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

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Abstract
For a class of piecewise monotone locally noncontracting maps f:X→X with small ‘‘traps’’ Yε⊆X (diam(Yε)≤ε), the existence of smooth conditionally f-invariant measures με are proved, corresponding to a limit as n→∞ conditional probabilities that fn+1x∈X\Yε if x,fx,...,fnx∈X\Yε and the point x is chosen at random. Also proven is the convergence of με to smooth f-invariant measures as ε→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.

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