A pathological random perturbation of the Cat Map

TL;DR

This paper presents a specific random perturbation of the Cat Map that results in a global statistical attractor, a line segment, with all initial distributions eventually supported on it, highlighting unique dynamical behavior.

Contribution

It introduces a novel random perturbation of the Cat Map that creates a global attractor in the form of a line segment, with partial smoothness in transition probabilities.

Findings

All initial distributions are attracted to the line segment.

The perturbation induces a global statistical attractor.

Transition probabilities are smooth in some directions.

Abstract

In this paper we give an example of a random perturbation of the Cat Map that produces a "global statistical attractor" in the form of a line segment. The transition probabilities for this random perturbation are smooth in some but not all directions. All initial distributions on $\mathbb{T}^2$ are attracted to distributions supported on this line segment.