  We introduce a class of billiards with chaotic unidirectional flows (or
non-chaotic unidirectional flows with "vortices") which go around a chaotic or
non-chaotic "core", where orbits can change their orientation. Moreover, the
corresponding billiard tables are simply connected in difference with many
attempts to build billiards with interesting and/or exotic dynamics by putting
inside billiard tables various "scatterers" with funny shapes. Therefore the
billiards in this new class are amenable to experimental studies in physics
labs as well as to the rigorous mathematical ones, which may shed a new light
on understanding of classical and quantum dynamics of Hamiltonian systems.
