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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 đ« the class of the Morse-Smale diffeomorphisms of 3-sphere S3 whose non-wondering set consists of the fixed source α, the fixed saddle Ï and the fixed sinks Ï1, Ï2. Class đ« diffeomorphism phase portrait is shown in Figure 1.
Fig. 1
The phase portrait of a diffeomorphism of class đ«
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_\sigma^s
, namely its closure is not locally flat at α. We call such diffeomorphism non-trivial (see Figure 2).
Fig. 2
The phase portrait of a non-trivial diffeomorphism of class đ«
Let đ«t 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 Ă S1 and the non-wandering set consists of exactly four periodic orbits đȘα, đȘÏ, đȘÏ1, đȘÏ2. Let
W_{{{\cal O}_\sigma}}^s
denote stable manifold of the saddle orbit. In the present paper we prove the following theorems.
Theorem 1. IfW_\sigma^sis a wild for f â đ« thenW_{{{\cal O}_\sigma}}^sis a wild for ft â đ«t.
Corollary 2. (Existence theorem) There is a flow ft with saddle orbit đȘÏ such thatcl(W_{{{\cal O}_\sigma}}^s)is wild.
Theorem 3. Two flows ft, fâČt â đ«t are topologically equivalent iff the diffeomorphisms f, fâČ â đ« are topologically conjugated.
The complete classification of diffeomorphisms from the class đ« 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 Ă 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.
Auxiliary facts
Dynamical concepts
Diffeomorphism f : Mn â Mn of smooth closed connected orientable n-manifold (n â„ 1)Mn is called Morse-Smale diffeomorphism (f â MS(Mn)) if:
Stable and unstable manifolds
W_p^s
,
W_q^u
intersect transversally for any periodic points p, q.
Two diffeomorphisms f, fâČ are called topologically conjugated if there exists a homeomorphism h : Mn â Mn such that f h = h fâČ.
Let f : Mn â Mn be a diffeomorphism. Let Ït be a flow on the manifold Mn Ă â generated by the unite vector field parallel to â and directed to +â, that is
{\varphi^t}(x,r) = (x,r + t).
Let g : Mn Ă â â Mn Ă â be a diffeomorphism given by the formula g(x, r) = (f (x), r â 1). Let G = {gk, k â â€} and W = (Mn Ă â)/G. Denote pW: Mn Ă â â W the natural projections. It is verified directly that gÏt = Ïtg. Then the map ft : W â W given by the formula
{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.
Two flows ft, fâČt are called topologically equivalent if exists a homeomorphism h : W â W which maps the trajectories of ft to trajectories of fâČt and preservs orientation on the trajectories.
Topological concepts
A closed subset X of a PL-manifold N is said to be tame if there is a homeomorphism h : N â N such that h(X) is a subpolyhedron; the other are called wild.
Let A be a closed subset of a metric space X. A is called locally k-co-connected in X at a â A (k-LCC at a) if each neighbourhood U of a in X contains a smaller neighbourhood V of a such that each map âIk+1 â V \ A extends to a map Ik+1 â 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 â A.
For example, Fox-Artin 2-sphere is not 1-LCC (see Exercise 2.8.1 [3]).
Let e: Mm â Nn be a topological embedding of m-dimensional manifold Mm with a boundary in n-manifold Nn (n â„ m). e is called locally flat at x â Mm (and e(Mm) is locally flat at e(x)) if there exist a neighbourhood U of e(x) â Nn and a homeomorphism h of U onto ân such that:
h(U \cap e({M^m})) = \mathbb {R}_+^m \subset \mathbb {R}^n
when x â â„Mm.
Since tameness implies local flatness for embeddings of manifolds in all co-dimensionals except two, we will say that e: Mm â Nn, m â n â 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 Mnâ1is locally flatly embedded in the n-manifold Nn. Then Mnâ1is k-LCC in Nn for all k â„ 1.
Proposition 5 (Proposition 1.3.6 [3]). Suppose Y is a locally contractible space and A â X. Then A is k-LCC in X iff A Ă Y is k-LCC in X Ă Y.
Notice that any manifold is a locally contractible space.
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_\sigma^s
of the saddle point Ï is a wild 2-sphere in S3 and it is not 1-LCC at a source α. By the construction the circle Ï Ă S1 in S3 Ă S1 coincides with the saddle periodic orbit đȘÏ for the suspension ft of the diffeomorphism f. Moreover, the closure of stable separatrice
W_{{{\cal O}_\sigma}}^s
of đȘÏ coincides with
cl(W_\sigma^s) \times {S^1}
and it is a 3-manifold homeomorphic to S2 Ă S1. Due to Proposition 5 the set
cl(W_{{{\cal O}_\sigma}}^s)
is not 1-LCC in S3 Ă S1. Thus, by Proposition 4,
W_{{{\cal O}_\sigma}}^s
is wild.
Topological classification of suspensions
Firstly we give a brief idea of the topological classification of diffeomorphisms from class đ«.
Classification of diffeomorphisms from đ«
Let f â đ« and
{V_f} = W_\alpha^u\backslash \alpha
. Denote by
{\hat V_f}
the orbit space with respect to f in Vf and by
{p_f}:{V_f} \to {\hat V_f}
the natural projection. According to [5], the space
{\hat V_f}
is diffeomorphic to S2 Ă S1 and the projection pf is a covering map which induces an epimorphism
{\eta_f}:{\pi_1}({\hat V_f}) \to \mathbb {Z}
. Let
\hat L_f^s = {p_f}(W_\sigma^s\backslash \sigma)
. According to [5],
\hat L_f^s
is a homotopically non-trivial 2-dimensional torus in
{\hat V_f}
(see Figure 4).
Fig. 3
The complete invariants for trivial and non-trivial Pixton's diffeomorphisms
Fig. 4
The vector field and
\tilde \Sigma
on đ3 Ă â
Proposition 6 (Theorem 4.5 [5]). Diffeomorphisms f, fâČ â đ« are topologically conjugated iff the tori\hat L_f^s
,
\hat L_{f'}^sare equivalent (that is there is a homeomorphism\hat h:{\hat V_f} \to {\hat V_{f'}}such that\hat h(\hat L_f^s) = \hat L_{f'}^sand ηf = ηfâČ Ä„*).
Proof of the sufficiency of Theorem 3
Let f, fâČ â đ«. Recall the notion of the suspensions of f, fâČ.
Let Ït be a flow on the manifold S3 Ă â generated by the unite vector field parallel to â and directed to +â, that is
{\varphi^t}(x,r) = (x,r + t).
Let g, gâČ : S3 Ă â â S3 Ă â be diffeomorphisms given by the formulas g(x, r) = (f (x), r â 1), gâČ(x, r) = (fâČ (x), r â 1). Let G = {gk, k â â€}, GâČ = {gâČk, k â â€} and W = (S3 Ă â)/G, WâČ = (S3 Ă â)/GâČ. Since f, f âČ preserve orientation of S3, W, WâČ are diffeomorphic to S3 Ă S1. Denote pW : S3 Ă â â W, pWâČ : S3 Ă â â W âČ the natural projections. It is verified directly that gÏt = Ïtg, gâČÏt = ÏtgâČ. Then maps ft : W â W, fâČt : WâČ â WâČ given by the formulas
{f^t}(x) = {p_W}({\varphi^t}(p_W^{- 1}(x)))
,
{f'^t}(x) = {p_{W'}}({\varphi^t}(p_{W'}^{- 1}(x)))
are well-defined flows on W, WâČ which are called the suspensions of f, fâČ, respectively, that is ft, fâČt â đ«t.
Now let f, fâČ â đ« be topologically conjugate by the homeomorphism h : S3 â S3. Define a homeomorphism
\tilde H:{S^3} \times \mathbb {R} \to {S^3} \times \mathbb {R}
by the formula
\tilde H(x,r) = (h(x),r),{\kern 1pt} (x,r) \in {S^3} \times \mathbb {R}.
Directly verifies that
\tilde Hg = g'\tilde H
, then
\tilde H
can be projected as a homeomorphism H : W â WâČ by the formula
H = {p_{W'}}\tilde Hp_W^{- 1}.
Since
\tilde H{\varphi^t} = {\varphi '^t}\tilde H
, then H ft = fâČt H. Thus H is a required homeomorphism which realizes an equivalency of the suspensions ft and fâČt.
Proof of necessity of Theorem 3
Let suspensions ft, fâČt be topologically equivalent by means of a homeomorphism H : S3 Ă S1 â S3 Ă S1. Let us prove that then the diffeomorphisms f, fâČ are topologically conjugate.
For this aim recall that the diffeomorphisms f, fâČ in the basins of sources α, αâČ are topologically conjugate by homeomorphisms
{h_\alpha}:W_\alpha^u \to \mathbb {R}^3
,
{h_{\alpha '}}:W_{\alpha '}^u \to \mathbb {R}^3
with the linear extension a: â3 â â3 given by the formula
a({x_1},{x_2},{x_3}) = (2{x_1},2{x_2},2{x_3}).
It follows from the definition of suspension that
\tilde \Sigma
,
\tilde \Sigma '
are sections for trajectories of Ït, ÏâČt passing through VÏt, VÏâČt, where
{V_{{\varphi^t}}} = W_{{O_\alpha}}^u\backslash {O_\alpha}
and
{V_{{{\varphi '}^t}}} = W_{{O_{\alpha '}}}^u\backslash {O_{\alpha '}}
and
W_{{O_\alpha}}^u,W_{{O_{\alpha '}}}^u
are unstable manifolds of orbits Oα, OαâČ of flows Ït, ÏâČt respectively. Let
{V_{{f^t}}} = W_{{{\cal O}_\alpha}}^u
,
{V_{{{f'}^t}}} = W_{{{\cal O}_{\alpha '}}}^u
. Then
\Sigma = {p_W}(\tilde \Sigma)
,
\Sigma ' = {p_{W'}}(\tilde \Sigma ')
are homeomorphic to S2 Ă S1 and are sections for trajectories of flows ft, fâČt in Vft, VfâČt, respectively.
Since H realizes an equivalence of the flows ft, fâČt then H(ÎŁ) is also a section for trajectories of the flows fâČt in VfâČt. Thus we can get ÎŁâČ from H(ÎŁ) by a continuous shift along the trajectories, that is there is a homeomorphism Ï : VfâČt â VfâČt which preserves the trajectories of fâČt in VfâČt and such that Ï(H(ÎŁ)) = ÎŁâČ. Let hÎŁ = ÏH|ÎŁ : ÎŁ â ÎŁâČ.
Then the homeomorphism hÎŁ has a lift
{h_{\tilde \Sigma}}:\tilde \Sigma \to \tilde \Sigma '
which is a homeomorphism such that
{h_\Sigma} = {p_{W'}}{h_{\tilde \Sigma}}p_W^{- 1}
. Let us introduce the canonical projection q : S3 Ă â â S3 by the formula q(x, r) = x and define a homeomorphism h : Vf â VfâČ by the formula
hq{|_{\tilde \Sigma}} = q{h_{\tilde \Sigma}}.
By the construction the homeomorphism h conjugates f|Vf with fâČ|VfâČ. Since
H(W_{{{\cal O}_\sigma}}^s) = W_{{{\cal O}_{\sigma '}}}^s
, then
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'}}
by the formula
\hat h{p_f} = {p_{f'}}h.
Then
\hat h(\hat L_f^s) = \hat L_{f'}^s
and ηf = ηfâČÄ„*. Thus, by Proposition 6, the diffeomorphisms f, fâČ are topologically conjugated.