Interacting shells in anti-de Sitter space revisited: an invariant measure, regular orbits, and chaos

Two concentric thin shells that cross transparently in anti-de Sitter space provide a simplemodel of confined, self-gravitating dynamics. We derive an explicit invariant measure for thePoincaré return map on head-on crossings, making it area preserving in suitable coordinates, anduse the map to organize noncollapsing dynamics. We revisit the example orbit and initial-datafamily for which Brito, Cardoso, and Rocha (BCR) reported evidence of chaos. At the example’sparameters, we numerically identify two elliptic fixed points and a saddle in the central partof the section. The reconstructed example and every sampled member of the family behaveregularly for up to 50 000 returns. For the example, nearby trajectories separate approximatelylinearly, with late-time separation orders of magnitude smaller than in the published separationcurve from which BCR estimated a positive Lyapunov exponent. The source of this discrepancyremains unresolved. For the family, we reproduce the bifurcation-diagram spread, showing thatthis diagnostic alone does not establish a transition toward chaos. Near the saddle, strongchaos contracts along sampled radial cuts as the energy content increases. A broader sectionsurvey finds chaotic orbits that travel between the section’s two ends, dominated by overtakingcrossings and exhibiting faster perturbation growth. At the largest sampled energy content, italso identifies two bands of collapsing initial data. These results support BCR’s identification ofchaos in the model while changing the interpretation of their selected initial data. The totalchaotic area and long-time transport remain unresolved.