Shadowing in linear skew products

TL;DR

This paper investigates shadowing in linear skew products with a full shift base and nonzero Lyapunov exponent, providing precise estimates and challenging existing conjectures through a novel reduction to a ruin problem.

Contribution

It offers a sharp estimate for shadowing accuracy in a high-dimensional setting and refutes a conjecture about shadowability intervals using a new probabilistic approach.

Findings

Sharp estimate for shadowing precision of pseudotrajectories

High-dimensional shadowing behavior differs from previous conjectures

Reduction of shadowing problem to a ruin problem for a random walk

Abstract

We consider a linear skew product with the full shift in the base and nonzero Lyapunov exponent in the fiber. We provide a sharp estimate for the precision of shadowing for a typical pseudotrajectory of finite length. This result indicates that the high-dimensional analog of Hammel-Yorke-Grebogi's conjecture concerning the interval of shadowability for a typical pseudotrajectory is not correct. The main technique is reduction of the shadowing problem to the ruin problem for a simple random walk.