## 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 *S*^{3} whose non-wondering set consists of the fixed source *α*, the fixed saddle *σ* and the fixed sinks *ω*_{1}, *ω*_{2}. Class 𝒫 diffeomorphism phase portrait is shown in Figure 1.

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
*α*. We call such diffeomorphism *non-trivial* (see Figure 2).

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*is diffeomorphic to

^{t}*S*

^{3}×

*S*

^{1}and the non-wandering set consists of exactly four periodic orbits 𝒪

*, 𝒪*

_{α}*, 𝒪*

_{σ}_{ω1}, 𝒪

_{ω2}. Let

**Theorem 1**. *If**is a wild for f* ∈ 𝒫 *then**is a wild for f ^{t}* ∈ 𝒫

*.*

^{t}**Corollary 2**. *(Existence theorem) There is a flow f ^{t} with saddle orbit* 𝒪

_{σ}such that*is wild*.

**Theorem 3**. *Two flows f ^{t}, f′^{t}* ∈ 𝒫

*∈ 𝒫*

^{t}are topologically equivalent iff the diffeomorphisms f, f′*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 *S*^{2} × *S*^{1}. 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* : *M ^{n}* →

*M*of smooth closed connected orientable

^{n}*n*-manifold (

*n*≥ 1)

*M*is called

^{n}*Morse-Smale diffeomorphism*(

*f*∈

*MS*(

*M*)) if:

^{n}- Non-wandering set Ω
_{f}is finite and hyperbolic; - Stable and unstable manifolds
,${W}_{p}^{s}$ intersect transversally for any periodic points${W}_{q}^{u}$ *p*,*q*.

Two diffeomorphisms *f*, *f*′ are called *topologically conjugated* if there exists a homeomorphism *h* : *M ^{n}* →

*M*such that

^{n}*f h*=

*h f*′.

*f*:

*M*→

^{n}*M*be a diffeomorphism. Let

^{n}*φ*be a flow on the manifold

^{t}*M*× ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is

^{n}*g*:

*M*× ℝ →

^{n}*M*× ℝ be a diffeomorphism given by the formula

^{n}*g*(

*x*,

*r*) = (

*f*(

*x*),

*r*− 1). Let

*G*= {

*g*,

^{k}*k*∈ ℤ} and

*W*= (

*M*× ℝ)/

^{n}*G*. Denote

*p*:

_{W}*M*× ℝ →

^{n}*W*the natural projections. It is verified directly that

*gφ*=

^{t}*φ*. Then the map

^{t}g*f*:

^{t}*W*→

*W*given by the formula

*W*which is called

*the suspension of f*.

When *f* ∈ *MS*(*M ^{n}*) the non-wandering set of the suspension

*f*consist of a finite number of periodic orbits composed by

^{t}*p*(Ω

_{W}_{f}× ℝ). The obtained flow is so-called

*non-singular*, what means it has no singular points.

Two flows *f ^{t}*,

*f′*are called

^{t}*topologically equivalent*if exists a homeomorphism

*h*:

*W*→

*W*which maps the trajectories of

*f*to trajectories of

^{t}*f′*and preservs orientation on the trajectories.

^{t}### 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}* →

*N*be a topological embedding of

^{n}*m*-dimensional manifold

*M*with a boundary in

^{m}*n*-manifold

*N*(

^{n}*n*≥

*m*).

*e*is called

*locally flat*at

*x*∈

*M*(and

^{m}*e*(

*M*) is

^{m}*locally flat*at

*e*(

*x*)) if there exist a neighbourhood

*U*of

*e*(

*x*) ∈

*N*and a homeomorphism

^{n}*h*of

*U*onto ℝ

^{n}such that:

- (1)
*h*(*U*∩*e*(*M*)) = ℝ^{m}^{m}⊂ ℝ^{n}when*x*∈*int M*or^{m} - (2)
when$h(U\cap e({M}^{m}\left)\right)={\mathbb{R}}_{+}^{m}\subset {\mathbb{R}}^{n}$ *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}* →

*N*,

^{n}*m*≠

*n*− 2 is

*wild*at

*e*(

*x*) when

*e*(

*M*) is fails to be locally flat at

^{m}*e*(

*x*).

**Proposition 4** (Proposition 1.3.1 [3]). *Suppose the manifold M*^{n−1}*is locally flatly embedded in the n-manifold N ^{n}*.

*Then M*

^{n−1}

*is k-LCC in N*≥ 1.

^{n}for all k**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

Let *f* be a non-trivial Pixton's diffeomorphism. Then the closure of the stable manifold
*σ* is a wild 2-sphere in *S*^{3} and it is not 1-LCC at a source *α*. By the construction the circle *σ* × *S*^{1} in *S*^{3} × *S*^{1} coincides with the saddle periodic orbit 𝒪* _{σ}* for the suspension

*f*of the diffeomorphism

^{t}*f*. Moreover, the closure of stable separatrice

*coincides with*

_{σ}*S*

^{2}×

*S*

^{1}. Due to Proposition 5 the set

*S*

^{3}×

*S*

^{1}. Thus, by Proposition 4,

## 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
*f* in *V _{f}* and by

*S*

^{2}×

*S*

^{1}and the projection

*p*is a covering map which induces an epimorphism

_{f}**Proposition 6** (Theorem 4.5 [5]). *Diffeomorphisms f, f*′ ∈ 𝒫 *are topologically conjugated iff the tori**are equivalent (that is there is a homeomorphism**such that**and 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*′.

*φ*be a flow on the manifold

^{t}*S*

^{3}× ℝ generated by the unite vector field parallel to ℝ and directed to +∞, that is

Let *g*, *g*′ : *S*^{3} × ℝ → *S*^{3} × 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*= (

*S*

^{3}× ℝ)/

*G*,

*W*′ = (

*S*

^{3}× ℝ)/

*G*′. Since

*f*,

*f*′ preserve orientation of

*S*

^{3},

*W*,

*W*′ are diffeomorphic to

*S*

^{3}×

*S*

^{1}. Denote

*p*:

_{W}*S*

^{3}× ℝ →

*W*,

*p*:

_{W′}*S*

^{3}× ℝ →

*W*′ the natural projections. It is verified directly that

*gφ*=

^{t}*φ*,

^{t}g*g*′

*φ*=

^{t}*φ*′. Then maps

^{t}g*f*:

^{t}*W*→

*W*,

*f′*:

^{t}*W*′ →

*W*′ given by the formulas

*W*,

*W*′ which are called the suspensions of

*f*,

*f*′, respectively, that is

*f*,

^{t}*f′*∈ 𝒫

^{t}*.*

^{t}*f*,

*f*′ ∈ 𝒫 be topologically conjugate by the homeomorphism

*h*:

*S*

^{3}→

*S*

^{3}. Define a homeomorphism

*H*:

*W*→

*W*′ by the formula

Since
*H f ^{t}* =

*f′*. Thus

^{t}H*H*is a required homeomorphism which realizes an equivalency of the suspensions

*f*and

^{t}*f′*.

^{t}### 4.3 Proof of necessity of Theorem 3

Let suspensions *f ^{t}*,

*f′*be topologically equivalent by means of a homeomorphism

^{t}*H*:

*S*

^{3}×

*S*

^{1}→

*S*

^{3}×

*S*

^{1}. Let us prove that then the diffeomorphisms

*f*,

*f*′ are topologically conjugate.

*f*,

*f*′ in the basins of sources

*α*,

*α*′ are topologically conjugate by homeomorphisms

*a*: ℝ

^{3}→ ℝ

^{3}given by the formula

It follows from the definition of suspension that
*φ ^{t}*,

*φ′*passing through

^{t}*V*,

_{φt}*V*, where

_{φ′t}*O*,

_{α}*O*of flows

_{α′}*φ*,

^{t}*φ′*respectively. Let

^{t}*S*

^{2}×

*S*

^{1}and are sections for trajectories of flows

*f*,

^{t}*f′*in

^{t}*V*,

_{ft}*V*, respectively.

_{f′t}Since *H* realizes an equivalence of the flows *f ^{t}*,

*f′*then

^{t}*H*(Σ) is also a section for trajectories of the flows

*f′*in

^{t}*V*. Thus we can get Σ′ from

_{f′t}*H*(Σ) by a continuous shift along the trajectories, that is there is a homeomorphism

*ψ*:

*V*→

_{f′t}*V*which preserves the trajectories of

_{f′t}*f*′

^{t}in

*V*and such that

_{f′t}*ψ*(

*H*(Σ)) = Σ′. Let

*h*

_{Σ}=

*ψH*|

_{Σ}: Σ → Σ′.

*h*

_{Σ}has a lift

*q*:

*S*

^{3}× ℝ →

*S*

^{3}by the formula

*q*(

*x*,

*r*) =

*x*and define a homeomorphism

*h*:

*V*→

_{f}*V*by the formula

_{f′}*h*conjugates

*f*|

*with*

_{Vf}*f*′|

*. Since*

_{Vf′}Then
*η _{f}* =

*η*

_{f′}

*ĥ*

_{*}. Thus, by Proposition 6, the diffeomorphisms

*f*,

*f*′ are topologically conjugated.

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