[Paper Review] On absolute nonshadowability of transitive maps

Sergey Tikhomirov|arXiv (Cornell University)|Jun 24, 2016

Mathematical Dynamics and Fractals19 references3 citations

TL;DR

This paper establishes a dichotomy for transitive maps and transitive attractors on compact metric spaces: either the map has the classical shadowing property, or almost every random infinite pseudotrajectory (under a natural stochastic process) fails to be shadowed with probability zero. The result shows that absolute nonshadowability is generic under transitivity, implying that for such systems, shadowing cannot be recovered probabilistically, even in the presence of dense orbits.

ABSTRACT

We study shadowing property for random infinite pseudotrajectories of a continuous map $f$ of a compact metric space. For the cases of transitive maps and transitive attractors we prove a dichotomy: either $f$ satisfies shadowing property or random pseudotrajectory is shadowable with probability 0.

Motivation & Objective

• To investigate whether random pseudotrajectories of transitive maps can be shadowed with positive probability.
• To determine whether the shadowing property can be recovered probabilistically in systems with dense orbits.
• To establish a dichotomy between full shadowing and absolute nonshadowability in transitive dynamical systems.
• To extend the understanding of shadowing beyond deterministic pseudotrajectories to stochastic settings.
• To show that for transitive maps and attractors, the absence of shadowing implies that almost all random pseudotrajectories are unshadowable.

Proposed method

• Models random pseudotrajectories as a Markov chain on the space of points, where each next point is chosen uniformly at random within a d-ball around f(y_n).
• Defines the probability p(y₀, d, ε) that a d-pseudotrajectory starting at y₀ can be ε-shadowed.
• Uses the concept of absolute nonshadowability: f is absolutely nonshadowable if p(y₀, d, ε) = 0 for some ε > 0 and all y₀, d > 0.
• Applies a recurrence argument based on the existence of a finite pseudotrajectory that cannot be ε-shadowed, combined with transitivity and compactness.
• Constructs a sequence of independent shadowing failures over time intervals, using the Markov property and uniform positive lower bounds on failure probability.
• Employs the Borel-Cantelli lemma and measure-theoretic arguments to show that the probability of eventual shadowing failure tends to 1.

Experimental results

Research questions

• RQ1 Can random infinite pseudotrajectories of a transitive map be shadowed with positive probability?
• RQ2 Is there a dichotomy such that either all pseudotrajectories are shadowable or almost none are, under transitivity?
• RQ3 Does the absence of the classical shadowing property imply that random pseudotrajectories are almost surely unshadowable?
• RQ4 Can the shadowing property be recovered probabilistically in systems with dense forward orbits?
• RQ5 What is the role of attractors and their transitivity in determining the shadowability of random pseudotrajectories?

Key findings

• For transitive maps, either the map has the classical shadowing property, or with probability zero can any random infinite pseudotrajectory be ε-shadowed for some ε > 0.
• In the case of transitive attractors, if the map does not satisfy the shadowing property on the attractor, then for any initial point in the domain of attraction, the probability of shadowing a random d-pseudotrajectory is zero.
• The probability of shadowing failure tends to 1 over time, due to repeated independent failures in overlapping time intervals.
• The proof relies on constructing a finite pseudotrajectory that cannot be ε-shadowed and using transitivity to embed it infinitely often in random trajectories.
• The result holds uniformly across initial conditions and small d > 0, showing that nonshadowability is robust under small perturbations of the pseudotrajectory.
• The key technical tool is a uniform lower bound η > 0 on the probability of failure in each recurrence cycle, ensuring that the infinite product of survival probabilities vanishes.