Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Probabilistic shadowing in linear skew products

Grigorii Monakov, Sergey Tikhomirov

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.

Probabilistic shadowing in linear skew products

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.

Paper Structure

This paper contains 9 sections, 12 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Proof of Theorem \ref{['skew_prod_thm']}
  4. Case 1: $a < 0$
  5. Case 2: $a > 0$
  6. Applications
  7. Bernoulli shift
  8. Doubling map on a circle
  9. Linear hyperbolic map on a torus

Key Result

Lemma 1

Consider two points $q = (w, x)$ and $\Tilde{q} = (w, 0)$. For arbitrary positive numbers $d, \varepsilon > 0$, $N \in \mathbb{N}$ the following equality holds:

Theorems & Definitions (22)

  • Definition
  • Definition
  • Lemma 1
  • Definition : Property I
  • Definition : Property II
  • Definition : Property III
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Remark 1
  • ...and 12 more