Marginal singularities, almost invariant sets, and small perturbations of chaotic dynamical systems.

For a class of piecewise monotone locally noncontracting maps f:X-->X with small "traps" Y( varepsilon ) subset, dbl equals X (diam(Y( varepsilon ))</= varepsilon ), the existence of smooth conditionally f-invariant measures &mgr;( varepsilon ) are proved, corresponding to a limit as n--> infinity conditional probabilities that f(n+1)x in X\Y( varepsilon ) if x,fx,.,f(nx) in X\Y( varepsilon ) and the point x is chosen at random. Also proven is the convergence of &mgr;( varepsilon ) to smooth f-invariant measures as varepsilon -->0. By means of this construction, the numerical phenomenon of the convergence of histograms of trajectories of maps with marginal singularities to densities of nonfinite smooth invariant measures in the computer modeling was investigated.