Attractors of Linear Maps with Bounded Noise

TL;DR

This paper studies the behavior of invertible linear maps affected by bounded noise, demonstrating the uniqueness and convexity of their minimal attractors and introducing a boundary map with globally attracting properties.

Contribution

It establishes the uniqueness, convexity, and smoothness of attractors for linear maps with bounded noise and introduces a boundary map with a globally attracting normal bundle.

Findings

Minimal attractors are unique and strictly convex.

Attractors have a continuously differentiable boundary.

A finite-dimensional boundary map is globally attracting.

Abstract

We consider invertible linear maps with additive spherical bounded noise. We show that minimal attractors of such random dynamical systems are unique, strictly convex and have a continuously differentiable boundary. Moreover, we present an auxiliary finite-dimensional deterministic boundary map for which the unit normal bundle of this boundary is globally attracting.