An initial value boundary problem for the Liouville equation with nonlinear dynamic boundary conditions which describes velocity of changing on time of the probability of particles at walls that confines the particles. These velocities are nonlinear functions of the density of the probability of particles to occupied the flat walls. The attractor of the problem has been constructed. This attractor contains periodic piecewise constant functions with finite , countable or uncountable points of discontinuities on a pe riod, which propagates along characteristics of the Liouville equation. We call such elements of the attractor by the distributions of relaxation, pre- turbulent and turbulent type, correspondingly — by the clas sification of Sharkovsky. There are also random distributio ns of particles, which can be produced by the nonlinear feedback on the walls. The results has been obtained by the reduction of the problem to dynamical system which is described by system of difference equations, depending on coordinates and momenta as of parameters. It is shown that the changing of these parameters leads to period doubling bifurcations of elements of the attractor on 4 - dimensional torus. The problem is solved in class of quasi-periodic functions.