass to compact subsets,
• conservative if for every measurable set $E$ of positive measure, the return set $\left\{ n \in N : n.x \in E \right\}$ is unbounded for $\mu$-almost every $x \in E$.

The non-wandering set for $N$ is $$\mathcal{E} = \{ g\Gamma \in G/\Gamma : g^+ \in \Lambda \}$$, where $g^+$ is the forward endpoint in $\partial \mathbb{H}^2$ and $\Lambda$ the limit set of $\Gamma$.

An $N$-minimal set is a closed, nonempty, $N$-invariant subset $Y \subset T^1\Sigma$ on which every $N$-orbit is dense in $Y$. For geometrically finite surfaces, $N$-minimal sets are either single horocycles or the full nonwandering set $\mathcal{E}$.

2. Construction of Loom Surfaces and Measures

The recent construction ("Weaving Geodesics and New Phenomena in Horocyclic Dynamics" (Dal'Bo et al., 28 Oct 2025)) produces explicit geometrically infinite hyperbolic surfaces, called loom surfaces $\Sigma_s$, exhibiting new dynamical phenomena:

• Begin with the band model $H$ of $\mathbb{H}^2$.
• Remove an infinite sequence of "half-planes" $D_{h_k}(s_k)$ at locations $s_k$ (described by explicit geometric data) and double the resulting surface along the boundary.
• The parameters $h_k$ and $s_k$ are chosen such that the injectivity radius diverges along all diverging geodesic rays; the sequence of "crossings" encodes a combinatorial "weaving pattern" with an associated sequence of slack parameters.

The arrangement ensures that certain horocyclic orbit closures can be prescribed and engineered to have tailored recurrence properties, controlled via combinatorial and geometric data.

3. Nontrivial Minimal Sets and Invariant Measures

Main Theorem (Dal'Bo et al., 28 Oct 2025):

There exists a hyperbolic surface $\Sigma$ such that $T^1\Sigma$ contains an $N$-minimal closed subset $Y = \overline{N x_0}$ that is neither a single orbit nor the full nonwandering set. This set supports an $N$-invariant, ergodic, infinite, locally-finite, conservative measure $\mu$ that is singular with respect to the geodesic flow, i.e., $a_t^* \mu \perp \mu$ for $t \neq 0$.

The measure construction relies on:

• An open section $\Psi$ for the $N$-action (Appendix, Prop A.1).
• Empirical averages along an $N$-orbit, leveraging tightness to extract a locally finite, invariant, conservative measure supported on $Y$.
• Disjointness property: for all $t \neq 0$, $$a_t \overline{N x_0} \cap \overline{N x_0} = \emptyset$$; thus, the measure $\mu$ is mutually singular under geodesic flow.

Key properties summarized:

Feature	Result
Support	Non-homogeneous, nontrivial $N$-minimal set, not a single orbit nor all of $\mathcal{E}$
Invariance	$N$-invariant, locally finite, conservative, ergodic
Singularity	$a_t^* \mu \perp \mu$ for $t \neq 0$
Hausdorff dim.	Arbitrary $\alpha\in(1,2)$ possible

This dynamical behavior sharply contrasts all previously known cases, where rigidity theorems (e.g., Ratner's classification, Babillot–Ledrappier, Burger–Roblin (Landesberg et al., 2019, Landesberg, 2020)) imply $N$-invariant, ergodic, locally finite measures are supported only on single orbits or are quasi-invariant under $A$.

4. Slack Function, Busemann-Type Function, and Dimension Control

The slack function $S(\alpha)$ measures the inefficiency of a curve $\alpha$ compared to the progress along a reference geodesic: $S(\alpha) = (t-s) - (\tau(a_t z) - \tau(a_s z)),$ with $S=0$ iff $\alpha$ is a true geodesic segment.

A Busemann-type function $\beta(y)$ is defined as: $$\beta(y) = \tau(y) - S(A_+y) = \lim_{t \to \infty} \tau(a_t y) - t.$$ The set $\{y : \beta(y) = 0\}$ is precisely the minimal $N$-invariant closure supporting $\mu$.

By controlling the "slack" and the sequence of crossings in the loom surface, the Hausdorff dimension of the support can be engineered to be any $\alpha \in (1,2)$, as in Theorem~\ref{thm:main distal surface} of (Dal'Bo et al., 28 Oct 2025). The dimension may even vary locally within the support.

5. Singularity with Respect to Geodesic Flow

A crucial property is that, for the constructed measure $\mu$,

$a_t^* \mu \perp \mu \quad \forall t \neq 0,$

which is a consequence of the Busemann function's equivariance ($\beta(a_t y) = \beta(y) + t$) and the fact that the translations $a_t$ move the support off itself completely. This singularity property violates the measure rigidity principles established for Radon measures invariant under unipotent flows in homogeneous spaces and geometrically finite hyperbolic surfaces (Landesberg, 2020, Landesberg et al., 2019).

6. Implications for Horocyclic Dynamics and Rigidity

This construction provides the first explicit demonstration of the breakdown of infinite measure rigidity for horocycle flows in the geometrically infinite setting. Notable implications:

• Minimal non-homogeneous sets: Nontrivial $N$-minimal sets exist that are not group-translates or homogeneous.
• Exotic invariant measures: There are measures that are infinite, locally finite, conservative, and ergodic, yet whose dynamics under the geodesic flow are maximally non-rigid, failing quasi-invariance entirely.
• Support on fractals: The support can be a fractal set of arbitrary dimension in $(1,2)$, unattainable by classical constructions.
• Tailored recurrence: The recurrence properties of horocycle orbits can be prescribed combinatorially through the weaving pattern of the underlying surface.

A summary of the spectrum of behaviors known in the literature:

Setting	Measure support	$A$-quasi-invariant?	Conservative?	Structure
Geometrically finite,	single orbit or Burger–Roblin	Yes	Yes	Homogeneous
covers, finite volume
Geometrically infinite	usually only standard	Yes	Yes/No	Homogeneous
Loom surface (Dal'Bo et al., 28 Oct 2025)	intermediate minimal set	No	Yes	Non-homogeneous, fractal

7. Summary and Significance

The existence of infinite, locally-finite, conservative horocyclic invariant measures that are singular with respect to the geodesic flow demonstrates an essential flexibility in the topological and measure-theoretic dynamics of horocycle flows on geometrically infinite hyperbolic surfaces. These measures are supported on minimal, non-homogeneous sets of arbitrary Hausdorff dimension and are not obtained via traditional boundary or conformal data.

This breaks a central dichotomy—previously, all nontrivial horocyclic invariant measures were either supported on closed orbits or were intertwined (via quasi-invariance) with the geodesic flow, and could be classified through boundary conformal measures (Landesberg, 2020, Landesberg et al., 2019). The new constructions reveal that, in the infinite-type, infinite-volume context, invariant sets and measures can be highly exotic, including minimal sets of fractional dimension and measures that are not only infinite and locally finite but dramatically non-rigid with respect to other flows. This opens a new paradigm for the paper of horocycle dynamics and measure ergodic theory on infinite-type hyperbolic manifolds (Dal'Bo et al., 28 Oct 2025).