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

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 α, the fixed saddle σ and the fixed sinks ω₁, ω₂. Class 𝒫 diffeomorphism phase portrait is shown in Figure 1.

The phase portrait of a diffeomorphism of class 𝒫

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 $W_{σ}^{s}$ W_\sigma^s , namely its closure is not locally flat at α. We call such diffeomorphism non-trivial (see Figure 2).

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

Let 𝒫^t be a set of flows which are suspensions on Pixton's diffeomorphisms. By the construction the ambient manifold for every such flow f^t is diffeomorphic to S³ × S¹ and the non-wandering set consists of exactly four periodic orbits 𝒪_α, 𝒪_σ, 𝒪_ω₁, 𝒪_ω₂. Let $W_{𝒪_{σ}}^{s}$ W_{{{\cal O}_\sigma}}^s denote stable manifold of the saddle orbit. In the present paper we prove the following theorems.

Theorem 1. If $W_{σ}^{s}$ W_\sigma^sis a wild for f ∈ 𝒫 then $W_{𝒪_{σ}}^{s}$ W_{{{\cal O}_\sigma}}^sis a wild for f^t ∈ 𝒫^t.

Corollary 2. (Existence theorem) There is a flow f^t with saddle orbit 𝒪_σ such that $cl (W_{𝒪_{σ}}^{s})$ cl(W_{{{\cal O}_\sigma}}^s)is wild.

Theorem 3. Two flows f^t, f′^t ∈ 𝒫^t are topologically equivalent iff the diffeomorphisms f, f′ ∈ 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 × 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 → Mⁿ of smooth closed connected orientable n-manifold (n ≥ 1)Mⁿ is called Morse-Smale diffeomorphism (f ∈ MS(Mⁿ)) if:

Non-wandering set Ω_f is finite and hyperbolic;

Stable and unstable manifolds $W_{p}^{s}$ W_p^s , $W_{q}^{u}$ W_q^u intersect transversally for any periodic points p, q.

Two diffeomorphisms f, f′ are called topologically conjugated if there exists a homeomorphism h : Mⁿ → Mⁿ such that f h = h f′.

Let f : Mⁿ → Mⁿ be a diffeomorphism. Let φ^t be a flow on the manifold Mⁿ × ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is $φ^{t} (x, r) = (x, r + t) .$ {\varphi^t}(x,r) = (x,r + t).

Let g : Mⁿ × ℝ → Mⁿ × ℝ be a diffeomorphism given by the formula g(x, r) = (f (x), r − 1). Let G = {g^k, k ∈ ℤ} and W = (Mⁿ × ℝ)/G. Denote p_W: Mⁿ × ℝ → W the natural projections. It is verified directly that gφ^t = φ^tg. Then the map f^t : W → W given by the formula $f^{t} (x) = p_{W} (φ^{t} (p_{W}^{- 1} (x)))$ {f^t}(x) = {p_W}({\varphi^t}(p_W^{- 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 f^t consist of a finite number of periodic orbits composed by p_W(Ω_f × ℝ). The obtained flow is so-called non-singular, what means it has no singular points.

Two flows f^t, f′^t are called topologically equivalent if exists a homeomorphism h : W → W which maps the trajectories of f^t to trajectories of f′^t 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 → 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 ∂I^k+1 → V \ A extends to a map I^k+1 → 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: M^m → Nⁿ be a topological embedding of m-dimensional manifold M^m with a boundary in n-manifold Nⁿ (n ≥ m). e is called locally flat at x ∈ M^m (and e(M^m) 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)

h(U ∩ e(M^m)) = ℝ^m ⊂ ℝⁿwhen x ∈ int M^m or

(2)

$h (U \cap e (M^{m})) = ℝ_{+}^{m} \subset ℝ^{n}$ h(U \cap e({M^m})) = \mathbb {R}_+^m \subset \mathbb {R}^n when x ∈ ∥M^m.

Since tameness implies local flatness for embeddings of manifolds in all co-dimensionals except two, we will say that e: M^m → Nⁿ, m ≠ n − 2 is wild at e(x) when e(M^m) is fails to be locally flat at e(x).

Proposition 4 (Proposition 1.3.1 [3]). Suppose the manifold Mⁿ⁻¹is locally flatly embedded in the n-manifold Nⁿ. Then Mⁿ⁻¹is k-LCC in Nⁿ for all k ≥ 1.

Proposition 5 (Proposition 1.3.6 [3]). Suppose Y is a locally contractible space and A ⊂ X. Then A is k-LCC in X iff A × Y is k-LCC in X × 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 $W_{σ}^{s}$ W_\sigma^s of the saddle point σ is a wild 2-sphere in S³ and it is not 1-LCC at a source α. By the construction the circle σ × S¹ in S³ × S¹ coincides with the saddle periodic orbit 𝒪_σ for the suspension f^t of the diffeomorphism f. Moreover, the closure of stable separatrice $W_{𝒪_{σ}}^{s}$ W_{{{\cal O}_\sigma}}^s of 𝒪_σ coincides with $cl (W_{σ}^{s}) \times S^{1}$ cl(W_\sigma^s) \times {S^1} and it is a 3-manifold homeomorphic to S² × S¹. Due to Proposition 5 the set $cl (W_{𝒪_{σ}}^{s})$ cl(W_{{{\cal O}_\sigma}}^s) is not 1-LCC in S³ × S¹. Thus, by Proposition 4, $W_{𝒪_{σ}}^{s}$ W_{{{\cal O}_\sigma}}^s 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 $V_{f} = W_{α}^{u} \ α$ {V_f} = W_\alpha^u\backslash \alpha . Denote by ${\hat{V}}_{f}$ {\hat V_f} the orbit space with respect to f in V_f and by $p_{f} : V_{f} \to {\hat{V}}_{f}$ {p_f}:{V_f} \to {\hat V_f} the natural projection. According to [5], the space ${\hat{V}}_{f}$ {\hat V_f} is diffeomorphic to S² × S¹ and the projection p_f is a covering map which induces an epimorphism $η_{f} : π_{1} ({\hat{V}}_{f}) \to ℤ$ {\eta_f}:{\pi_1}({\hat V_f}) \to \mathbb {Z} . Let ${\hat{L}}_{f}^{s} = p_{f} (W_{σ}^{s} \ σ)$ \hat L_f^s = {p_f}(W_\sigma^s\backslash \sigma) . According to [5], ${\hat{L}}_{f}^{s}$ \hat L_f^s is a homotopically non-trivial 2-dimensional torus in ${\hat{V}}_{f}$ {\hat V_f} (see Figure 4).

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

The vector field and $\tilde{Σ}$ \tilde \Sigma on 𝕊³ × ℝ

Proposition 6 (Theorem 4.5 [5]). Diffeomorphisms f, f′ ∈ 𝒫 are topologically conjugated iff the tori ${\hat{L}}_{f}^{s}$ \hat L_f^s , ${\hat{L}}_{f^{'}}^{s}$ \hat L_{f'}^sare equivalent (that is there is a homeomorphism $\hat{h} : {\hat{V}}_{f} \to {\hat{V}}_{f^{'}}$ \hat h:{\hat V_f} \to {\hat V_{f'}}such that $\hat{h} ({\hat{L}}_{f}^{s}) = {\hat{L}}_{f^{'}}^{s}$ \hat h(\hat L_f^s) = \hat L_{f'}^sand η_f = η_f′ ĥ_*).

4.2

Proof of the sufficiency of Theorem 3

Let f, f′ ∈ 𝒫. Recall the notion of the suspensions of f, f′.

Let φ^t be a flow on the manifold S³ × ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is $φ^{t} (x, r) = (x, r + t) .$ {\varphi^t}(x,r) = (x,r + t).

Let g, g′ : S³ × ℝ → S³ × ℝ be diffeomorphisms given by the formulas g(x, r) = (f (x), r − 1), g′(x, r) = (f′ (x), r − 1). Let G = {g^k, k ∈ ℤ}, G′ = {g′^k, k ∈ ℤ} and W = (S³ × ℝ)/G, W′ = (S³ × ℝ)/G′. Since f, f ′ preserve orientation of S³, W, W′ are diffeomorphic to S³ × S¹. Denote p_W : S³ × ℝ → W, p_W′ : S³ × ℝ → W ′ the natural projections. It is verified directly that gφ^t = φ^tg, g′φ^t = φ^tg′. Then maps f^t : W → W, f′^t : W′ → W′ given by the formulas $f^{t} (x) = p_{W} (φ^{t} (p_{W}^{- 1} (x)))$ {f^t}(x) = {p_W}({\varphi^t}(p_W^{- 1}(x))) , ${f^{'}}^{t} (x) = p_{W'} (φ^{t} (p_{W'}^{- 1} (x)))$ {f'^t}(x) = {p_{W'}}({\varphi^t}(p_{W'}^{- 1}(x))) are well-defined flows on W, W′ which are called the suspensions of f, f′, respectively, that is f^t, f′^t ∈ 𝒫^t.

Now let f, f′ ∈ 𝒫 be topologically conjugate by the homeomorphism h : S³ → S³. Define a homeomorphism $\tilde{H} : S^{3} \times ℝ \to S^{3} \times ℝ$ \tilde H:{S^3} \times \mathbb {R} \to {S^3} \times \mathbb {R} by the formula $\tilde{H} (x, r) = (h (x), r), (x, r) \in S^{3} \times ℝ .$ \tilde H(x,r) = (h(x),r),{\kern 1pt} (x,r) \in {S^3} \times \mathbb {R}.

Directly verifies that $\tilde{H} g = g^{'} \tilde{H}$ \tilde Hg = g'\tilde H , then $\tilde{H}$ \tilde H can be projected as a homeomorphism H : W → W′ by the formula $H = p_{W'} \tilde{H} p_{W}^{- 1} .$ H = {p_{W'}}\tilde Hp_W^{- 1}.

Since $\tilde{H} φ^{t} = {φ^{'}}^{t} \tilde{H}$ \tilde H{\varphi^t} = {\varphi '^t}\tilde H , then H f^t = f′^t H. Thus H is a required homeomorphism which realizes an equivalency of the suspensions f^t and f′^t.

4.3

Proof of necessity of Theorem 3

Let suspensions f^t, f′^t be topologically equivalent by means of a homeomorphism H : S³ × S¹ → S³ × S¹. Let us prove that then the diffeomorphisms f, f′ are topologically conjugate.

For this aim recall that the diffeomorphisms f, f′ in the basins of sources α, α′ are topologically conjugate by homeomorphisms $h_{α} : W_{α}^{u} \to ℝ^{3}$ {h_\alpha}:W_\alpha^u \to \mathbb {R}^3 , $h_{α^{'}} : W_{α^{'}}^{u} \to ℝ^{3}$ {h_{\alpha '}}:W_{\alpha '}^u \to \mathbb {R}^3 with the linear extension a: ℝ³ → ℝ³ given by the formula $a (x_{1}, x_{2}, x_{3}) = (2 x_{1}, 2 x_{2}, 2 x_{3}) .$ a({x_1},{x_2},{x_3}) = (2{x_1},2{x_2},2{x_3}).

Let $S^{r} = {(x_{1}, x_{2}, x_{3}) \in ℝ^{3} : x_{1}^{2} + x_{2}^{2} + x_{3}^{2} {= 2}^{r}, r \in ℝ}$ {S^r} = \{({x_1},{x_2},{x_3}) \in \mathbb {R}^3:x_1^2 + x_2^2 + x_3^2{= 2^r},r \in \mathbb {R}\} , $S_{α}^{r} = h_{α}^{- 1} (S^{r})$ S_\alpha^r = h_\alpha^{- 1}({S^r}) and $S_{α^{'}}^{r} = h_{α^{'}}^{- 1} (S^{r})$ S_{\alpha '}^r = h_{\alpha '}^{- 1}({S^r}) . Define cylinders $\tilde{Σ}, {\tilde{Σ}}^{'} \subset S^{3} \times ℝ$ \tilde \Sigma,\tilde \Sigma ' \subset {S^3} \times \mathbb {R} by the formulas $\tilde{Σ} = {(x, r) \in S^{3} \times ℝ : x \in S_{α}^{r}, r \in ℝ}, {\tilde{Σ}}^{'} = {(x, r) \in S^{3} \times ℝ : x \in S_{α^{'}}^{r}, r \in ℝ} .$ \tilde \Sigma = \{(x,r) \in {S^3} \times \mathbb {R}:x \in S_\alpha^r,r \in \mathbb {R}\},{\kern 1pt} \tilde \Sigma ' = \{(x,r) \in {S^3} \times \mathbb {R}:x \in S_{\alpha '}^r,r \in \mathbb {R}\}.

It follows from the definition of suspension that $\tilde{Σ}$ \tilde \Sigma , ${\tilde{Σ}}^{'}$ \tilde \Sigma ' are sections for trajectories of φ^t, φ′^t passing through V_φ^t, V_φ′^t, where $V_{φ^{t}} = W_{O_{α}}^{u} \ O_{α}$ {V_{{\varphi^t}}} = W_{{O_\alpha}}^u\backslash {O_\alpha} and $V_{{φ^{'}}^{t}} = W_{O_{α^{'}}}^{u} \ O_{α^{'}}$ {V_{{{\varphi '}^t}}} = W_{{O_{\alpha '}}}^u\backslash {O_{\alpha '}} and $W_{O_{α}}^{u}, W_{O_{α^{'}}}^{u}$ W_{{O_\alpha}}^u,W_{{O_{\alpha '}}}^u are unstable manifolds of orbits O_α, O_α′ of flows φ^t, φ′^t respectively. Let $V_{f^{t}} = W_{𝒪_{α}}^{u}$ {V_{{f^t}}} = W_{{{\cal O}_\alpha}}^u , $V_{{f^{'}}^{t}} = W_{𝒪_{α^{'}}}^{u}$ {V_{{{f'}^t}}} = W_{{{\cal O}_{\alpha '}}}^u . Then $Σ = p_{W} (\tilde{Σ})$ \Sigma = {p_W}(\tilde \Sigma) , $Σ^{'} = p_{W'} ({\tilde{Σ}}^{'})$ \Sigma ' = {p_{W'}}(\tilde \Sigma ') are homeomorphic to S² × S¹ and are sections for trajectories of flows f^t, f′^t in V_f^t, V_f′^t, respectively.

Since H realizes an equivalence of the flows f^t, f′^t then H(Σ) is also a section for trajectories of the flows f′^t in V_f′^t. Thus we can get Σ′ from H(Σ) by a continuous shift along the trajectories, that is there is a homeomorphism ψ : V_f′^t → V_f′^t which preserves the trajectories of f′^t in V_f′^t and such that ψ(H(Σ)) = Σ′. Let h_Σ = ψH|_Σ : Σ → Σ′.

Then the homeomorphism h_Σ has a lift $h_{\tilde{Σ}} : \tilde{Σ} \to {\tilde{Σ}}^{'}$ {h_{\tilde \Sigma}}:\tilde \Sigma \to \tilde \Sigma ' which is a homeomorphism such that $h_{Σ} = p_{W'} h_{\tilde{Σ}} p_{W}^{- 1}$ {h_\Sigma} = {p_{W'}}{h_{\tilde \Sigma}}p_W^{- 1} . Let us introduce the canonical projection q : S³ × ℝ → S³ by the formula q(x, r) = x and define a homeomorphism h : V_f → V_f′ by the formula $h q |_{\tilde{Σ}} = q h_{\tilde{Σ}} .$ hq{|_{\tilde \Sigma}} = q{h_{\tilde \Sigma}}.

By the construction the homeomorphism h conjugates f|_{V_f} with f′|_{V_f′}. Since $H (W_{𝒪_{σ}}^{s}) = W_{𝒪_{σ^{'}}}^{s}$ H(W_{{{\cal O}_\sigma}}^s) = W_{{{\cal O}_{\sigma '}}}^s , then $h (W_{σ}^{s} \ σ) = W_{σ^{'}}^{s} \ σ^{'}$ h(W_\sigma^s\backslash \sigma) = W_{\sigma '}^s\backslash \sigma ' . Let us define a homeomorphism $\hat{h} : {\hat{V}}_{f} \to {\hat{V}}_{f^{'}}$ \hat h:{\hat V_f} \to {\hat V_{f'}} by the formula $\hat{h} p_{f} = p_{f'} h .$ \hat h{p_f} = {p_{f'}}h.

Then $\hat{h} ({\hat{L}}_{f}^{s}) = {\hat{L}}_{f^{'}}^{s}$ \hat h(\hat L_f^s) = \hat L_{f'}^s and η_f = η_f′ĥ_*. Thus, by Proposition 6, the diffeomorphisms f, f′ are topologically conjugated.

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

Pubblicato online: 13 nov 2020

Pagine: 261 - 266

Ricevuto: 24 dic 2019

Accettato: 25 mar 2020

DOI: https://doi.org/10.2478/amns.2020.2.00049

Parole chiave
Suspension, classification, Pixton diffeomorphism, wild embedding

© 2020 Olga V. Pochinka et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

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

Olga V. Pochinka

Danila D. Shubin

Pubblicato online: 13 nov 2020

Pagine: 261 - 266

Ricevuto: 24 dic 2019

Accettato: 25 mar 2020

DOI: https://doi.org/10.2478/amns.2020.2.00049

Parole chiaveSuspension, classification, Pixton diffeomorphism, wild embedding

© 2020 Olga V. Pochinka et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Parole chiave
Suspension, classification, Pixton diffeomorphism, wild embedding