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

TL;DR

This paper investigates the inverse problem of reconstructing geometric features from a return map associated with a convex core, revealing how curvature and thickness influence the system and discussing conditions for uniqueness.

Contribution

It characterizes how the return map encodes the gradient and curvature structure of the thickness function, highlighting non-uniqueness and conditions for recoverability.

Findings

Return map determines the gradient structure of the thickness function d.

Geometry is encoded through a curvature-dependent operator on the Hessian of d.

Intrinsic non-uniqueness arises from scaling and dynamical equivalences.

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.