In Hamiltonian systems, regular islands are enclosed by smooth invariant curves — unbreakable transport barriers that separate ordered from chaotic motion. A trajectory inside the island stays inside forever. The curves guarantee this because the map is continuous and area-preserving; a smooth, measure-preserving map cannot teleport a trajectory across a closed curve.
Break the continuity, and the islands fragment.
In a discontinuous area-preserving map — still exactly conservative, still deterministic, but with a spatial seam where the map jumps — regular islands undergo hierarchical fragmentation. The smooth invariant curves shatter into smaller regular components connected by chaotic channels. Trajectories initialized near elliptic fixed points get trapped for long times, exhibiting the telltale quasiperiodic motion of regular dynamics, but eventually escape through the chaotic channels into the surrounding sea.
The fragmentation is hierarchical: each regular remnant contains smaller remnants within it, connected by finer chaotic channels, down to scales set by the discontinuity. The structure resembles the intact Hamiltonian island viewed through a cracked lens — recognizable but penetrated at every scale.
The proof that discontinuity is the cause is direct. The authors restore continuity in a modified version of the same map, and smooth invariant curves immediately reappear. All other features — the nontwist condition, the area preservation, the specific potential — remain identical. Only the seam matters.
Continuity is the load-bearing property of Hamiltonian phase space. It is what makes invariant curves into transport barriers. Conservation of area is necessary but not sufficient — without continuity, the area is preserved but the topology is destroyed. The container holds the same volume of phase space but can no longer prevent flow between compartments. Fragmentation is what conservation without continuity looks like.