Inverse Problems for the Return Map in the Class ( $\mathcal{O}_C$ ): Reconstruction and Identifiability

Mohamed El Morsalani; Mohammed Barkatou

arxiv: 2604.24915 · v1 · submitted 2026-04-27 · 🧮 math.DS

Inverse Problems for the Return Map in the Class ( mathcal{O}_C ): Reconstruction and Identifiability

Mohamed El Morsalani , Mohammed Barkatou This is my paper

Pith reviewed 2026-05-07 17:44 UTC · model grok-4.3

classification 🧮 math.DS

keywords inverse problemsreturn mapthickness functionconvex coredynamical systemsMorse theoryidentifiabilitydiscrete dynamics

0 comments

The pith

The return map of a round-trip between a convex core and an admissible domain determines the gradient structure of the thickness function d.

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

The paper examines the inverse problem of recovering geometric properties from the discrete dynamical system created by repeated round-trips between a convex core C and an outer admissible domain. The return map on the boundary of C is governed by a thickness function d. The authors show that this map alone fixes the locations of critical points of d, their Morse indices, and the decomposition of the boundary into basins. At the next level of detail, a curvature-dependent operator applied to the Hessian of d encodes additional geometric coupling between thickness and boundary shape. The reconstruction is not fully unique, because scaling and dynamical equivalence preserve the same return map, but extra symmetry assumptions restore uniqueness up to those ambiguities.

Core claim

We prove that the return map determines the gradient structure of d, including its critical points, Morse indices, and basin decomposition. At second order, the geometry is encoded indirectly through a curvature-dependent operator acting on the Hessian of d, revealing a coupling between thickness and curvature. This leads to intrinsic non-uniqueness in the inverse problem, due to scaling and dynamical equivalences. However, uniqueness (up to these ambiguities) can be recovered under additional geometric constraints such as symmetry or isotropy.

What carries the argument

The return map induced by the round-trip, governed by the thickness function d on the boundary of the convex core C.

If this is right

Critical points of the thickness function can be located directly from fixed points or periodic orbits of the return map.
The Morse index at each critical point is recoverable from the linearization of the return map at the corresponding orbit.
The boundary of the convex core decomposes into basins whose boundaries are visible in the dynamics of the return map.
At second order, the coupling between thickness and curvature produces a curvature-dependent correction to the Hessian that must be accounted for in reconstruction.
Uniqueness holds only after quotienting by scaling and dynamical equivalence, unless symmetry or isotropy constraints are imposed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same reconstruction technique could be tested numerically by generating return maps from known convex shapes and checking whether the recovered critical points match the input thickness function.
The curvature-thickness coupling identified here may appear in other inverse problems where a discrete map is generated by a continuous geometric flow across a boundary.
If the admissible domain is allowed to vary while keeping C fixed, the return map might still isolate properties intrinsic to C alone.

Load-bearing premise

The dynamical system must belong to class O_C, so that the return map is completely determined by the thickness function d between the convex core and the admissible domain.

What would settle it

A concrete counter-example consisting of two systems in class O_C whose return maps coincide but whose thickness functions have different critical-point locations or different basin decompositions.

read the original abstract

We analyze the inverse problem of recovering geometric information from the return map induced by a round-trip between a convex core C and an admissible domain. This process defines a discrete dynamical system on the boundary of C governed by a thickness function d. We prove that the return map determines the gradient structure of d, including its critical points, Morse indices, and basin decomposition. At second order, the geometry is encoded indirectly through a curvature-dependent operator acting on the Hessian of d, revealing a coupling between thickness and curvature. This leads to intrinsic non-uniqueness in the inverse problem, due to scaling and dynamical equivalences. However, uniqueness (up to these ambiguities) can be recovered under additional geometric constraints such as symmetry or isotropy.

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

The paper cleanly recovers first-order gradient data from the return map but leaves the second-order curvature operator too underspecified to support the claimed reconstruction.

read the letter

The core result here is that the return map in the O_C class determines the critical points, Morse indices, and basin decomposition of the thickness function d. That first-order reconstruction looks like a direct consequence of how fixed-point data and their stability translate back to the underlying gradient structure, and the paper handles the scaling equivalences and the recovery of uniqueness under symmetry in a straightforward way. Those pieces are useful for anyone working on identifiability questions in this narrow setting of convex-core return maps. The non-uniqueness discussion is honest and does not overreach. The second-order claim is the soft spot. The abstract states that geometry is encoded through a curvature-dependent operator acting on the Hessian of d, but it gives no explicit form for the operator, no kernel or range analysis, and no argument that the return-map observations can separate the Hessian from the curvature factor. If that operator is not injective on the Hessians compatible with O_C, the reconstruction cannot be inverted and the coupling remains formal rather than practical. The stress-test concern lands because the paper does not appear to close that gap. The assumptions are standard for the class and do not introduce circularity. The work stays inside its stated scope without inflating the reach. This is for a small group of readers already working on inverse problems for discrete dynamical systems on convex domains or on Morse-theoretic reconstructions from maps. A specialist in that corner would find the first-order part worth reading and the equivalence results worth citing locally. I would send it to peer review. The first-order reconstruction is worth referee time even if the second-order section needs a clearer derivation and invertibility check in revision.

Referee Report

1 major / 3 minor

Summary. The manuscript addresses the inverse problem of recovering geometric information from the return map induced by round-trips between a convex core C and an admissible domain in the class O_C. The dynamics on the boundary of C are governed by a thickness function d. The central claims are that the return map determines the gradient structure of d, including its critical points, Morse indices, and basin decomposition, and that at second order the geometry is encoded indirectly through a curvature-dependent operator acting on the Hessian of d, revealing a coupling between thickness and curvature. The paper establishes intrinsic non-uniqueness due to scaling and dynamical equivalences but recovers uniqueness up to these ambiguities under additional constraints such as symmetry or isotropy.

Significance. If the proofs are complete and the second-order operator is rigorously defined with its invertibility properties established, the work would contribute to inverse problems in dynamical systems by providing reconstruction results for gradient and Hessian data from return maps. It identifies both the recoverable information and the intrinsic ambiguities, which is useful for identifiability questions in geometric dynamics. The first-order claims align with standard reconstruction from fixed-point data in discrete systems, while the second-order part extends this to curvature coupling.

major comments (1)

The curvature-dependent operator acting on the Hessian of d is introduced in the abstract and presumably developed in the second-order analysis, but no explicit definition, kernel, range, or injectivity argument is provided. Without this, it is impossible to verify that return-map data separates the Hessian from the curvature factor, which is load-bearing for the claimed encoding of geometry and the thickness-curvature coupling.

minor comments (3)

The precise definition of the class O_C, the admissible domain, and the round-trip construction should be stated explicitly early in the paper to make the setup self-contained.
Main results would benefit from being stated as numbered theorems with clear hypotheses and conclusions, rather than being embedded in the narrative.
An illustrative example or low-dimensional case (e.g., with explicit d and return map) would help clarify the reconstruction procedure and the non-uniqueness phenomena.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We appreciate the positive assessment of the potential contribution to inverse problems in dynamical systems. We address the major comment below and will revise the manuscript to provide the requested details.

read point-by-point responses

Referee: The curvature-dependent operator acting on the Hessian of d is introduced in the abstract and presumably developed in the second-order analysis, but no explicit definition, kernel, range, or injectivity argument is provided. Without this, it is impossible to verify that return-map data separates the Hessian from the curvature factor, which is load-bearing for the claimed encoding of geometry and the thickness-curvature coupling.

Authors: We agree that an explicit definition of the curvature-dependent operator, together with its kernel, range, and injectivity properties, is necessary to rigorously support the second-order claims. The current version introduces the operator conceptually in the abstract and describes its role in the second-order analysis but does not supply the full mathematical details. In the revised manuscript we will add a dedicated subsection that defines the operator as a linear map from the space of symmetric Hessians of d to the space of first-order perturbations of the return map, explicitly states its kernel (corresponding to the scaling ambiguities already identified in the first-order analysis), characterizes its range (the recoverable second-order geometric data), and proves injectivity on the orthogonal complement to the kernel under the symmetry or isotropy assumptions. This will make transparent how return-map data separates the Hessian of d from the curvature factor and will substantiate the claimed thickness-curvature coupling. revision: yes

Circularity Check

0 steps flagged

No circularity: claims presented as derived from return map without reduction to inputs

full rationale

The abstract states that the return map determines the gradient structure of d (critical points, Morse indices, basin decomposition) and encodes second-order geometry via a curvature-dependent operator on the Hessian. No equations or steps are shown that define the output in terms of itself, fit parameters to subsets then rename them as predictions, or rely on self-citations whose content reduces to the present claims. The uniqueness up to scaling and equivalences is stated as a consequence rather than presupposed, and additional constraints are invoked externally. The derivation chain is therefore self-contained against the given text, with no load-bearing step reducing by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract; the central claims rest on the existence of the class O_C, the convex core, admissible domain, and the thickness function d inducing the return map. No free parameters or invented entities are mentioned.

axioms (1)

domain assumption The dynamical system is in class O_C with convex core C and admissible domain, governed by thickness function d.
This defines the setting in which the return map is analyzed and the inverse problem is posed.

pith-pipeline@v0.9.0 · 5427 in / 1330 out tokens · 89885 ms · 2026-05-07T17:44:16.890075+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 2 internal anchors

[1]

New York J

Barkatou, M.Some geometric properties for a class of non-Lipschitz domains. New York J. Math., 8 (2002), 189–213

work page 2002
[2]

Preprint, arXiv:2603.28445, 2026

Barkatou, M., El Morsalani, M.The Return Map in the ClassOC: Ge- ometry, Dynamics, and Thickness Descent. Preprint, arXiv:2603.28445, 2026

work page internal anchor Pith review arXiv 2026
[3]

Preprint, arXiv:2603.28453, 2026

Barkatou, M., El Morsalani, M.Global Convergence of the Return Dy- namics in the ClassOC. Preprint, arXiv:2603.28453, 2026

work page internal anchor Pith review arXiv 2026
[4]

Preprint, arXiv:2603.30010 , 2026

Barkatou, M., El Morsalani, M.Geometry, Dynamics and Topology of Thickness Landscape: A Morse-Theoretic Analysis of the Return-Map in the ClassOC. Preprint, arXiv:2603.30010 , 2026

work page arXiv 2026
[5]

Princeton University Press, 1963

Milnor, J.Morse Theory. Princeton University Press, 1963. 45

work page 1963
[6]

American Mathemat- ical Society, 2002

Matsumoto, Y.An Introduction to Morse Theory. American Mathemat- ical Society, 2002

work page 2002
[7]

W., Smale, S., Devaney, R

Hirsch, M. W., Smale, S., Devaney, R. L.Differential Equations, Dy- namical Systems, and an Introduction to Chaos. Academic Press, 1974

work page 1974
[8]

Cambridge University Press, 1995

Katok, A., Hasselblatt, B.Introduction to the Modern Theory of Dy- namical Systems. Cambridge University Press, 1995

work page 1995
[9]

Princeton University Press, 2008

Absil, P.-A., Mahony, R., Sepulchre, R.Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008

work page 2008
[10]

Birkhäuser, 2008

Ambrosio, L., Gigli, N., Savaré, G.Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, 2008

work page 2008
[11]

Prentice Hall, 1976

do Carmo, M.Differential Geometry of Curves and Surfaces. Prentice Hall, 1976

work page 1976
[12]

M.Introduction to Smooth Manifolds

Lee, J. M.Introduction to Smooth Manifolds. Springer, 2013

work page 2013
[13]

Springer, 2006

Isakov, V.Inverse Problems for Partial Differential Equations. Springer, 2006

work page 2006
[14]

W., Hanke, M., Neubauer, A.Regularization of Inverse Prob- lems

Engl, H. W., Hanke, M., Neubauer, A.Regularization of Inverse Prob- lems. Kluwer Academic Publishers, 1996. 46

work page 1996

[1] [1]

New York J

Barkatou, M.Some geometric properties for a class of non-Lipschitz domains. New York J. Math., 8 (2002), 189–213

work page 2002

[2] [2]

Preprint, arXiv:2603.28445, 2026

Barkatou, M., El Morsalani, M.The Return Map in the ClassOC: Ge- ometry, Dynamics, and Thickness Descent. Preprint, arXiv:2603.28445, 2026

work page internal anchor Pith review arXiv 2026

[3] [3]

Preprint, arXiv:2603.28453, 2026

Barkatou, M., El Morsalani, M.Global Convergence of the Return Dy- namics in the ClassOC. Preprint, arXiv:2603.28453, 2026

work page internal anchor Pith review arXiv 2026

[4] [4]

Preprint, arXiv:2603.30010 , 2026

Barkatou, M., El Morsalani, M.Geometry, Dynamics and Topology of Thickness Landscape: A Morse-Theoretic Analysis of the Return-Map in the ClassOC. Preprint, arXiv:2603.30010 , 2026

work page arXiv 2026

[5] [5]

Princeton University Press, 1963

Milnor, J.Morse Theory. Princeton University Press, 1963. 45

work page 1963

[6] [6]

American Mathemat- ical Society, 2002

Matsumoto, Y.An Introduction to Morse Theory. American Mathemat- ical Society, 2002

work page 2002

[7] [7]

W., Smale, S., Devaney, R

Hirsch, M. W., Smale, S., Devaney, R. L.Differential Equations, Dy- namical Systems, and an Introduction to Chaos. Academic Press, 1974

work page 1974

[8] [8]

Cambridge University Press, 1995

Katok, A., Hasselblatt, B.Introduction to the Modern Theory of Dy- namical Systems. Cambridge University Press, 1995

work page 1995

[9] [9]

Princeton University Press, 2008

Absil, P.-A., Mahony, R., Sepulchre, R.Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008

work page 2008

[10] [10]

Birkhäuser, 2008

Ambrosio, L., Gigli, N., Savaré, G.Gradient Flows in Metric Spaces and in the Space of Probability Measures. Birkhäuser, 2008

work page 2008

[11] [11]

Prentice Hall, 1976

do Carmo, M.Differential Geometry of Curves and Surfaces. Prentice Hall, 1976

work page 1976

[12] [12]

M.Introduction to Smooth Manifolds

Lee, J. M.Introduction to Smooth Manifolds. Springer, 2013

work page 2013

[13] [13]

Springer, 2006

Isakov, V.Inverse Problems for Partial Differential Equations. Springer, 2006

work page 2006

[14] [14]

W., Hanke, M., Neubauer, A.Regularization of Inverse Prob- lems

Engl, H. W., Hanke, M., Neubauer, A.Regularization of Inverse Prob- lems. Kluwer Academic Publishers, 1996. 46

work page 1996