Probabilistic aspects of shadowing
Sergey B. Tikhomirov (St. Petersburg/IMPA)


It is well-known that shadowing holds in a neighborhood of a hyperbolic set. It is known that shadowing can hold for non hyperbolic systems, but due to results of Sakai, Abdenur, Diaz, Pilyugin, Tikhomirov shadowing is "almost"equivalent to structural stability. At the same time numerical experiments by Hammel-Grebogi-Yorke for logistics and Henon maps shows that shadowing holds for relatively long pseudotrajectories. It poses a question which type of shadowing holds for systems, which are not necessarily hyperbolic. I consider probabilistic approach for the shadowing. I show that for infinite pseudotrajectories it does not change the notion. At the same time we introduce approach in the spirit of Hammel-Grebogy-Yorke and shows that in some cases it works for non-uniformly hyperbolic cocycles. The talk is based on joint works with G. Monakov.


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