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The separating mechanisms occuring in a class of Poincar6 halfm aps that is induced by the flow of a saddle-focus are described quantitatively. The "critical spiral" is calculated explicitly inside the domain o f the map. A complete hierarchy, consisting of three different types of separating mechanisms, is demonstrated. It is characterized in terms o f the properties o f the critical spiral, as it depends on the canonical param eters o f the system.

Julia sets are self-similar separatrices of the coast-line type found in noninvertible 2-D maps. The same class of maps also generates hyperchaos (chaos with two mixing directions). Smale’s notion of a “nontrivial basic set” provides a connection. These sets arise when a chaotic (or hyperchaotic) attractor “explodes”. In the case of more than one escape route, this set becomes a “fuzzy boundary” (Mira). Its projection as the map becomes noninvertible (1-D ) is a “Julia set in 1 D ”. In the analogous hyperchaos case the 2-D limiting map contains a classical Julia set of the continuous type. An identically looking set can also be obtained within a non-exploded hyperchaotic attractor, however, as a “cloud”. Julia-like attractors therefore exist. The theory also predicts Mandelbrot sets for 4-D flows. Julia-like behavior is a new, numerically easy-to-test for property o f most nontrivial dynamical systems.