Attractors of direct products

TL;DR

The paper constructs smooth flows on the 2-sphere demonstrating that the attractor of their Cartesian square can be smaller than the square of the attractor, challenging assumptions about attractor behavior under product operations.

Contribution

It provides explicit examples of smooth flows on $S^2$ with non-intuitive properties of attractors under Cartesian products, including differences in physical measures.

Findings

Constructed flows with smaller product attractors

Demonstrated non-coincidence of physical measures under squaring

Challenged existing assumptions about attractor behavior

Abstract

For Milnor, statistical, and minimal attractors, we construct examples of smooth flows $\varphi$ on $S^2$ for which the attractor of the Cartesian square of $\varphi$ is smaller than the Cartesian square of the attractor of $\varphi$. In the example for the minimal attractors, the flow $\varphi$ also has a global physical measure such that its square does not coincide with the global physical measure of the square of $\varphi$.