On 4-dimensional flows with wildly embedded invariant manifolds of a periodic orbit

Olga V. Pochinka 1 Β and Danila D. Shubin 1
  • 1 , 25/12, Bolshaya Pecherskaya St., Nizhny Novgorod, Russia
Olga V. Pochinka and Danila D. Shubin

Abstract

In the present paper we construct an example of 4-dimensional flows on π•Š3 Γ— π•Š1 whose saddle periodic orbit has a wildly embedded 2-dimensional unstable manifold. We prove that such a property has every suspension under a non-trivial Pixton's diffeomorphism. Moreover we give a complete topological classification of these suspensions.

1 Introduction and statement of results

Qualitative study of dynamical systems reveals various topological constructions naturally emerged in the modern theory. For example, the Cantor set with cardinality of continuum and Lebesgue measure zero as an expanding attractor or an contracting repeller. Also, a curve in 2-torus with an irrational rotation number, which is not a topological submanifold but is an injectively immersed subset, can be found being invariant manifold of the Anosov toral diffeomorphism's fixed point.

Another example of linkage between topology and dynamics is the Fox-Artin arc [4] appeared in work by D. Pixton [9] as the closure of a saddle separatrix of a Morse-Smale diffeomorphism on the 3-sphere. A wild behaviour of the Fox-Artin arc complicates the classification of dynamical systems, there is no combinatorial description as Peixoto's graph [8] for 2-dimensional Morse-Smale flows.

It is well known that there are no wild arcs in dimension 2. They exist in dimension 3 and can be realized as invariant sets for discrete dynamics, unlike regular 3-dimensional flows, which do not possess wild invariant sets. The dimension 4 is very rich. Here appear wild objects for both discrete and continuous dynamics. Although there are no wild arcs in this dimension, there are wild objects of co-dimension 1 and 2. So, the closure of 2-dimensional saddle separatrix can be wild for 4-dimensional Morse-Smale system (a diffeomorphism or a flow). Such examples have been recently constructed by V. Medvedev and E. Zhuzoma [6]. T. Medvedev and O. Pochinka [7] have shown that the wild Fox-Artin 2-dimension sphere appears as closure of heteroclinic intersection of Morse-Smale 4-diffeomorphism.

In the present paper we prove that the suspension under a non-trivial Pixton's diffeomorphism provides a 4-flow with wildly embedded 3-dimensional invariant manifold of a periodic orbit. Moreover, we show that there are countable many different wild suspensions. In more details.

Denote by 𝒫 the class of the Morse-Smale diffeomorphisms of 3-sphere S3 whose non-wondering set consists of the fixed source Ξ±, the fixed saddle Οƒ and the fixed sinks Ο‰1, Ο‰2. Class 𝒫 diffeomorphism phase portrait is shown in Figure 1.

Fig. 1
Fig. 1

The phase portrait of a diffeomorphism of class 𝒫

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00049

As the Pixton's example belongs to this class we call it the Pixton class. That example is characterized by the wild embedding of the stable manifold Wσs , namely its closure is not locally flat at α. We call such diffeomorphism non-trivial (see Figure 2).

Fig. 2
Fig. 2

The phase portrait of a non-trivial diffeomorphism of class 𝒫

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00049

Let 𝒫t be a set of flows which are suspensions on Pixton's diffeomorphisms. By the construction the ambient manifold for every such flow ft is diffeomorphic to S3 Γ— S1 and the non-wandering set consists of exactly four periodic orbits π’ͺΞ±, π’ͺΟƒ, π’ͺΟ‰1, π’ͺΟ‰2. Let Wπ’ͺΟƒs denote stable manifold of the saddle orbit. In the present paper we prove the following theorems.

Theorem 1. IfWΟƒsis a wild for f ∈ 𝒫 thenWπ’ͺΟƒsis a wild for ft ∈ 𝒫t.

Corollary 2. (Existence theorem) There is a flow ft with saddle orbit π’ͺΟƒ such thatcl(Wπ’ͺΟƒs)is wild.

Theorem 3. Two flows ft, fβ€²t ∈ 𝒫t are topologically equivalent iff the diffeomorphisms f, fβ€² ∈ 𝒫 are topologically conjugated.

The complete classification of diffeomorphisms from the class 𝒫 has been done by Ch. Bonatti and V. Grines [1]. They proved that a complete invariant for Pixton's diffeomorphism is an equivalent class of the embedding of a knot in S2 Γ— S1. In section 4 we briefly give another idea to classify such systems. It was described in [5] and led to complete classification on Morse-Smale 3-diffeomorphisms in [2].

Acknowledgement: 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. No. 075-15-2019-1931. The auxiliary facts was implemented in the framework of the Basic Research Program at the National Research University Higher School of Economics (HSE University) in 2019.

2 Auxiliary facts

2.1 Dynamical concepts

Diffeomorphism f : Mn β†’ Mn of smooth closed connected orientable n-manifold (n β‰₯ 1)Mn is called Morse-Smale diffeomorphism (f ∈ MS(Mn)) if:

  1. Non-wandering set Ξ©f is finite and hyperbolic;
  2. Stable and unstable manifolds Wps , Wqu intersect transversally for any periodic points p, q.

Two diffeomorphisms f, fβ€² are called topologically conjugated if there exists a homeomorphism h : Mn β†’ Mn such that f h = h fβ€².

Let f : Mn β†’ Mn be a diffeomorphism. Let Ο†t be a flow on the manifold Mn Γ— ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is
Ο†t(x,r)=(x,r+t).
Let g : Mn Γ— ℝ β†’ Mn Γ— ℝ be a diffeomorphism given by the formula g(x, r) = (f (x), r βˆ’ 1). Let G = {gk, k ∈ β„€} and W = (Mn Γ— ℝ)/G. Denote pW: Mn Γ— ℝ β†’ W the natural projections. It is verified directly that gΟ†t = Ο†tg. Then the map ft : W β†’ W given by the formula
ft(x)=pW(Ο†t(pWβˆ’1(x)))
is a well-defined flow on W which is called the suspension of f.

When f ∈ MS(Mn) the non-wandering set of the suspension ft consist of a finite number of periodic orbits composed by pW(Ξ©f Γ— ℝ). The obtained flow is so-called non-singular, what means it has no singular points.

Two flows ft, fβ€²t are called topologically equivalent if exists a homeomorphism h : W β†’ W which maps the trajectories of ft to trajectories of fβ€²t and preservs orientation on the trajectories.

2.2 Topological concepts

A closed subset X of a PL-manifold N is said to be tame if there is a homeomorphism h : N β†’ N such that h(X) is a subpolyhedron; the other are called wild.

For example, Fox-Artin arc is wild (see [4]).

Let A be a closed subset of a metric space X. A is called locally k-co-connected in X at a ∈ A (k-LCC at a) if each neighbourhood U of a in X contains a smaller neighbourhood V of a such that each map βˆ‚Ik+1 β†’ V \ A extends to a map Ik+1 β†’ U \ A.

We say that A is locally k-co-connected (k-LCC in X) if A is k-LCC at a for each a ∈ A.

For example, Fox-Artin 2-sphere is not 1-LCC (see Exercise 2.8.1 [3]).

Let e: Mm β†’ Nn be a topological embedding of m-dimensional manifold Mm with a boundary in n-manifold Nn (n β‰₯ m). e is called locally flat at x ∈ Mm (and e(Mm) is locally flat at e(x)) if there exist a neighbourhood U of e(x) ∈ Nn and a homeomorphism h of U onto ℝn such that:

  1. (1)h(U ∩ e(Mm)) = ℝm βŠ‚ ℝnwhen x ∈ int Mm or
  2. (2)h(U∩e(Mm))=ℝ+mβŠ‚β„n when x ∈ βˆ₯Mm.

Since tameness implies local flatness for embeddings of manifolds in all co-dimensionals except two, we will say that e: Mm β†’ Nn, m β‰  n βˆ’ 2 is wild at e(x) when e(Mm) is fails to be locally flat at e(x).

Proposition 4 (Proposition 1.3.1 [3]). Suppose the manifold Mnβˆ’1is locally flatly embedded in the n-manifold Nn. Then Mnβˆ’1is k-LCC in Nn for all k β‰₯ 1.

Proposition 5 (Proposition 1.3.6 [3]). Suppose Y is a locally contractible space and A βŠ‚ X. Then A is k-LCC in X iff A Γ— Y is k-LCC in X Γ— Y.

Notice that any manifold is a locally contractible space.

3 Wildness of the stable manifold of the saddle periodic orbit for the suspension

Proof of Theorem 1.

Let f be a non-trivial Pixton's diffeomorphism. Then the closure of the stable manifold WΟƒs of the saddle point Οƒ is a wild 2-sphere in S3 and it is not 1-LCC at a source Ξ±. By the construction the circle Οƒ Γ— S1 in S3 Γ— S1 coincides with the saddle periodic orbit π’ͺΟƒ for the suspension ft of the diffeomorphism f. Moreover, the closure of stable separatrice Wπ’ͺΟƒs of π’ͺΟƒ coincides with cl(WΟƒs)Γ—S1 and it is a 3-manifold homeomorphic to S2 Γ— S1. Due to Proposition 5 the set cl(Wπ’ͺΟƒs) is not 1-LCC in S3 Γ— S1. Thus, by Proposition 4, Wπ’ͺΟƒs is wild.

4 Topological classification of suspensions

Firstly we give a brief idea of the topological classification of diffeomorphisms from class 𝒫.

4.1 Classification of diffeomorphisms from 𝒫

Let f ∈ 𝒫 and Vf=WΞ±u\Ξ± . Denote by V^f the orbit space with respect to f in Vf and by pf:Vfβ†’V^f the natural projection. According to [5], the space V^f is diffeomorphic to S2 Γ— S1 and the projection pf is a covering map which induces an epimorphism Ξ·f:Ο€1(V^f)β†’β„€ . Let L^fs=pf(WΟƒs\Οƒ) . According to [5], L^fs is a homotopically non-trivial 2-dimensional torus in V^f (see Figure 4).

Fig. 3
Fig. 3

The complete invariants for trivial and non-trivial Pixton's diffeomorphisms

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00049

Fig. 4
Fig. 4

The vector field and Σ˜ on π•Š3 Γ— ℝ

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00049

Proposition 6 (Theorem 4.5 [5]). Diffeomorphisms f, fβ€² ∈ 𝒫 are topologically conjugated iff the toriL^fs , L^fβ€²sare equivalent (that is there is a homeomorphismh^:V^fβ†’V^fβ€²such thath^(L^fs)=L^fβ€²sand Ξ·f = Ξ·fβ€² Δ₯*).

4.2 Proof of the sufficiency of Theorem 3

Let f, fβ€² ∈ 𝒫. Recall the notion of the suspensions of f, fβ€².

Let Ο†t be a flow on the manifold S3 Γ— ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is
Ο†t(x,r)=(x,r+t).

Let g, gβ€² : S3 Γ— ℝ β†’ S3 Γ— ℝ be diffeomorphisms given by the formulas g(x, r) = (f (x), r βˆ’ 1), gβ€²(x, r) = (fβ€² (x), r βˆ’ 1). Let G = {gk, k ∈ β„€}, Gβ€² = {gβ€²k, k ∈ β„€} and W = (S3 Γ— ℝ)/G, Wβ€² = (S3 Γ— ℝ)/Gβ€². Since f, f β€² preserve orientation of S3, W, Wβ€² are diffeomorphic to S3 Γ— S1. Denote pW : S3 Γ— ℝ β†’ W, pWβ€² : S3 Γ— ℝ β†’ W β€² the natural projections. It is verified directly that gΟ†t = Ο†tg, gβ€²Ο†t = Ο†tgβ€². Then maps ft : W β†’ W, fβ€²t : Wβ€² β†’ Wβ€² given by the formulas ft(x)=pW(Ο†t(pWβˆ’1(x))) , fβ€²t(x)=pW'(Ο†t(pW'βˆ’1(x))) are well-defined flows on W, Wβ€² which are called the suspensions of f, fβ€², respectively, that is ft, fβ€²t ∈ 𝒫t.

Now let f, fβ€² ∈ 𝒫 be topologically conjugate by the homeomorphism h : S3 β†’ S3. Define a homeomorphism H˜:S3×ℝ→S3×ℝ by the formula
H˜(x,r)=(h(x),r),β€Š(x,r)∈S3×ℝ.
Directly verifies that H˜g=gβ€²H˜ , then H˜ can be projected as a homeomorphism H : W β†’ Wβ€² by the formula
H=pW'H˜pWβˆ’1.

Since HΛœΟ†t=Ο†β€²tH˜ , then H ft = fβ€²t H. Thus H is a required homeomorphism which realizes an equivalency of the suspensions ft and fβ€²t.

4.3 Proof of necessity of Theorem 3

Let suspensions ft, fβ€²t be topologically equivalent by means of a homeomorphism H : S3 Γ— S1 β†’ S3 Γ— S1. Let us prove that then the diffeomorphisms f, fβ€² are topologically conjugate.

For this aim recall that the diffeomorphisms f, fβ€² in the basins of sources Ξ±, Ξ±β€² are topologically conjugate by homeomorphisms hΞ±:WΞ±u→ℝ3 , β€ŠhΞ±β€²:WΞ±β€²u→ℝ3 with the linear extension a: ℝ3 β†’ ℝ3 given by the formula
a(x1,x2,x3)=(2x1,2x2,2x3).
Let Sr={(x1,x2,x3)βˆˆβ„3:x12+x22+x32=2r,rβˆˆβ„} , SΞ±r=hΞ±βˆ’1(Sr) and SΞ±β€²r=hΞ±β€²βˆ’1(Sr) . Define cylinders Σ˜,Ξ£Λœβ€²βŠ‚S3×ℝ by the formulas
Σ˜={(x,r)∈S3×ℝ:x∈SΞ±r,rβˆˆβ„},β€ŠΞ£Λœβ€²={(x,r)∈S3×ℝ:x∈SΞ±β€²r,rβˆˆβ„}.

It follows from the definition of suspension that Σ˜ , Ξ£Λœβ€² are sections for trajectories of Ο†t, Ο†β€²t passing through VΟ†t, VΟ†β€²t, where VΟ†t=WOΞ±u\OΞ± and VΟ†β€²t=WOΞ±β€²u\OΞ±β€² and WOΞ±u,WOΞ±β€²u are unstable manifolds of orbits OΞ±, OΞ±β€² of flows Ο†t, Ο†β€²t respectively. Let Vft=Wπ’ͺΞ±u , Vfβ€²t=Wπ’ͺΞ±β€²u . Then Ξ£=pW(Σ˜) , β€ŠΞ£β€²=pW'(Ξ£Λœβ€²) are homeomorphic to S2 Γ— S1 and are sections for trajectories of flows ft, fβ€²t in Vft, Vfβ€²t, respectively.

Since H realizes an equivalence of the flows ft, fβ€²t then H(Ξ£) is also a section for trajectories of the flows fβ€²t in Vfβ€²t. Thus we can get Ξ£β€² from H(Ξ£) by a continuous shift along the trajectories, that is there is a homeomorphism ψ : Vfβ€²t β†’ Vfβ€²t which preserves the trajectories of fβ€²t in Vfβ€²t and such that ψ(H(Ξ£)) = Ξ£β€². Let hΞ£ = ψH|Ξ£ : Ξ£ β†’ Ξ£β€².

Then the homeomorphism hΞ£ has a lift hΣ˜:Ξ£Λœβ†’Ξ£Λœβ€² which is a homeomorphism such that hΞ£=pW'hΣ˜pWβˆ’1 . Let us introduce the canonical projection q : S3 Γ— ℝ β†’ S3 by the formula q(x, r) = x and define a homeomorphism h : Vf β†’ Vfβ€² by the formula
hq|Σ˜=qhΣ˜.
By the construction the homeomorphism h conjugates f|Vf with fβ€²|Vfβ€². Since H(Wπ’ͺΟƒs)=Wπ’ͺΟƒβ€²s , then h(WΟƒs\Οƒ)=WΟƒβ€²s\Οƒβ€² . Let us define a homeomorphism h^:V^fβ†’V^fβ€² by the formula
h^pf=pf'h.

Then h^(L^fs)=L^fβ€²s and Ξ·f = Ξ·fβ€²Δ₯*. Thus, by Proposition 6, the diffeomorphisms f, fβ€² are topologically conjugated.

References

  • [1]↑

    C. Bonatti and V. Grines (2000), Knots as topological invariant for gradient-like diffeomorphisms of the sphere S3, // J. Dyn. Control Syst. 6:4, 579–602.

    • Crossref
    • Export Citation
  • [2]↑

    Bonatti C., Grines V. (2019), Pochinka O. Topological classification of Morse-Smale diffeomorphisms on 3-manifolds // Duke Mathematical Journal. 2019. Vol. 168. No. 13. P. 2507–2558.

    • Crossref
    • Export Citation
  • [3]↑

    R. J. Daverman G. A. Venema (2009), Embedding in Manifolds. Graduate studies in mathematics, v. 106. http://www.calvin.edu/~venema/embeddingsbook/

  • [4]↑

    Fox R, Artin E (1948). Some wild cells and spheres in three-dimensional space. // Ann of Math. 1948, 979–990.

  • [5]↑

    V.Z. Grines, T.V. Medvedev, O.V. Pochinka. (2016), Dynamical Systems on 2- and 3-Manifolds. Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp.

  • [6]↑

    V.S. Medvedev, E.V. Zhuzhoma (2013), Morse–Smale systems with few non-wandering points. Topology and its Applications, 160 (2013) 498–507.

    • Crossref
    • Export Citation
  • [7]↑

    T.V. Medvedev, O.V. Pochinka (2018). The wild Fox-Artin arc in invariant sets of dynamical systems // Dynamical Systems. 2018. Vol. 33. No. 4. P. 660–666.

    • Crossref
    • Export Citation
  • [8]↑

    M.M. Peixoto (1973). On the classification of flows on 2-manifolds, Dynamical systems, Proc. Sympos. (Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 389–419

  • [9]↑

    Pixton D (1977). Wild unstable manifolds. Topology. 1977, 16(2), 167–172.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    C. Bonatti and V. Grines (2000), Knots as topological invariant for gradient-like diffeomorphisms of the sphere S3, // J. Dyn. Control Syst. 6:4, 579–602.

    • Crossref
    • Export Citation
  • [2]

    Bonatti C., Grines V. (2019), Pochinka O. Topological classification of Morse-Smale diffeomorphisms on 3-manifolds // Duke Mathematical Journal. 2019. Vol. 168. No. 13. P. 2507–2558.

    • Crossref
    • Export Citation
  • [3]

    R. J. Daverman G. A. Venema (2009), Embedding in Manifolds. Graduate studies in mathematics, v. 106. http://www.calvin.edu/~venema/embeddingsbook/

  • [4]

    Fox R, Artin E (1948). Some wild cells and spheres in three-dimensional space. // Ann of Math. 1948, 979–990.

  • [5]

    V.Z. Grines, T.V. Medvedev, O.V. Pochinka. (2016), Dynamical Systems on 2- and 3-Manifolds. Dev. Math., 46, Springer, Cham, 2016, xxvi+295 pp.

  • [6]

    V.S. Medvedev, E.V. Zhuzhoma (2013), Morse–Smale systems with few non-wandering points. Topology and its Applications, 160 (2013) 498–507.

    • Crossref
    • Export Citation
  • [7]

    T.V. Medvedev, O.V. Pochinka (2018). The wild Fox-Artin arc in invariant sets of dynamical systems // Dynamical Systems. 2018. Vol. 33. No. 4. P. 660–666.

    • Crossref
    • Export Citation
  • [8]

    M.M. Peixoto (1973). On the classification of flows on 2-manifolds, Dynamical systems, Proc. Sympos. (Univ. Bahia, Salvador, 1971), Academic Press, New York, 1973, 389–419

  • [9]

    Pixton D (1977). Wild unstable manifolds. Topology. 1977, 16(2), 167–172.

    • Crossref
    • Export Citation
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  • View in gallery

    The phase portrait of a diffeomorphism of class 𝒫

  • View in gallery

    The phase portrait of a non-trivial diffeomorphism of class 𝒫

  • View in gallery

    The complete invariants for trivial and non-trivial Pixton's diffeomorphisms

  • View in gallery

    The vector field and Σ˜ on π•Š3 Γ— ℝ