Probabilistic aspects of shadowing

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 Sakao, 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 topic. I show that for infinite pseudotrajectories it does not change the notion. At the same time it shows that relatively long pseudotrajectories can be shadowed by exact trajectory with high probability. The main technique is a reduction to special form of gambler's ruin problems and mild form of large deviation principle for random walks. We show that our approach works for several examples -- skew product maps.

The talk is based on joint works with G. Monakov