Abstract
We construct new substantive examples of nonautonomous vector fields on a \(3\)-dimensional sphere having simple dynamics but nontrivial topology. The construction is based on two ideas : the theory of diffeomorphisms with wild separatrix embedding and the construction of a nonautonomous suspension over a diffeomorphism. As a result, we obtain periodic, almost periodic, or even nonrecurrent vector fields that have a finite number of special integral curves possessing exponential dichotomy on \(\mathbb R\) such that among them there is one saddle integral curve (with a \((3,2)\) dichotomy type) with a wildly embedded \(2\)-dimensional unstable separatrix and a wildly embedded \(3\)-dimensional stable manifold. All other integral curves tend to these special integral curves as \(t\to \pm \infty\). We also construct other vector fields having \(k\geqslant 2\) special saddle integral curves with the tamely embedded \(2\)-dimensional unstable separatrices forming mildly wild frames in the sense of Debrunner–Fox. In the case of periodic vector fields, the corresponding specific integral curves are periodic with the period of the vector field, and are almost periodic in the case of an almost periodic vector field.
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Notes
Two diffeomorphisms \(f\) and \(g\) of a smooth manifold \(M\) are diffeotopic if they can be joined by a continuous arc \(F_t\) (\(F_0 = f\), \(F_1 = g\)) such that each \(F_t\) is a diffeomorphism of \(M\).
The term “fiber-wise” means the existence of a diffeomorphism \(\Psi\colon M_{f} \to M\times S^1\) acting as \((x,s)\to (\psi(x,s),s)\) .
We take the homotopy class \([c]\in\pi_1(S^2\times S^1)\) of a loop \(c\colon \mathbb R/\mathbb Z \to S^2\times S^1\). Then \(c\colon [0,1]\to S^2\times S^1\) lifts to a curve \(\bar{c}\colon [0,1]\to \mathbb R^3\setminus \{O\}\) joining \(x\) with the point \((a^{s})^n(x)\) for some \(n\in \mathbb Z\), where \(n\) is independent of the lift. We then set \(\eta^s_{_{S^2\times S^1}}([c])=n\).
A topological space is regular if and only if for any its point \(x\) and any its neighborhood \(U\), there is a closed neighborhood \(V\) of \(x\) such that \(V\subset U\).
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Funding
This research was supported by the Laboratory of Dynamical Systems and Applications of NRU HSE, Ministry of Science and Higher Education of the Russian Federation, grant No. 075-15-2019-1931. L. M. Lerman was also partially supported by the Ministry of Science and Higher Education of the Russian Federation under grant No. 0729-2020-0036.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 15–32 https://doi.org/10.4213/tmf10216.
Appendix Elements of uniform topology
For the convenience of the reader, we recall some notions from uniform topology. The main definitions of the theory of uniform spaces, with all the necessary details, can be found in [14].
A set \(X\) is called a uniform space if a collection \(\mathcal U\) of subsets is defined on \(X\times X\) such that the following conditions are satisfied (and \(\mathcal U\) is then called the uniformity).
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1.
Each element of \(\mathcal U\) contains the diagonal \(\Delta = \cup_{x\in X}\{(x,x)\}\).
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2.
If \(U\in \mathcal U\), then \(U^{-1}\in \mathcal U\), where \(U^{-1}\) is the set of all pairs \((y,x)\) for which \((x,y)\in U\).
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3.
For any \(U\in \mathcal U\), some \(V\in \mathcal U\) exists such that \(V\circ V \subset \mathcal U\), where \(V\circ V\) denotes composition: \((x,z)\in V\circ V\) if there is \(y\in X\) such that \((x,y)\in V\) and \((y,z)\in V\).
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4.
If \(U,V \in \mathcal U\), then \(U\cap V \in \mathcal U\).
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5.
If \(U \in \mathcal U\) and \(U\subset V \subset X\times X,\) then \(V\in \mathcal U\).
If \(X\) is a metric space with a metric \(d\), then item 1 corresponds to the property \(d(x,x)=0\), and item 2 corresponds to the symmetry of \(d\): \(d(x,y)=d(y,x)\). Property 3 is of the triangle inequality type: for each point of \(x\in X\) and any ball of radius \(r\) with centered at \(x\), a ball of the radius \(r/2\) centered at the same point must exist. Conditions 3 and 5 are similar to the axioms of neighborhoods near a point for the topology defined by uniformity.
The uniformity \(\mathcal U\) on a given set \(X\) can be defined in many ways, which gives different uniform spaces. This was used above where different uniform structures were defined on the set \(M\times \mathbb R\). if \((X,\mathcal U)\), \((Y, \mathcal V)\) are two uniform spaces, then the notion of a uniformly continuous map \(h\colon X\to Y\) is defined. Namely, a map \(h\colon X\to Y\) is uniformly continuous with respect to \(\mathcal U, \mathcal V\), if for any \(V\in \mathcal V\) the set \(\{(x,y)\mid (h(x),h(y))\in V\}\) belongs to \(\mathcal U\). If \(h\colon X\to Y\) is one-to-one and both \(h\) and \(h^{-1}\) are uniformly continuous, then \(h\) is called an equimorphism. In this case, the uniform spaces \((X,\mathcal U)\) and \((Y, \mathcal V)\) are said to be uniformly equivalent or equimorphic.
A uniformity \(\mathcal U\) on a set \(X\) making it a uniform space \((X,\mathcal U)\) endows \(X\) with a certain topology, making it a topological space. This space can have various topological properties. Conversely, each regularFootnote
A topological space is regular if and only if for any its point \(x\) and any its neighborhood \(U\), there is a closed neighborhood \(V\) of \(x\) such that \(V\subset U\).
topology \(\mathcal T\) on \(X\) is a uniform topology that corresponds to some uniformity, but such a uniformity is not unique in general. But if the topological space is compact and regular, then there is a unique uniformity generating the topology \(\mathcal T\).A topological space is regular if and only if for any its point \(x\) and any its neighborhood \(U\), there is a closed neighborhood \(V\) of \(x\) such that \(V\subset U\).
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Grines, V.Z., Lerman, L.M. Nonautonomous vector fields on \(S^3\): Simple dynamics and wild embedding of separatrices. Theor Math Phys 212, 903–917 (2022). https://doi.org/10.1134/S0040577922070029
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DOI: https://doi.org/10.1134/S0040577922070029