Shadowing for finite pseudotrajectories with decreasing size of error

We consider the shadowing property of pseudotrajectories with decreasing errors for a linear skew product. The probabilistic properties of finite pseudotrajectories are studied. It is shown that for pseudotrajectories with errors decreasing exponentially, the typical dependence between the length of the pseudotrajectory and the shadowing accuracy is polynomial. The proof is based on the large deviation principle and the gambler's ruin problem. The talk starts with the overview of shadowing for finite pseudotrajectories and its probabilistic aspects. The work is supported by RSCF grant 21-11-00047.