# On the attractors of step skew products over the Bernoulli shift

## TL;DR

This paper investigates the properties of attractors in step skew products over Bernoulli shifts, showing their stability and describing their structure for different fiber types, with implications for dynamical systems theory.

## Contribution

It proves that for generic cases, statistical and Milnor attractors coincide and are Lyapunov stable, and characterizes attractors for fibers as a circle or segment.

## Key findings

- Statistical and Milnor attractors coincide in generic cases.
- Attractors are Lyapunov stable for circle fibers.
- Milnor attractor is described as a union of graphs for segment fibers.

## Abstract

The statistical and Milnor attractors of step skew products over the Bernoulli shift are studied. For the case of the fiber a circle we prove that for a topologically generic step skew product the statistical and the Milnor attractor coincide and are Lyapunov stable. For this end we study some properties of the projection of the attractor onto the fiber, which might be of independent interest. For the case of the fiber being a segment we give a description of the Milnor attractor as the closure of the union of graphs of finitely many almost everywhere defined functions from the base of the skew product to the fiber.