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

Abstract

We study the inverse problem associated with the return map $F$ on the boundary of a convex core $C$. By decomposing the return displacement into tangential and normal components, we derive an exact nonlinear reconstruction system for the unknown thickness $d$, yielding one-point uniqueness. Analyzing the linearization at fixed points introduces an operator $Q_{\ast }$: when $Q_{\ast }\ne 0$, the thickness and outer shape are uniquely recoverable via global propagation. In the degenerate regime $Q_{\ast }=0$, local ambiguity is resolved using nonlinear Ehresmann-type transport and holonomy maps along closed paths. Finally, we establish conditional realizability and identify key topological and dynamical quantities from the return data, characterizing the commuting regime as a singular limit.