Probabilistic shadowing in linear skew products

TL;DR

This paper studies the likelihood of accurately shadowing pseudotrajectories in linear skew products, establishing conditions for polynomial precision shadowing with high probability, using large deviation theory.

Contribution

It introduces general conditions under which pseudotrajectories can be shadowed with polynomial accuracy in linear skew products, expanding understanding of shadowing phenomena.

Findings

Conditions for polynomial shadowing probability are established.

Applicable to Bernoulli shift, doubling map, and Anosov linear maps.

Uses Cramer's large deviation theorem as a key tool.

Abstract

We investigate the probability of shadowing of a random finite pseudotrajectory by an exact trajectory for linear skew products. We describe general conditions under which a random pseudotrajectory can be shadowed with polynomial (with respect to its length) precision with high probability. Examples satisfying that general condition are continuous linear skew products over Bernoulli shift, doubling map on a circle, and any Anosov linear map on a torus. The main tool used in the proof is Cramer's large deviation theorem.