Multiplane gravitational lenses with an abundance of images.

The article focuses on the problem of determining the maximum number of gravitationally lensed images produced by a system of point masses distributed across multiple lens planes, a question unresolved in the multiplane setting. Building on known results for single-plane lensing—where the maximum number of images formed by \( g \) point masses is \( 5g - 5 \)—the authors construct explicit examples of \( K \)-plane lens systems with \( g_i \) masses in the \( i \)-th plane that produce \(\prod_{i=1}^K (5g_i - 5)\) images of a single background source. Their method uses Rhie's single-plane extremal configurations combined with a parameter-rescaling algorithm and a stability principle from differential topology to perturb nonphysical preliminary systems into physically meaningful ones without losing solutions. Numerical simulations illustrate these constructions and reveal a "caustic of multiplicity" phenomenon in the nonphysical case, which splits into distinct caustics upon perturbation, affecting the caustic structure and image multiplicity. The work advances understanding of multiplane gravitational lensing by providing lower bounds on the maximum number of images achievable, though the exact maximum remains an open problem.