Attractors of Direct Products

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 a global physical measure of the square of $\varphi$.
We are interested in attractors definitions of which rely on a natural (in our case, Lebesgue) measure on the phase space, which allows these attractors to capture asymptotic behavior of most points while possibly neglecting what happens with a set of orbits of zero measure. One type of such attractors was introduced by J. Milnor in [M] under the name “the likely limit set”. We refer to it as the Milnor attractor.
The Milnor attractor $A_{\mathrm{Mil}}(\varphi)$ of a dynamical system $\varphi$ is the smallest closed set that contains the $\omega$-limit sets of $\mu$-almost all orbits.
Another way to define an attractor is via an “attracting” invariant measure: the attractor is its support. Most suitable are the notions of physical and natural measures, which may be viewed as analogues of SRB-measures for general, non-hyperbolic dynamical systems (see, e.g., [BB]; we adapt the definitions from [BB] to the case of flows). Physical measures describe the distribution of $\mu$-a.e. orbit, while natural measures capture the limit behaviour of the reference measure.
For a flow $\varphi$ on a compact manifold $X$ with measure $\mu$, a probability measure $\nu$ is called physical if there is a set $B$ with $\mu(B) > 0$ such that for any $x \in B$ and any continuous function $f\in C(X, \mathcal R)$ the Birkhoff time average over the orbit of $x$ is equal to the space average w.r.t. $\nu$:
$$\lim\limits_{T \to +\infty}\frac{1}{T}\int_0^T f(\varphi^t(x)) dt = \int_X f\, d\nu.$$

A measure $\nu$ is called natural for a flow $\varphi$ if there exists an open subset $U \subset X$ such that for any probability measure $\tilde{\mu}$ absolutely continuous with respect to $\mu$ and with $\mathrm{supp}(\tilde{\mu}) \subset U$ one has weak-$*$ convergence
$$\frac{1}{T}\int_0^T \varphi^t_*\tilde{\mu} \;dt \to \nu , T \to +\infty.$$

Statistical and minimal attractors are supports of these measures, respectively.
When constructing examples of dynamical systems with required properties, it is not uncommon to utilize, at least as a piece of the construction, direct products of systems in lower dimensions. It is tempting to think that the attractor of the direct product of two systems always coincides with the direct product of their attractors. Although this holds, indeed, for so-called maximal attractors, this is not true for several other types of attractors, namely, for Milnor, statistical, and minimal attractors, and also for the supports of physical measures, when the latter exist. We present examples of smooth flows on $S^2$ that exhibit such non-coincidence. Our examples are mostly Cartesian squares of flows. Although the square of a flow is always a flow of infinite codimension, it is interesting to find, for every type of attractor, the least codimension for the flow itself in which one can have non-coincidence between the attractor of the square and the square of the attractor.
[BB] Blank, M., Bunimovich L.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity, 16:1, 387–401 (2003)
[M] Milnor, J.: On the concept of attractor. Comm. Math. Phys. 99, 177–195 (1985)