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

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;

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 φt(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 ft(x)=pW(φt(pW−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.

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

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.

Let f ∈ 𝒫 and Vf=Wαu\α{V_f} = W_\alpha^u\backslash \alpha . Denote by V^f{\hat V_f} the orbit space with respect to f in V_f and by pf:Vf→V^f{p_f}:{V_f} \to {\hat V_f} the natural projection. According to [5], the space V^f{\hat V_f} is diffeomorphic to S² × S¹ and the projection p_f is a covering map which induces an epimorphism ηf:π1(V^f)→ℤ{\eta_f}:{\pi_1}({\hat V_f}) \to \mathbb {Z} . Let L^fs=pf(Wσs\σ)\hat L_f^s = {p_f}(W_\sigma^s\backslash \sigma) . According to [5], L^fs\hat L_f^s is a homotopically non-trivial 2-dimensional torus in V^f{\hat V_f} (see Figure 4).

Fig. 3

Fig. 4

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

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 φt(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 ft(x)=pW(φt(pW−1(x))){f^t}(x) = {p_W}({\varphi^t}(p_W^{- 1}(x))) , f′t(x)=pW'(φt(pW'−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 H˜:S3×ℝ→S3×ℝ\tilde H:{S^3} \times \mathbb {R} \to {S^3} \times \mathbb {R} by the formula H˜(x,r)=(h(x),r), (x,r)∈S3×ℝ.\tilde H(x,r) = (h(x),r),{\kern 1pt} (x,r) \in {S^3} \times \mathbb {R}.

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

Since H˜φt=φ′tH˜\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.

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 hα:Wαu→ℝ3{h_\alpha}:W_\alpha^u \to \mathbb {R}^3 , hα′:Wα′u→ℝ3{h_{\alpha '}}:W_{\alpha '}^u \to \mathbb {R}^3 with the linear extension a: ℝ³ → ℝ³ given by the formula a(x1,x2,x3)=(2x1,2x2,2x3).a({x_1},{x_2},{x_3}) = (2{x_1},2{x_2},2{x_3}).

Let Sr={(x1,x2,x3)∈ℝ3:x12+x22+x32=2r,r∈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}\} , Sαr=hα−1(Sr)S_\alpha^r = h_\alpha^{- 1}({S^r}) and Sα′r=hα′−1(Sr)S_{\alpha '}^r = h_{\alpha '}^{- 1}({S^r}) . Define cylinders Σ˜,Σ˜′⊂S3×ℝ\tilde \Sigma,\tilde \Sigma ' \subset {S^3} \times \mathbb {R} by the formulas Σ˜={(x,r)∈S3×ℝ:x∈Sαr,r∈ℝ}, Σ˜′={(x,r)∈S3×ℝ:x∈Sα′r,r∈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 Σ˜\tilde \Sigma , Σ˜′\tilde \Sigma ' are sections for trajectories of φ^t, φ′^t passing through V_φ^t, V_φ′^t, where Vφt=WOαu\Oα{V_{{\varphi^t}}} = W_{{O_\alpha}}^u\backslash {O_\alpha} and Vφ′t=WOα′u\Oα′{V_{{{\varphi '}^t}}} = W_{{O_{\alpha '}}}^u\backslash {O_{\alpha '}} and WOαu,WOα′uW_{{O_\alpha}}^u,W_{{O_{\alpha '}}}^u are unstable manifolds of orbits O_α, O_α′ of flows φ^t, φ′^t respectively. Let Vft=W𝒪αu{V_{{f^t}}} = W_{{{\cal O}_\alpha}}^u , Vf′t=W𝒪α′u{V_{{{f'}^t}}} = W_{{{\cal O}_{\alpha '}}}^u . Then Σ=pW(Σ˜)\Sigma = {p_W}(\tilde \Sigma) , Σ′=pW'(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 hΣ˜:Σ˜→Σ˜′{h_{\tilde \Sigma}}:\tilde \Sigma \to \tilde \Sigma ' which is a homeomorphism such that hΣ=pW'hΣ˜pW−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 hq|Σ˜=qhΣ˜.hq{|_{\tilde \Sigma}} = q{h_{\tilde \Sigma}}.

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

Then h^(L^fs)=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.