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Introduction
A-flows were introduced by Smale [15] (see basic definitions bellow). This class of flows contains structurally stable flows including Morse-Smale flows and Anosov flows. Recall that a Morse-Smale flow has a non-wandering set consisting of finitely many hyperbolic periodic trajectories and hyperbolic singularities, while any Anosov flow has hyperbolic structure on the whole supporting manifold. A-flows have hyperbolic non-wandering sets that are the topological closure of periodic trajectories. In a sense, A-flows with nontrivial and trivial pieces (basic sets) of non-wandering sets take an intermediate place between Morse-Smale and Anosov flows. We see that A-flows form an important class containing flows with regular and chaotic dynamics.
Due to Smale's Spectral Theorem, a non-wandering set of A-flow is a disjoint union of closed transitive invariant pieces called basic sets. A basic set is called trivial if it is either an isolated fixed point or isolated periodic trajectory.
A nontrivial basic set Ω is called expanding if its topological dimension coincides with the dimension of unstable manifold at each point of Ω. Due to Williams [16], an expanding attractor consists of unstable manifolds of its points. Moreover, the unstable manifolds of points of expanding attractor form a lamination whose leaves are planes and cylinders. In addition, this lamination is locally homeomorphic to the product of Cantor set and Euclidean space of dimension at least two. Thus, the minimal topological dimension of expanding attractor equals two. Therefore, the minimal dimension of manifold supporting two-dimension expanding attractors equals three. It is natural to study the following question : what manifolds admit A-flows with 2-dimensional expanding attractors ?
In the paper, we consider closed 3-manifolds supporting A-flows with 2-dimensional expanding attractors. The main result of the paper is the following statement.
Theorem 1
Given any closed 3-manifold M3, there is an A-flow fton M3such that the non-wandering set NW (ft) consists of a two-dimensional non-orientable expanding attractor and trivial basic sets.
This result contrasts with the case for 3-dimensional A-diffeomorphisms. To be precise, it follows from [8,9,12,17] that if a closed 3-manifold M3 admits an A-diffeomorphism with 2-dimensional expanding attractor, then π1(M3) ≠ 0. Note that though there are Anosov flows with 2-dimensional expanding attractors [1], the A-flow to be constructed in Theorem 1 will not be necessarily Anosov. In fact, Margulis [11] proved that the fundamental group π1(M3) of closed 3-manifold M3 supporting Anosov flows has an exponential growth. In the end of the paper, we discuss the result and formulate some conjectures.
Acknowledgments. We would like to thank Misha Malkin. The authors are partially supported by Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF, grant ag. 075-15-2019-1931.
Basic definitions
Let ft be a smooth flow on a closed n-manifold Mn, n ≥ 3. A subset L ⊂ Mn = M is invariant provided L consists of trajectories of ft. An invariant nonsingular set Λ ⊂ M is called hyperbolic if the sub-bundle TΛM of the tangent bundle T M can be represented as a Dft-invariant continuous splitting
E_\Lambda ^{ss} \oplus E_\Lambda ^t \oplus E_\Lambda ^{uu}
such that
E_\Lambda ^t
is the line bundle tangent to the trajectories of the flow ft);
there are Cs > 0, Cu > 0, 0 < λ < 1 such that
\parallel d{f^t}(v)\parallel \le {C_s}{\lambda ^t}\parallel v\parallel ,\quad v \in E_\Lambda ^{ss},\quad t > 0,\parallel d{f^{ - t}}(v)\parallel \le {C_u}{\lambda ^t}\parallel v\parallel ,\quad v \in E_\Lambda ^{uu},\quad t > 0.
If x ∈ Λ is a fixed point of hyperbolic L, then x is an isolated hyperbolic equilibrium state. The topological structure of flow near x is described by Grobman-Hartman theorem, see for example [13]. In this case
E_x^t = 0
and
\dim E_\Lambda ^{ss} + \dim E_\Lambda ^{uu} = n
.
If hyperbolic L does not contain fixed points, then the bundles
E_\Lambda ^{uu} \oplus E_\Lambda ^1 = E_\Lambda ^u,\quad E_\Lambda ^{ss} \oplus E_\Lambda ^1 = E_\Lambda ^s,\quad E_\Lambda ^{uu},\quad E_\Lambda ^{ss}
are uniquely integrable [5], [15]. The corresponding leaves
{W^u}(x),\quad {W^s}(x),\quad {W^{uu}}(x),\quad {W^{ss}}(x)
through a point x ∈ λ are called unstable, stable, strongly unstable, and strongly stable manifolds.
Given a set U ⊂ Mn, denote by ft0 (U) the shift of U along the trajectories of ft on the time t0. Recall that a point x is non-wandering if given any neighborhood U of x and a number T0, there is t0 ≥ T0 such that U ∩ ft0 (U) ≠ ∅. The non-wandering set NW (ft) of ft is the union of all non-wandering point.
Denote by Fix (ft) the set of fixed points of flow ft. Following Smale [15], we call ft an A-flow provided its non-wandering set NW (ft) is hyperbolic and the periodic trajectories are dense in NW (ft) \ Fix (ft). It is well known [10, 15] that that the non-wandering set NW (ft) of A-flow ft is a disjoint union of closed, and invariant, and transitive sets called basic sets. Following Williams [16], we’ll call a basic set Ω an expanding attractor provided Ω is an attractor and its topological dimension equals the dimension of unstable manifold Wu(x) for every points x ∈ Ω. A basic set L is called orientable provided the fiber bundles
E_\Lambda ^{ss}
and
E_\Lambda ^{uu}
are orientable. Note that if
E_\Lambda ^{ss}
and
E_\Lambda ^{uu}
are one-dimensional, then the orientability of L means that the both
E_\Lambda ^{ss}
and
E_\Lambda ^{uu}
can be embedded in vector fields on Mn.
Recall that an A-flow is a Morse-Smale flow provided its non-wandering set is the union of finitely many singularities and periodic trajectories. Clearly that all basic sets of Morse-Smale flow are trivial.
Proof of the main result
We begin with previous results which are interesting itself. Recall that any closed manifold admits a Morse-Smale flow with a source that is a repelling fixed point. In particular, any closed manifold admits a gradient-like Morse-Smale flow having at least one source and sink [14]. The following result says that any closed 3-manifold admits a Morse-Smale flow with one-dimensional repelling periodic trajectory.
Lemma 2
Given any closed 3-manifolds M3, there is a Morse-Smale flow ftwith repelling isolated periodic trajectory on M3.
Sketch of the proof
Take a gradient-like Morse-Smale flow
f_0^t
with a sink, say ω, on M3. Let U(ω) be a neighborhood of ω. Without loss of generality, we can suppose that U(ω) is a ball. Since ω is a hyperbolic fixed point, one can assume that the boundary ∂U(ω) is a smooth sphere that is transversal to the trajectories. This means that the vector field inducing the flow
f_0^t
is directed inside of U(ω). Let us introduce coordinates (x,y,z) and the corresponding cylinder coordinates (∂, φ, z) in U(ω) smoothly connected with the original coordinates in U(ω). Consider the system
\dot \rho = \rho \cdot (1 - \rho ),\quad \dot \phi = 1,\quad \dot z = - z.
It is easy to check that this system has an attractive hyperbolic trajectory and the saddle fixed point at the origin. Reversing time, one gets the repelling trajectory.
Lemma 3
There is an A-flow on S2 × S1such that the spectral decomposition of ftconsists of two-dimensional (non-orientable) expanding attractor Λaand four isolated hyperbolic repelling trajectories. Moreover, there is a neighborhood P of Λahomeomorphic to the solid torus S1 × D2such that the boundary ∂ P = S1 × S1is transversal to the trajectories which enter inside of P as the time parameter increases.
Proof
Take an A-diffeomorphism f : S2 → S2 whose the spectral decomposition consists of a Plykin attractor Λ0 and four hyperbolic sources. Due to Plykin [12], such diffeomorphism exists. Contemporary construction of Plykin attractor can be found in [7]. Since Plykin attractor is one-dimensional, Λ0 is an expanding attractor. Without loss of generality, one can assume that f is a preserving orientation diffeomorphism (otherwise, one takes f2).
Let sust(f) be the dynamical suspension over f. Since f is an A-diffeomorphism, sust(f) is an A-flow. Obviously, the spectral decomposition of f corresponds to the spectral decomposition of sust(f). Because of f preserves orientation, the supporting manifold for sust(f) is homeomorphic to S2 × S1. Since Λ0 is a one-dimensional expanding attractor, sust(f) has a two dimensional expanding attractor denoted by Λa.
Take an isolated hyperbolic repelling trajectory γ that corresponds to some source of f. Since γ is a repelling trajectory, there is a neighborhood V (γ) homeomorphic to a solid torus such that the boundary ∂V (γ) is transversal to the trajectories of sust(f), so that the trajectories move outside of V (γ) as the time parameter increases. This follows that the interior of (S2 × S1) \ V (γ) is the neighborhood, say P, of Λa such that the boundary ∂ P = S1 × S1 is transversal to the trajectories which enter inside of P as a time parameter increases.
Since f is an orientation preserving diffeomorphism of the sphere S2, f is homotopic to the identity. This implies that P is homeomorphic to the solid torus S1 × D2. Hence, sust(f) = ft is a desired flow.
Proof of Theorem 1
Let M3 be a closed 3-manifold. According to Lemma 2, there is a Morse-Smale flow ϕt with repelling isolated periodic trajectory l on M3. Hence, there is a neighborhood U(l) of l such that U(l) is homeomorphic to the interior of the solid torus S1 × D2, and the boundary ∂U(l) is transversal to the trajectories of ϕt, so that the trajectories move outside of U(l) as a time parameter increases. This follows that U(l) can be replaced by the solid torus P satisfying Lemma 3. As a consequence, one gets the A-flow with two-dimensional (non-orientable) expanding attractor Λa. This completes the proof.
Conclusions
It follows from the proof of the main result that the 2-dimensional expanding attractor satisfying Theorem 1 is non-orientable. We suggest the following conjecture.
Conjecture 4.Given any closed 3-manifold M3, there is an A-flow fton M3such that the non-wandering set NW (ft) contains an orientable two-dimensional expanding attractor.
Two-dimensional expanding attractors are evidence of chaotic dynamics. However, it seems that the following conjecture holds because of Plykin diffeomorphism is structurally stable.
Conjecture 5.Given any closed 3-manifold M3, there is a structurally stable flow fton M3such that the non-wandering set NW (ft) consists of a two-dimensional expanding attractor and trivial basic sets.
Note that a structurally stable flow automatically is an A-flow.
