Strict concavity of the growth indicator function for relatively Anosov groups

Andrew Zimmer; Dongryul M. Kim; Hee Oh

Strict concavity of the growth indicator function for relatively Anosov groups

T0 review · 0 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves strict concavity of the growth indicator function for every Zariski dense relatively Anosov group, completing the regularity picture of these directional growth functions.

desk verdict Kim–Oh–Zimmer proves the missing strict concavity piece for relatively Anosov groups via a genuinely new Cartan displacement observable, and the argument holds up under scrutiny; the main risk is the imported projectively visible model, but I found no internal gap. read the letter →

arxiv 2607.13760 v1 pith:EXVDAUIP submitted 2026-07-15 math.DG math.DSmath.GRmath.GT

Dongryul M. Kim , Hee Oh , Andrew Zimmer This is my paper · ORCID

classification math.DGmath.DSmath.GRmath.GT MSC 22E4037D4053C35

keywords growthindicatorfunctionstrictconcavityManhattanhypersurfacerelativelyAnosovgroupsCartanprojectionBowen–Margulis–SullivanmeasurescriticalgapatinfinitysemisimpleLie

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that for a Zariski dense relatively Anosov subgroup of a semisimple real algebraic group, the growth indicator function is strictly concave between non-collinear directions. This was the last missing piece after differentiability and infinite-slope properties had been established in prior work. The proof works by showing that the dual object, the Manhattan hypersurface, is a C^1 hypersurface. The key new tool is a bounded continuous observable on the associated flow space whose orbit integrals coarsely recover the partial Cartan displacement of recurrent orbit segments. A careful reader should care because strict concavity yields uniqueness of dominant asymptotic directions in weighted counting problems and a clean duality between tangent linear forms and interior rays of the limit cone.

What carries the argument

The central new object is the Cartan displacement observable: a Γ-invariant bounded continuous map f from the flow space into the partial Cartan subspace a_θ, constructed from a projectively visible model of the transverse group. Its orbit integrals over recurrent segments differ from the corresponding partial Cartan projection by a uniformly bounded error (Proposition 5.1). This observable turns Cartan displacement into a continuous cocycle, allowing the derivative of the local convex graph of the Manhattan hypersurface to be expressed as a ratio of integrals against Bowen–Margulis–Sullivan measures. Continuity of those averages, guaranteed by the critical-gap-at-infinity condition, then up

What would settle it

Take any relatively θ-Anosov group covered by the theorems and compute ψ_Γ^θ at two non-collinear vectors with finite values; if equality ψ(v+w) = ψ(v)+ψ(w) holds for any such pair, strict concavity is false and the main theorem collapses. More directly, one could look for a θ-transverse group with a boundary point that is positive on the limit cone and has a critical gap at infinity but where the Manhattan hypersurface is not C^1.

Watch

Extended reading notes

Core claim

The main results are Theorem 1.1 and Theorem 1.4. For a non-elementary relatively θ-Anosov group Γ in a connected semisimple real algebraic group, the θ-growth indicator function ψ_Γ^θ is strictly concave: for non-collinear v and w with finite values, ψ(v+w) > ψ(v)+ψ(w). This is proved by establishing the equivalent dual fact that the θ-Manhattan hypersurface ∂Q^θ(Γ) is a C^1 hypersurface. The argument is local and more general: for any non-elementary θ-transverse group, the Manhattan hypersurface is C^1 near every point that is positive on the limit cone and has a critical gap at infinity; in the Zariski dense case, it is locally strictly convex there.

Load-bearing premise

The whole argument depends on the existence of a projectively visible model for the transverse group, with a coarse equivalence between Hilbert distance and the norm of the partial Cartan projection; if that imported construction failed, the Cartan displacement observable and the subsequent derivative formulas would lose their metric control.

Editorial extensions

If this is right

For every relatively θ-Anosov group, the growth indicator function is strictly concave on non-collinear directions, completing the desired regularity picture: differentiability, infinite slope at the boundary, and now strict concavity.
The θ-Manhattan hypersurface is globally C^1; in the Zariski dense case it is locally strictly convex.
Each linear form has at most one ray in the limit cone along which it is tangent to the growth indicator, so weighted counting problems have a unique dominant asymptotic direction.
The tangent linear forms are in one-to-one correspondence with rays in the interior of the limit cone, with the correspondence given by the gradient of the growth indicator.
For general transverse groups, local C^1 regularity of the Manhattan hypersurface holds at every point with positive-on-limit-cone and a critical gap at infinity, extending the Anosov result to a broader class.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The Cartan displacement observable is likely reusable beyond this paper: it converts partial Cartan displacement into a bounded continuous cocycle whenever a projectively visible model exists, so similar C^1 regularity and strict-concavity arguments may apply to other geometrically finite settings, such as cusped Hitchin representations.
The uniqueness of tangent rays suggests a stronger form of uniqueness for equilibrium measures in the associated flow space: each linear form should give a unique measure of maximal weighted entropy, which may simplify local mixing results.
Theorem 1.6 implies that for a general transverse group, failure of global C^1 regularity can only occur at boundary points that are not positive on the limit cone or lack a critical gap at infinity; one could test whether such points actually produce corner-like singularities.
Because the proof links strict convexity of the Manhattan hypersurface to Zariski density, a natural test is whether any non-Zariski-dense relatively Anosov group has a flat segment in its Manhattan hypersurface; the paper's method does not settle this directly.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know two things about this paper. First, it settles the last open piece of the regularity picture for growth indicators of relatively Anosov groups: strict concavity, obtained via C^1 regularity of the Manhattan hypersurface. Second, the proof is not a routine transplant of the Anosov case. The new Cartan displacement observable (Prop 5.1) is a real idea: it bypasses the missing flow reparametrization, and the estimate (5.7)–(5.9) is the load-bearing step. I checked the slice-summing in Prop 6.1 and the continuity of BMS averages in Thm 7.1; the Borel–Cantelli and uniform multiplicity arguments are coherent, and the critical-gap stability argument in Lemma 7.2 is sound. The paper is honest in its imports: Theorem 3.4 (projectively visible model) is explicitly flagged as coming from CZZ24, and the critical-gap finiteness result is outsourced to Wen. Those are external, but they are cited precisely and the reduction is sketched, not hand-waved. The strict-convexity assertion for Zariski dense groups is a short, correct use of [CZZ24, Cor 13.2]. The weakest point remains what the reader flagged: if the coarse equivalence in Thm 3.4 failed, the observable bounds collapse. But that is an imported theorem, not a gap in this paper, and the comparison is plausible. I also note the paper's own Remark 1.2 and 1.5 acknowledging concurrent work by Reyes–Wang; that is the right scholarly posture. Self-citations here are to prior work the authors actually rely on, not padding. The central derivation is not circular: C^1 regularity is proved first, and strict concavity is deduced by Quint's duality. I disagree with any suspicion that this is fitting: there are no free parameters, and the theorem applies to a natural class, not to a constructed example. Who is this for: anyone working on higher-rank discrete subgroups, Patterson–Sullivan theory, or regularity of growth indicators. A serious referee should engage with the proof of Prop 5.1 and the passage from Theorem 7.1 to the derivative formula in Section 8; those are the sections worth the most referee time. I would send it to peer review without hesitation. Colloquium-worthy, not just citable; I expect it to be cited within a year.

Referee Report

0 major / 5 minor

Summary. The paper proves that for a Zariski dense relatively θ-Anosov subgroup Γ of a connected semisimple real algebraic group, the θ-growth indicator function ψ_Γ^θ is strictly concave on non-collinear directions (Theorem 1.1), equivalently that the θ-Manhattan hypersurface ∂Q^θ(Γ) is C^1 (Theorem 1.4). The engine is a local regularity theorem (Theorem 1.6): for a non-elementary θ-transverse group, ∂Q^θ(Γ) is C^1 near every boundary point that is positive on the θ-limit cone and has a critical gap at infinity; in the Zariski dense case it is locally strictly convex. The proof introduces a bounded continuous Cartan displacement observable f: G → a_θ whose orbit integrals coarsely recover the partial Cartan projection of recurrent orbit segments (Proposition 5.1). Using this observable, the derivative of the convex graphing function of the Manhattan hypersurface is identified with ratios of BMS averages (Corollary 5.2, Proposition 6.1), and the critical-gap condition is used to obtain uniform control and continuity of these averages (Theorem 7.1). A standard convexity argument then upgrades almost-everywhere differentiability to C^1 regularity.

Significance. If correct, the paper fills the last missing piece in the regularity picture for growth indicator functions of relatively Anosov groups: strict concavity. The authors correctly note that differentiability and infinite-slope properties were already known from [CZZ25], while strict concavity was open; the paper also records an application to the local mixing result [KOP26]. The proof is organized as a sequence of explicit, checkable lemmas, and the main novelty — the Cartan displacement observable — is a genuine technical contribution. A notable strength is that no free parameters or fitted constants enter; the argument reduces to established external results (Patterson–Sullivan theory, projectively visible models, Wen's finiteness theorem), and the dependency on these inputs is clearly stated. The authors also acknowledge the independent work of Reyes–Wang. If the result stands, it settles the concavity question and provides a useful local regularity theorem for general transverse groups.

minor comments (5)

[Section 3.2, proof of Theorem 3.4] The notation 'Φ∘ρ = id_{Γ0}' is type-incorrect: ρ maps Γ0 to Γ and Φ maps G to PSL(d,R), so the composition is not the identity on Γ0. The intended meaning is presumably that Φ identifies Γ with Γ0 under ρ, i.e. Φ(ρ(γ)) = γ after identifying Γ0 with its image. Please rephrase.
[Section 5, equation (5.6)] The line 'by (5.3)' is slightly terse: (5.3) gives an inequality for P_α(g^t v), and one must take logarithms and use the equivalence of norms to obtain the displayed Lipschitz bound for Q. This is clear but could be spelled out in one sentence.
[Section 6, proof of Proposition 6.1] The auxiliary function f(s) is defined for s ≠ 0 but the proof only uses s > 0 and s < 0 with the conditions ε − f(s) > 0 and ε + f(−s) > 0. This is harmless, but the notation would be cleaner if the two one-sided cases were separated.
[Section 7, Lemma 7.2] In the proof of Lemma 7.2, the assertion that the set {φ(κ(ρ(γ))) : γ ∈ Γ0} is bounded below follows from positivity of φ on the limit cone together with θ-discreteness; the current phrasing may make it look like an immediate consequence of positivity alone. A short justification would improve readability.
[References] There is a typographical issue in the reference '[R W26]', where an unintended space appears in the author name. Please correct to 'Reyes–Wang'.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular step found: the C^1 regularity proof does not assume strict concavity, and no fitted input is relabeled as a prediction.

full rationale

The central derivation is not circular. The proof of Theorem 1.6 constructs a Cartan displacement observable (Proposition 5.1), proves a Borel–Cantelli-based derivative formula for the convex graphing function Φ (Proposition 6.1), establishes continuity of BMS averages under the critical-gap hypothesis (Theorem 7.1), and then applies a standard convexity argument in Section 8. None of these steps assumes the target strict-concavity or C^1 conclusion, and no fitted parameter is renamed as a prediction. The imported inputs from [CZZ24], [CZZ25], and [Wen26] are prior theorems on projectively visible models, Patterson–Sullivan measures, and finiteness of BMS measures; although a number of these are by the same authors, they are used as tools rather than as a restatement of the theorem being proved. The step 'Theorem 1.4 implies Theorem 1.1 by Quint’s duality' invokes an external duality theorem, while the local strict-convexity addendum cites [CZZ24, Corollary 13.2] rather than deriving strict convexity from the new C^1 result. The weakest non-reproved input is the projectively visible model of Theorem 3.4, but the paper supplies the additional comparison argument and this is a correctness risk, not circularity. Overall, the derivation chain is self-contained in the sense required here: no equation is used as both hypothesis and conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No fitted numerical constants appear in the paper. The central claim depends on a network of established results, several co-authored by the present authors: projectively visible models (CZZ24), PS measure theory for transverse groups (KOW25), differentiability and infinite slopes for relatively Anosov groups (CZZ25), and Wen's critical-gap finiteness (Wen26). The genuinely new object, the Cartan displacement observable, is proven to exist within the paper.

assumptions (4)

domain assumption Gamma is a non-elementary theta-transverse (or relatively theta-Anosov where needed) discrete subgroup of a connected semisimple real algebraic group; Zariski density is assumed for the strict-concavity theorem.
Standing hypothesis introduced in Section 1; needed for limit sets, Patterson-Sullivan measures, convergence action, and nonempty interior of the limit cone.
domain assumption Projectively visible model theorem (CZZ24, Theorem 6.2) gives an isomorphic projectively visible subgroup Gamma0 of Aut(Omega) and a coarse equivalence between Hilbert distance and Cartan projection.
Used throughout Sections 3-8; not reproved in this paper.
domain assumption Wen's critical-gap-at-infinity finiteness result (Theorem 4.3 / A.2) applies to the GPS system and yields finite BMS measures.
Required for Theorem 7.1; the reduction is outlined in Appendix A but the general GPS theorem is imported from Wen26.
standard math Quint's duality identifies Q^theta with the set of linear forms dominating the growth indicator, and identifies its boundary with tangent forms to psi.
Deductive bridge from C^1 regularity of the Manhattan hypersurface to strict concavity of psi (Section 1.2).

invented entities (1)

Cartan displacement observable f: G -> a_theta
purpose: Bounded Gamma0-invariant continuous function whose orbit integrals recover partial Cartan displacement of recurrent orbit segments up to uniform error; turns Cartan geometry into a flow observable.
Constructed and proved in Proposition 5.1; it is an internal mathematical tool, not an empirical entity, and carries no outside falsifiable prediction.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Strict concavity of the growth indicator function for relatively Anosov groups." pith.science (2026). https://pith.science/paper/EXVDAUIP

@misc{pith2026260713760,
  author       = {Pith},
  title        = {Pith review of: Strict concavity of the growth indicator function for relatively Anosov groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXVDAUIP}},
  note         = {Machine review of arXiv:2607.13760}
}

abstract

Let $\Gamma$ be a discrete subgroup of a connected semisimple real algebraic group of higher rank. The growth indicator function $\psi_\Gamma$ records the directional exponential growth of the Cartan projections of elements of $\Gamma$ in the positive Weyl chamber $\mathfrak a^+$. We prove that if $\Gamma$ is a non-elementary relatively Borel Anosov group, then $\psi_\Gamma$ is strictly concave on non-collinear directions. We prove this by establishing the $\mathcal C^1$-smoothness of the Manhattan hypersurface, defined as the unit level set of the critical-exponent map $\phi\mapsto\delta^\phi(\Gamma)$. More generally, for a non-elementary $\theta$-transverse group, we prove local $\mathcal C^1$-regularity near every point of the $\theta$-Manhattan hypersurface that is positive on the $\theta$-limit cone and has a critical gap at infinity. In particular, the $\theta$-Manhattan hypersurface is globally $\mathcal C^1$ for relatively $\theta$-Anosov groups, and their $\theta$-growth indicator functions are strictly concave on non-collinear directions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 14 canonical work pages

[1]

Floyd maps for relatively hyperbolic groups , url =

Gerasimov, Victor , date-added =. Floyd maps for relatively hyperbolic groups , url =. Geom. Funct. Anal. , mrclass =. 2012 , bdsk-url-1 =. doi:10.1007/s00039-012-0175-6 , fjournal =

doi:10.1007/s00039-012-0175-6 2012
[2]

, TITLE =

Smilga, I. , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2018 , NUMBER =. doi:10.4171/GGD/447 , URL =

work page doi:10.4171/ggd/447 2018
[3]

Preprint , Year =

Reyes, Eduardo and Wang, Tianqi , Title =. Preprint , Year =
[4]

, TITLE =

Sambarino, A. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/etds.2012.170 , URL =

doi:10.1017/etds.2012.170 2014
[5]

and Sambarino, A

Potrie, R. and Sambarino, A. , TITLE =. Invent. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00222-017-0721-9 , URL =

doi:10.1007/s00222-017-0721-9 2017
[6]

and Canary, R

Bridgeman, M. and Canary, R. and Labourie, F. and Sambarino, A. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00039-015-0333-8 , URL =

doi:10.1007/s00039-015-0333-8 2015
[7]

Dongryul and Oh, Hee and Pan, Wenyu , Title =

Kim, M. Dongryul and Oh, Hee and Pan, Wenyu , Title =. Preprint , Year =
[8]

Blayac, Pierre-Louis and Zhu, Feng , TITLE =. J. Mod. Dyn. , FJOURNAL =. 2023 , PAGES =. doi:10.3934/jmd.2023026 , URL =

doi:10.3934/jmd.2023026 2023

Show all 32 references

[9]

arXiv , author =:2404.09713 , journal =

doi:10.48550/arXiv.2404.09713 , eid =. arXiv , author =:2404.09713 , journal =

doi arXiv
[10]

, TITLE =

Tits, J. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 1971 , PAGES =. doi:10.1515/crll.1971.247.196 , URL =

1971 doi
[11]

arXiv e-prints , keywords =

Vector-valued horofunction boundaries and Patterson--Sullivan measures. arXiv e-prints , keywords =. doi:10.48550/arXiv.2603.28693 , archivePrefix =. 2603.28693 , primaryClass =

doi
[12]

Cantrell, Stephen and Tanaka, Ryokichi , TITLE =. Geom. Topol. , FJOURNAL =. 2025 , NUMBER =. doi:10.2140/gt.2025.29.1851 , URL =

2025 doi
[13]

Pit, Vincent and Schapira, Barbara , TITLE =. Ann. Inst. Fourier (Grenoble) , FJOURNAL =. 2018 , NUMBER =. doi:10.5802/aif.3167 , URL =

2018 doi
[14]

and Oh, Hee and Wang, Yahui , TITLE =

Kim, Dongryul M. and Oh, Hee and Wang, Yahui , TITLE =. Math. Ann. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00208-025-03292-6 , URL =

2025 doi
[15]

Internat

Burger, Marc , TITLE =. Internat. Math. Res. Notices , FJOURNAL =. 1993 , NUMBER =. doi:10.1155/S1073792893000236 , URL =

1993 doi
[16]

L'indicateur de croissance des groupes de

Quint, Jean-Fran. L'indicateur de croissance des groupes de. Ergodic Theory Dynam. Systems , FJOURNAL =. 2003 , NUMBER =. doi:10.1017/S0143385702001268 , URL =

2003 doi
[17]

Divergence exponentielle des sous-groupes discrets en rang sup\'erieur , JOURNAL =

Quint, Jean-Fran. Divergence exponentielle des sous-groupes discrets en rang sup\'erieur , JOURNAL =. 2002 , NUMBER =. doi:10.1007/s00014-002-8352-0 , URL =

2002 doi
[18]

Canary, Richard and Zhang, Tengren and Zimmer, Andrew , TITLE =. Math. Ann. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00208-025-03137-2 , URL =

2025 doi
[19]

, TITLE =

Benoist, Y. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 1997 , NUMBER =. doi:10.1007/PL00001613 , URL =

1997 doi
[20]

Ergodic Theory Dynam

Sambarino, Andr\'es , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2024 , NUMBER =. doi:10.1017/etds.2023.13 , URL =

2024 doi
[21]

Relatively

Zhu, Feng and Zimmer, Andrew , journal=. Relatively. 2022 , note=

2022
[22]

and Oh, Hee , TITLE =

Kim, Dongryul M. and Oh, Hee , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2025 , PAGES =. doi:10.1515/crelle-2025-0040 , URL =

2025 doi
[23]

Convex cocompact actions in real projective geometry , JOURNAL =

Danciger, Jeffrey and Gu\'eritaud, Fran. Convex cocompact actions in real projective geometry , JOURNAL =. 2024 , NUMBER =

2024
[24]

arXiv e-prints , keywords =

Counting, mixing and equidistribution for GPS systems with applications to relatively Anosov groups. arXiv e-prints , keywords =. doi:10.48550/arXiv.2404.09718 , archivePrefix =. 2404.09718 , primaryClass =

doi arXiv
[25]

, TITLE =

Quint, J.-F. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2002 , NUMBER =. doi:10.1007/s00039-002-8266-4 , URL =

2002 doi
[26]

Canary, Richard and Zhang, Tengren and Zimmer, Andrew , TITLE =. J. Mod. Dyn. , FJOURNAL =. 2024 , PAGES =. doi:10.3934/jmd.2024009 , URL =

2024 doi
[27]

Bray, Harrison , TITLE =. J. Mod. Dyn. , FJOURNAL =. 2020 , PAGES =. doi:10.3934/jmd.2020011 , URL =

2020 doi
[28]

Duke Math

Tholozan, Nicolas , TITLE =. Duke Math. J. , FJOURNAL =. 2017 , NUMBER =. doi:10.1215/00127094-00000010X , URL =

2017 doi
[29]

and Manoussos, A

Abels, H. and Manoussos, A. and Noskov, G. , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/jlms/jdq091 , URL =

2011 doi
[30]

arXiv e-prints , keywords =

Finiteness of Bowen-Margulis-Sullivan Measure for Gromov-Patterson-Sullivan Systems. arXiv e-prints , keywords =. doi:10.48550/arXiv.2604.03982 , archivePrefix =. 2604.03982 , primaryClass =

doi arXiv
[31]

Canary, Richard and Zhang, Tengren and Zimmer, Andrew , TITLE =. J. Topol. , FJOURNAL =. 2026 , NUMBER =. doi:10.1112/topo.70064 , URL =

2026 doi
[32]

Algebraic groups and arithmetic , PAGES =

Benoist, Yves , TITLE =. Algebraic groups and arithmetic , PAGES =. 2004 , ISBN =

2004

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Floyd maps for relatively hyperbolic groups , url =

Gerasimov, Victor , date-added =. Floyd maps for relatively hyperbolic groups , url =. Geom. Funct. Anal. , mrclass =. 2012 , bdsk-url-1 =. doi:10.1007/s00039-012-0175-6 , fjournal =

doi:10.1007/s00039-012-0175-6 2012

[2] [2]

, TITLE =

Smilga, I. , TITLE =. Groups Geom. Dyn. , FJOURNAL =. 2018 , NUMBER =. doi:10.4171/GGD/447 , URL =

work page doi:10.4171/ggd/447 2018

[3] [3]

Preprint , Year =

Reyes, Eduardo and Wang, Tianqi , Title =. Preprint , Year =

[4] [4]

, TITLE =

Sambarino, A. , TITLE =. Ergodic Theory Dynam. Systems , FJOURNAL =. 2014 , NUMBER =. doi:10.1017/etds.2012.170 , URL =

doi:10.1017/etds.2012.170 2014

[5] [5]

and Sambarino, A

Potrie, R. and Sambarino, A. , TITLE =. Invent. Math. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00222-017-0721-9 , URL =

doi:10.1007/s00222-017-0721-9 2017

[6] [6]

and Canary, R

Bridgeman, M. and Canary, R. and Labourie, F. and Sambarino, A. , TITLE =. Geom. Funct. Anal. , FJOURNAL =. 2015 , NUMBER =. doi:10.1007/s00039-015-0333-8 , URL =

doi:10.1007/s00039-015-0333-8 2015

[7] [7]

Dongryul and Oh, Hee and Pan, Wenyu , Title =

Kim, M. Dongryul and Oh, Hee and Pan, Wenyu , Title =. Preprint , Year =

[8] [8]

Blayac, Pierre-Louis and Zhu, Feng , TITLE =. J. Mod. Dyn. , FJOURNAL =. 2023 , PAGES =. doi:10.3934/jmd.2023026 , URL =

doi:10.3934/jmd.2023026 2023