Limit of small exits in open Hamiltonian systems

The nature of open Hamiltonian systems is analyzed, when the size of the exits decreases and tends to zero. Fractal basins appear typically in open Hamiltonian systems, but we claim that in the limit of small exits, the invariant sets tend to fill up the whole phase space with the strong consequence that a new kind of basin appears, where the unpredictability grows indefinitely. This means that for finite, arbitrarily small accuracy, we can find uncertain basins, where any information about the future of the system is lost. This total indeterminism had only been reported in dissipative systems, in particular in the so-called intermingled riddled basins, as well as in the riddledlike basins. We show that this peculiar, behavior is a general feature of open Hamiltonian systems.