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 \(\varphi \).
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Notes
Generic flows are Morse–Smale for any orientable two-dimensional surface and also the projective plane, Klein bottle, and torus with a cross-cap, with respect to any \(C^r\)-topology on the space of vector fields, see [9] and [15]. This holds for other non-orientable surfaces if \(r = 1\) (which can be seen by combining the discussion in [9] with Pugh’s closing lemma), but remains an open question for \(r > 1\).
Synchronization is a process where some dynamical properties of two systems (or of two trajectories of one system) evolve in an almost identical way, usually due to some interaction between the systems. In our case synchronization takes the following form. We have two systems such that generic orbits spend almost all the time near two points: A, B for the first system and \({\tilde{A}}, {\tilde{B}}\) for the second, but a generic orbit of the product-system spends almost all the time near the diagonal points \((A, {\tilde{A}})\) and \((B, {\tilde{B}})\) and neglects the pairs \((A, {\tilde{B}})\) and \((B, {\tilde{A}})\). Synchronization in a similar sense was discussed in [8]; more classical definitions of synchronization can be found in, e.g., [19].
We do not distinguish flows and semi-flows. In each case it will be clear from the context whether the flow is invertible or not.
Here \(\delta _x\) stands for the \(\delta \)-measure at \(x \in X\) and \(\int _0^T \delta _{\varphi ^t(x)}dt\) stands for a Borel measure \(\mu _{x,T}\) defined, due to the Riesz representation theorem, by the requirement that for any continuous function \(f\in C(X, {\mathbb {R}})\) one must have
$$\begin{aligned} \int _X f d\mu _{x, T} = \int _0^T(\delta _{\varphi ^t(x)}, f) dt = \int _0^T f(\varphi ^t(x)) dt. \end{aligned}$$That is, a semi-neighborhood where trajectories make winds near the polycycle. It exists because the saddle-node was chosen to be contracting.
Obviously, the Milnor attractor will coincide with the whole polycycle, since every trajectory in the semi-neighborhood is attracted to the polycycle.
Here we refer to the order that the field has when restricted to the central manifold of the saddle-node. “Order two” means that the quadratic term of the restriction is non-zero.
By ‘generic’ we mean that the characteristic number (minus ratio of eigenvalues; the negative one goes to the numerator) is not equal to 1, and for the loop to be attracting, as it is well-known, we need the characteristic number to be \(\ge 1\).
The same argument applies to a loop whose saddle has characteristic number 1 if the loop happens to be attracting.
The subscript s in \(\Delta _s\) stands for ‘saddle’. Below we will also consider a similar map \(\Delta _{sn}\) for a saddle-node.
Indeed, there exist \(C^1\)-smooth linearizing coordinates in a neighborhood of the saddle (see [14]) that induce charts on the transversals in which the monodromy map is just \(x \mapsto x^\nu \), and the claim holds. Then it is straightforward to check that a \(C^1\) change of coordinate (that preserves the origin) on any of the two transversals does not change the limit.
In particular, if a vector field has a contracting saddle-node with a homoclinic curve, then the saddle-node belongs to the statistical attractor, but the points of the homoclinic curve do not, and our argument will not distinguish this case from the case where there is no homoclinic curve.
We consider the closed semi-neighborhood that includes the polycycle and, in particular, the two saddles.
Since z and \({\tilde{z}}\) are taken from the transversals \(\Sigma _A\) and \(\Sigma _{{\tilde{A}}}\) in our construction, here we mean the product of the one-dimensional Lebesgue measures on these transversals.
This means that we replace \(\varphi \) and \({\tilde{\varphi }}\) by their restrictions to these sets.
As usual, we are assuming that our measure is finite and normalized to total measure one.
In what follows, we will identify points of the circle and their angular coordinates.
Here the point \(({\theta }_r, -1) \in S^1 \times [-1, 1] = {\mathcal {C}}_{[-1, 1]}\) is at the lower boundary of the cylinder. By default we will use the original coordinates \(({\theta }, z)\) on \({\mathcal {C}}_{[-1, 1]}\) to specify points and subsets of the cylinder.
The field on the ball constructed above can be rectified near the boundary sphere and so can be glued smoothly to the field on the manifold. For the resulting flow, any set inside a ball that contains almost every its point will intersect almost every orbit of the flow.
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Acknowledgements
The idea of the paper was conceived in a discussion with Alexey Okunev, to whom we are very grateful. We also want to thank Andrey Dukov whose comments on the draft of this paper greatly helped us in improving it. We are deeply indebted to Yulij Ilyashenko for his heartening words and remarks. Both authors were partially supported by the RFBR grant 20-01-00420-a.
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Both authors are partially supported by the RFBR grant 20-01-00420-a.
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Minkov, S., Shilin, I. Attractors of Direct Products. Qual. Theory Dyn. Syst. 20, 77 (2021). https://doi.org/10.1007/s12346-021-00510-9
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DOI: https://doi.org/10.1007/s12346-021-00510-9