Bellis strong stable sets on infinite hyperbolic surfaces

Fran\c{c}oise Dal'Bo; Sergi Burniol Clotet; Sergio Herrero Vila

arxiv: 2604.02649 · v1 · submitted 2026-04-03 · 🧮 math.DS

Bellis strong stable sets on infinite hyperbolic surfaces

Sergi Burniol Clotet , Fran\c{c}oise Dal'Bo , Sergio Herrero Vila This is my paper

Pith reviewed 2026-05-13 19:18 UTC · model grok-4.3

classification 🧮 math.DS

keywords hyperbolic surfacesstrong stable setshorocyclic orbitsgeodesic flowclosed geodesicsinfinite surfacesdynamical systemsstable manifolds

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The pith

On infinite hyperbolic surfaces, strong stable sets do not coincide with horocyclic orbits for vectors whose geodesic rays encounter arbitrarily short closed geodesics.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies a corrected proof for a theorem originally due to A. Bellis. The theorem asserts that in the unit tangent bundle of certain infinite hyperbolic surfaces, vectors whose geodesic rays hit arbitrarily short closed geodesics have strong stable sets that are larger than their associated horocyclic orbits. This is shown by constructing geodesic rays that wind around infinitely many closed geodesics, which forces additional points into the strong stable set. A reader would care because it clarifies when the usual identification of stable manifolds with horocycles breaks down in non-compact settings. The result has implications for the dynamics of the geodesic flow on these surfaces.

Core claim

The theorem states that, for vectors whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set in the dynamical sense does not coincide with the associated horocyclic orbit. The proof is based on constructing geodesic rays that wind around infinitely many closed geodesics.

What carries the argument

Construction of geodesic rays that wind around infinitely many closed geodesics to show the strong stable set exceeds the horocyclic orbit.

Load-bearing premise

Geodesic rays that wind around infinitely many closed geodesics can be constructed on the infinite hyperbolic surfaces under consideration.

What would settle it

Observing a vector on an infinite hyperbolic surface whose geodesic ray encounters arbitrarily short closed geodesics but for which the strong stable set exactly equals the horocyclic orbit would falsify the claim.

Figures

Figures reproduced from arXiv: 2604.02649 by Fran\c{c}oise Dal'Bo, Sergi Burniol Clotet, Sergio Herrero Vila.

**Figure 1.** Figure 1: Winding around a closed geodesic in the universal covering, where t0 the time such that ˜u(t0) ∈ (γ−, γ+). Proof. Let p be the intersection point between (γ −, γ+) and u˜(R +). As p belongs to ˜u(R +), we have Bu˜(+∞)(p, u˜(0)) = −d(p, u˜(0)). In addition, as p ∈ (γ −, γ+) we know that d(γ −1p, p) = ℓ(γ). Thus Bu˜(+∞)(γ −1u˜(0), u˜(0)) = Bu˜(+∞)(γ −1u˜(0), γ−1 p)+ + Bu˜(+∞)(γ −1 p, p) + Bu˜(+∞)(p, u˜(0)) ≤… view at source ↗

**Figure 2.** Figure 2: grn β −1 n v˜n ∈ hRu˜. The next step of the construction is to ensure that the time spent to wind around all the geodesics (α − n , α+ n ) is finite. Proposition 4 (Convergence of the time to wind around closed geodesics). Let rn = B∞(β −1 n i, i)). If for every n ≥ 0, ℓ(αn) < 1 2n , then the sequence (rn)n∈N converges towards a real number rα. Proof. We have |rn+1 − rn| = [PITH_FULL_IMAGE:figures/full_fi… view at source ↗

read the original abstract

We provide a corrected proof of a theorem of A. Bellis on strong stable sets in the unit tangent bundle of certain hyperbolic surfaces. The theorem states that, for vectors whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set in the dynamical sense does not coincide with the associated horocyclic orbit. The proof is based on Bellis' idea of constructing geodesic rays that wind around infinitely many closed geodesics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This corrects the proof of Bellis' theorem on non-coincidence of strong stable sets and horocycles, but the winding ray construction needs checking to confirm it hits arbitrarily short geodesics.

read the letter

This paper supplies a corrected proof for a theorem originally due to A. Bellis. The theorem says that on certain infinite hyperbolic surfaces, if a vector's geodesic ray hits arbitrarily short closed geodesics, then its strong stable set in the flow does not match the horocyclic orbit. The new part is the detailed proof. It uses Bellis' idea to construct geodesic rays that wind around infinitely many closed geodesics. The construction is meant to ensure the lengths of those geodesics go to zero. What the paper does well is lay out the steps without circular reasoning. It relies on the geometry of the surface and the properties of hyperbolic metrics to build the rays explicitly. This seems like a straightforward fix to whatever gap was in the original argument. The soft spot is in the construction itself. The stress-test note points out that the rays need to wind infinitely often while always finding shorter geodesics. On surfaces with funnels or ends, it is possible that after some windings the ray escapes to a region where all closed geodesics have length bounded below. The paper claims to handle this by the choice of sequence, but without the full details it is hard to see if the trapping works for all cases. If it does, the result holds; if not, the statement might need restriction to surfaces where such rays exist. The math looks standard for this area, with no free parameters or invented objects. Citations are to the original work and basic references in hyperbolic dynamics. This is for people who care about the precise behavior of stable manifolds in non-compact settings. It is not a major new theorem, but the correction is useful. I would send it for peer review so that experts can check the construction details.

Referee Report

2 major / 2 minor

Summary. The manuscript provides a corrected proof of a theorem of A. Bellis asserting that, on certain infinite hyperbolic surfaces, for vectors in the unit tangent bundle whose geodesic rays encounter arbitrarily short closed geodesics, the strong stable set (in the dynamical sense) does not coincide with the associated horocyclic orbit. The argument rests on an explicit construction, following Bellis' idea, of geodesic rays that wind around infinitely many closed geodesics.

Significance. If the construction is made fully rigorous, the result would clarify the relationship between dynamical stable manifolds and geometric horocycles for the geodesic flow on infinite-volume hyperbolic surfaces, with potential implications for orbit closures and ergodicity questions in this setting.

major comments (2)

[Construction of geodesic rays] Construction section: the argument that geodesic rays can be chosen to wind around infinitely many closed geodesics while encountering lengths tending to zero is stated at a high level but lacks an explicit verification that the winding sequence keeps the ray trapped near successively shorter geodesics rather than escaping into funnels or ends where no short geodesics exist. This step is load-bearing for the non-coincidence claim.
[Proof of non-coincidence] Proof of non-coincidence: the demonstration that the strong stable set properly contains (or differs from) the horocyclic orbit for the constructed vectors relies on the rays encountering arbitrarily short geodesics; a concrete estimate or lemma showing the resulting divergence of orbits under the flow would be needed to make the distinction rigorous.

minor comments (2)

[Introduction] The class of 'certain hyperbolic surfaces' is not defined until late; an early precise statement of the geometric assumptions (e.g., presence of funnels, cusp structure, or curvature bounds) would improve readability.
[Notation and preliminaries] Notation for the unit tangent bundle, geodesic flow, and strong stable sets should be introduced once and used consistently; occasional shifts between dynamical and geometric language obscure the argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough reading of the manuscript and for providing constructive feedback that will help improve the clarity of our arguments. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: Construction section: the argument that geodesic rays can be chosen to wind around infinitely many closed geodesics while encountering lengths tending to zero is stated at a high level but lacks an explicit verification that the winding sequence keeps the ray trapped near successively shorter geodesics rather than escaping into funnels or ends where no short geodesics exist. This step is load-bearing for the non-coincidence claim.

Authors: We agree that additional details are needed to make this verification explicit. In the revised manuscript, we will expand the construction section by adding a detailed argument, including estimates on the hyperbolic distances and the choice of winding numbers, to show that the geodesic rays remain in neighborhoods of the short closed geodesics and do not escape to the funnels. This will involve specifying the sequence of geodesics more carefully and proving that the total length of the ray segments ensures trapping. revision: yes
Referee: Proof of non-coincidence: the demonstration that the strong stable set properly contains (or differs from) the horocyclic orbit for the constructed vectors relies on the rays encountering arbitrarily short geodesics; a concrete estimate or lemma showing the resulting divergence of orbits under the flow would be needed to make the distinction rigorous.

Authors: We acknowledge that a more concrete estimate would strengthen the proof. We will add a new lemma in the revised version that provides an explicit lower bound on the distance between points in the strong stable set and the horocyclic orbit, based on the shortness of the encountered geodesics. This lemma will quantify how the geodesic flow causes divergence when passing near short geodesics, thereby rigorously establishing the non-coincidence. revision: yes

Circularity Check

0 steps flagged

No circularity in corrected proof of Bellis theorem

full rationale

The paper supplies a corrected proof of an external theorem by A. Bellis, relying on an explicit construction of geodesic rays that wind around infinitely many closed geodesics to show non-coincidence of strong stable sets and horocyclic orbits. No step reduces by definition or by construction to its own inputs; the construction is the independent method used to establish the claim for the stated vectors. The reference is to prior work by a different author, not a self-citation load-bearing chain, and no fitted parameters, ansatzes smuggled via citation, or renamings of known results appear. The derivation remains self-contained against the geometric assumptions of infinite hyperbolic surfaces.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard domain assumptions from hyperbolic geometry and dynamical systems; no free parameters, new axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption Hyperbolic surfaces admit a geodesic flow with well-defined stable sets and horocycles
Standard background from hyperbolic geometry and the theory of geodesic flows on manifolds.

pith-pipeline@v0.9.0 · 5363 in / 1162 out tokens · 61141 ms · 2026-05-13T19:18:04.619840+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Ballmann, M

W. Ballmann, M. Gromov and V. Schroeder,Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkh¨ auser Boston, Boston, MA, 1985

work page 1985
[2]

A. F. Beardon,The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer, New York, 1983

work page 1983
[3]

Bellis, ´Etude topologique du flot horocyclique: le cas des surfaces g´ eom´ etriquement infinies, PhD thesis, Institut de Recherche Math´ ematique de Rennes, 2018

A. Bellis, ´Etude topologique du flot horocyclique: le cas des surfaces g´ eom´ etriquement infinies, PhD thesis, Institut de Recherche Math´ ematique de Rennes, 2018

work page 2018
[4]

Dal’Bo,Trajectoires g´ eod´ esiques et horocycliques, Savoirs Actuels, EDP Sciences, Les Ulis, 2007

F. Dal’Bo,Trajectoires g´ eod´ esiques et horocycliques, Savoirs Actuels, EDP Sciences, Les Ulis, 2007

work page 2007
[5]

Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2)10(1958), 338–354. IMERL, CMAT, Universidad de la Rep ´ublica, IRL-CNRS-IFUMI 2030, Uruguay Email address:sergi.burniol@gmail.com IRMAR, Universit´e de Rennes, France, IRL-CNRS-IFUMI 2030, Uruguay Email address:francoise.dalbo@univ-rennes.fr IRMAR, Unive...

work page 1958

[1] [1]

Ballmann, M

W. Ballmann, M. Gromov and V. Schroeder,Manifolds of nonpositive curvature, Progress in Mathematics, 61, Birkh¨ auser Boston, Boston, MA, 1985

work page 1985

[2] [2]

A. F. Beardon,The geometry of discrete groups, Graduate Texts in Mathematics, 91, Springer, New York, 1983

work page 1983

[3] [3]

Bellis, ´Etude topologique du flot horocyclique: le cas des surfaces g´ eom´ etriquement infinies, PhD thesis, Institut de Recherche Math´ ematique de Rennes, 2018

A. Bellis, ´Etude topologique du flot horocyclique: le cas des surfaces g´ eom´ etriquement infinies, PhD thesis, Institut de Recherche Math´ ematique de Rennes, 2018

work page 2018

[4] [4]

Dal’Bo,Trajectoires g´ eod´ esiques et horocycliques, Savoirs Actuels, EDP Sciences, Les Ulis, 2007

F. Dal’Bo,Trajectoires g´ eod´ esiques et horocycliques, Savoirs Actuels, EDP Sciences, Les Ulis, 2007

work page 2007

[5] [5]

Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math

S. Sasaki, On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. (2)10(1958), 338–354. IMERL, CMAT, Universidad de la Rep ´ublica, IRL-CNRS-IFUMI 2030, Uruguay Email address:sergi.burniol@gmail.com IRMAR, Universit´e de Rennes, France, IRL-CNRS-IFUMI 2030, Uruguay Email address:francoise.dalbo@univ-rennes.fr IRMAR, Unive...

work page 1958