Dynamics of topological flows and homeomorphisms with a finite hyperbolic chain-recurrent set on n-manifolds Текст научной статьи по специальности «Математика»

Динамические системы, 2019, том 9(37), №3, 289-296 MSC 2010: 37D15

Dynamics of topological flows and homeomorphisms with a finite hyperbolic chain-recurrent set on n-manifolds1

0. Pochinka, S. Zinina

National Research University Higher School of Economics, Mordovian State Univetsity, Russian Federation

E-mail:opochinka@hse.ru, suddenbee@gmail.com

Abstract. Starting from dimension 4, so-called non-smoothed manifolds, manifolds that do not allow triangulation and other obstacles that prevent the use of the technique of smooth manifolds for the study of multidimensional manifolds appear. In addition, all methods for studying smooth dynamical systems on multidimensional manifolds are based on the approximation of all subsets by piecewise linear or topological objects. In this regard, the idea of consideration of dynamical systems on multidimensional manifolds that do not use the concept of smoothness in their definition is completely natural. So homeomorphisms and topological Morse-Smale flows, which are also firmly connected with the topology of the ambient manifold, as well as their smooth analogues, have already entered into scientific usage. In the present paper we investigate general dynamical properties of homeomorphisms and topological flows with a finite hyperbolic chain recurrent set. Keywords: topological flow, chain-recurrent set, hyperbolic set

1. Introduction and formulation of results

Let Mn be a closed n-dimensional manifold with metric d. A topological flow on Mn is a family of homeomorphisms ft: Mn ^ Mn that continuously depends on t E R and satisfies the following conditions:

1) f 0(x) = x for any point x E Mn;

2) ft(fs(x)) = ft+s(x) for any s,t E R, x E Mn.

The trajectory or the orbit of a point x E Mn is the set Ox = {ft(x),t E R(Z)}. It is believed that the trajectories of the flow (homeomorphism) are oriented in accordance with an increase in the parameter t. Any two trajectories of a dynamical system either coincide or do not intersect, therefore, the phase space is represented as a union of pairwise disjoint trajectories. There are three types of trajectories:

1) fixed point Ox = {x};

1rThe authors is partially supported by Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. № 075-15-2019-1931, except Statements 1 and 2 whose proofs were supported by the Foundation for the Advancement of Theoretical Physics and Mathematics "BASIS" ag. № 19-7-1-15-1

© O. POCHINKA, S. ZININA

2) periodic trajectory (orbit) Ox for which there exists a number per(x) > 0 (per(x) E N) such that fper(x)(x) = x, but ft(x) = x for all real (natural) numbers 0 < t < per(x). The number per(x) is called period of a periodic orbit and does not depend on the choice of a point in orbit;

3) regular trajectory Ox — a trajectory that is not a fixed point or a periodic orbit.

To characterize the wandering of the trajectories of a dynamical system, the concept of chain recurrence is traditionally used.

The e-chain of length T connecting the point x with the point y for the flow ft is called a sequence of points x = x0,... ,xn = y for which there exists a sequence of times t\,... ,tn such that d(fti(xi-i),xi) < e, ti > 1 for 1 < i < n and ti + • • • + tn = T.

The e-chain of length n connecting the point x with the point y for a homeomorphism f is called a sequence of points x = x0,...,xn = y, such that d(f (xi-i),xi) < e for 1 < i < n.

A point x E Mn is said to be chain recurrent for the flow ft (cascade f), if for any e > 0 there is T(n), which depends on e > 0, and there is an e-chain of the length T(n) from the point x to itself. The set of chain recurrent points of ft (f) is called the chain recurrent .set of ft (f) denoted by Rft (Rf) and its connected components are called chain components. The set Rft (Rf) is ft (f) - invariant, that is, it consists of the orbits of the flow (homeomorphism) ft ( f), which are called chain recurrent. It is obvious that fixed points and periodic orbits are chain recurrent.

As a model behavior of flow (homeomorphism) in a neighborhood of a fixed point, we consider a linear flow (homeomorphism) atx : Rn ^ Rn (a\ : Rn ^ Rn),A E {0,1,..., n} of the following form:

at\(xl, ...,x\,x\+l, ...,xn) = (2txl,r2x\,) 2 tx\+l,..., 2 *xn)

(ax(xi, ...,x\,xx+i, ...,xn) = (±2xi, 2x2,..., 2xx, ±2-ixx+i, 2-ixx+2,..., 2-ixn)).

A fixed point p of a flow (homeomorphism) f (f) is called is topologically hyperbolic if there exists a neighborhood Up C Mn, a number A E {0,1,...,n} and a homeomorphism hp : Up ^ Rn such that hpft\Up = a\php\Up (hpf \Up = axphp\Up) whenever the left and right sides are defined. Let

EX = {(xi,...,xn) E Rn : xi = • • • = xx = 0},

EX = {(xi, ...,xn) E Rn : xx+i = • • • = xn = 0}.

The number Ap is called the index of the fixed hyperbolic point p. A point of indexes n and 0 will be called source and sink, respectively; any point p such that Ap E {1, • • • ,n — 1} is called saddle. For a topologically hyperbolic fixed point p of the flow (homeomorphism) ft (f) the sets h-i(ESp),h-i(EXp) are called local stable, unstable manifolds.

The sets

= U ftK'Eir)), = U ))

teR teR

is called stable and unstable invariant manifolds of the point p.

If p is a periodic point of a period k for diffeomorphism f then its invariant manifolds and the index are defined as for fixed point fk (p) with respect to the homeomorphism fk. The number XOp which equals Xp is called index of the orbit Op of the periodic point p.

Statement 1. The unstable Wp and the stable Ws manifolds of the hyperbolic fixed point p are independent of the choice of the local homeomorphism hp and are defined

in topological terms as follows: WU = {y E Mn : lim f-t(y) = p} u WS = {y E Mn : lim f t(y)= p}.

It follows from Statement 1 that W^ Pi W^1 = 0 and Pi W£ = 0 for any different hyperbolic points p, q.

Denote by G a class of homeomorphisms and topological flows given on Mn with a finite hyperbolic chain recurrent set.

Let F E G. The dynamics of systems of this class are close in their properties to gradient-like systems (see, for example, [4], [2]). Namely, similar to S. Smale's order [5], we introduce a partial order relation on the set of chain-recurrent orbits of the dynamical system F by the condition

Oi — Oj ^ WOi P = 0,

where Oi, Oj are orbits from the set and WO. = (J W £, = U Wpu.

' P&Oi i p&Oi

A m-cycle (m > 1) is a collection Oi, O2, ••• , Om of pairwise disjoint chain recurrent orbits that satisfy the condition O1 -< O2 -<•••* Om -< O1.

Statement 2. Every dynamical .system F E G has no cycles.

Due to Statement 2 the introduced relation can be continued (not uniquely) to a complete order relation, that is for every chain recurrent orbits Oi, Oj either Oi Oj, or Oj — Oi. Thus, let us consider the orbits of a dynamical system F E G numbered in accordance with the introduced order:

Oi — ••• — Ok.

In addition, without loss of generality, we assume that any sink orbit is located in this order below any saddle orbit, and any source orbit is higher than any saddle one. The main result of the present paper is the following fact.

Theorem 1. Let F E G. Then

(1) Mn = U WO = U WO i;

i=1 i=1

(2) WO). (WO.) s a topological submanifold of Mn, homeomorphic to RA°i (Rn_A°i);

i-1 k (3) (ci(wu.) \ wu) С и WZ ((ci(wSi) \ wSi) С и WS,).

j

j = 1 j=i+1

Notice that a similar result for Morse-Smale diffeomorphisms was proved in [3] and for Morse-Smale homeomorphism was proved in [1].

2. Auxilary facts

In this section we prove announced statements. Proof of Statement 1.

Proof. Let us prove that if F E G and p is a fixed point of the system F, then WpU and Wp are independent of the choice of the local homeomorphism hp.

Suppose for definiteness that F is a homeomorphism f (for a flow the proof is similar). Let hp : Up ^ Rn be a homeomorphism different from hp and such that hpf \(jp = aXphp\^p whenever the left and right sides are defined. Then in a neighborhood UO of the original point O in Rn is well-defined a homeomorphism h = hph-1 which conjugates aXp with aXp. As conjugating homeomorphism preserves the invariant manifolds then \p = \p and h(E^ ) = Esx , h(EU ) = . Thus,

h-1(EsXp) = h-1(h(ESp)) = h-1 (ESp). It is the same for E^. □

Proof of Statement 2.

Proof. We will prove that every dynamical system F E G has no cycles.

Suppose the contrary: there exists a sequence of orbits O1 -<■■■< Om < O1. By

m

construction, any point of the set (J (W Si ^ WUi+1), where Om+1 = O1, is a chain

i=1

recurrent. It immediately contradicts with the finiteness of the chain recurrent set of the system F. □

Statement 3. Every homeomorphism f = f1, which is the one-time shift of a flow

Tu(h = Wpu(f), Wp;\

f E G belongs to the class G. Moreover, Rft = Rf and Wu(ft) = Wu(f), Ws(ft) =

WpS(f) for every chain recurrent point p.

Proof. It immediately follows from the definition of hyperbolic point that Rft C Rf. Let us show that Rf C Rft. Let p E Rf, then there exists a e-chain of length n connecting the point p with itself. Then the point p will also be a chain recurrence point for the flow ft, since it has a e-chain of length T = n, and ti = 1 (i = 1; n) connecting the point p with itself.

It follows from the uniqueness of the invariant manifolds of a chain recurrent point, proved in Statement 1, that WpU(ft) = W^(f), W^(ft) = Wp;(f) for every chain recurrent point p. □

3. General dynamical properties of systems from class G

In this section, we prove Theorem 1 on the embedding and asymptotic behaviour of invariant manifolds of chain recurrent points of a dynamical system from class G. Due to Statement 3, it is enough to prove the theorem for the case when the dynamical system F is a homeomorphism f. Moreover we can suppose (by changing of f by some power of f) that chain recurrent set consists of the fixed points, that is Oi = pi and f \Up is conjugated with the diffeomorphism a\p, (x-\_, ...,x\,x\+1, ...,xn) = (2x1, 2x2,..., 2x\, 2-1x\p,+1, 2-1x\p, +2,..., 2-1 xn)) by means a homeomorphism hpi.

A fixed point p of a flow (homeomorphism) ft (f) is called is topologically hyperbolic if there exists a neighborhood Up C Mn, a number A E {0,1,...,n} and a homeomorphism hp : Up ^ Rn such that hpff\Up = atXphp\Up (hpf \Up = aXphp\Up) whenever the left and right sides are defined..

Below we prove each item of the theorem in a separate subsection. All statements formulated for unstable manifolds hold for stable manifolds as well. One gets them if one formally changes "u" to "s" because Rf = Rf-1 and stable manifolds of chain recurrent points for f are the unstable manifolds of the chain recurrent points for f-1.

3.1. Representation of the ambient manifold as the union of the invariant manifolds of the periodic points

Proof of the item (1) of Theorem 1.

k

Proof. Now we prove that Mn = (J W^ for every homeomorphism f E G.

i=1 i

Let x E Mn. Let us recall that a point y E Mn is called an a-limit point for the point x if there is a sequence tn ^ —x>, tn E Z such that

lim d(ftn(x),y) = 0.

The set a(x) of all a-limit points for the point x is called the a-limit set of x. As Mn is compact then the set a(x) is not empty. Let us show that a(x) C Rf. Indeed, as f is uniformly continuous and lim d(ftn(x),y) = 0, for every e > 0 there is

tn

ne E N such that d(ftn(x),y) < e and d(ftn+1(x),f(y)) < e for every n > n£. Thus, y, f (y), ftn+1(x),ftn+2(x),..., ftn+1 (x),y is the e-chain connected y with itself.

We show that a(x) consists of exactly one fixed point which depends on x. Assume the contrary i.e there are distinct fixed points pv, pw E a(x). Since Rf is finite there is a p > 0 such that d(pi,pj) > p whenever i = j. Denote Vi = {y E Mn : d(y,pi) < P}. Since all the points pi, i = 1,k are fixed there is a neighborhood Ui such that cl(Ui) C Vi and f-1(cl(Ui)) P Vj = 0 for every j = i. By the assumption there is an increasing sequence qe of the iterations of f-1 such that f-q2m (x) E Uv, f -q2m+1 (x) E Uw and q2m+1 — q2m > 2. We pick the sequence nm so that nm is the maximal natural number belonging to the interval (q2m,q2m+1) for each f-(nm-1\x) E cl(Uv). Then

f-nm (x) G cl(Uv). On the other hand f-nm (x) = f-1(f-(nm-1) (x)) G Vj for j = v and

k

hence f-nm (x) G (Mn \ (J Ui). But then a(x) is not a subset of Rf and we have a

i=1

contradiction.

Thus for each point x G Mn there is the unique point pv (x) G Rf such that a(x) = pv(x), i.e. there is a sequence kn ^ such that lim d(f-kn(x),pv(x)) = 0.

It follows from the definition of the hyperbolic fixed point that f-kn (x) G W^^ for all n greater then some n0. Then x G W^^ because the unstable manifold is invariant. □

3.2. Embedding of the invariant manifolds of periodic points into the ambient manifold

To prove item (2) of Theorem 1 we need the following lemma.

Lemma 1. Let a be a hyperbolic .saddle fixed point of a diffeomorphism f G G, let TS C WS be a compact neighborhood of the point a and £ G TS. Then for every sequence of points {£m} C (Mn \ TS) converging to the point £ there are a subsequence {£m.}, a sequence of natural numbers kmj ^ and a point f G (W" \ a) such that the sequence of points {fkmj (£mj)} converges to the point f.

Proof. Without loss of generality one assumes (US fl WS) C TS, £ G (US fl f (US)) and {£m} C (U S f f (U S)). We pick a number r > 0 so that the ball Br(O) = {(x\,... ,xn) G Rn : (xf + • • • + x"n) < r} would be a subset of the set hS(US).

Let hs (£m) = £m = (£1,m, . . . ,£\v ,m,£\a+1,m, . . . > £n,m). The set K" =

{(x\,..., x\a) G Ox1... x\a : y < xf + • • • + x\a < r2} is a fundamental domain of the restriction of the diffeomorphism a\a to Ox1... ,x\a \ O. Then for every m G N there is the unique integer km such that < 4km ((^l,™)2 + • • • + m)2) < r2. Let fjm = akm+1(£m). Since lim £m = ha (£) G (OxXa+1 ...xn \ O) for every i G {1,..., ACT}

the limit lim £i>m equals 0 and hence lim km = Furthermore the sequence {£i,m}

is bounded for every i G {AS + 1,... ,n} and hence fjim = (f) m £i,m ^ 0 for m ^ and i G {AS + 1,..., n}.

Therefore the coordinates of the points fjm = (fj1,m,..., fjn,m) satisfy < (n1,m)2 + • • • + (nj\a,m)2 < r2 for i G {1,..., AS} and fji,m 0 as m ^ <x> for i G {AS + 1,..., n}, i.e. the points nm are inside some compact subset of Rn. Since any sequence of points of a compact set has a converging subsequence, there are a subsequence {kmj} of the sequence {km} and a point fj G (WO \ O) such that lim fjm = f. Then £m. =

Imj

km

('Imj)) is the desired subsequence. □

Proof of item (2) of Theorem 1

Proof. Here we prove that W'U. is a submanifold of the manifold Mn, homeomorphic to

ru ' Pi

Let Tpi = hpi (E'U ). Then for every point x E Wu. there is a natural number nx such that x E f-n* (Tp.). Let TPi (x) = f-n* (TPi) then there is a chart fa : U ^ Rn of the manifold Mn such that fa(Ux n Tp.(x)) = RA«. If \Pi = n or \i = 0 then ^x(Ux n Tpi(x)) = ^x(Ux n Wp.). Therefore the unstable manifold of every node point is a smooth submanifold.

Now we show that Wpi is a submanifold of Mn homeomorphic to RXpi for every saddle point pi as well. Suppose the contrary: Wu. is not a submanifold of Mn. Then there is a point x E Wu. such that (Ux \ Tpi(x)) n Wu. = 0 for every chart ^x : Ux ^ Rn of the manifold Mn for which ^x(Ux n TPi(x)) = RXpi. Hence there is a sequence {xm} C (Wpi \ TPi (x)) such that d(xm,x) ^ 0 for m ^

Lemma 1 gives us that there is a subsequence xmj and there is a sequence kj such that the sequence yj = f-kj (xmj) C Wpi converges to a point y E (W^i \ pi).

According to the item (1) of Theorem 1 there is a point pv E Rf such that y E W^v. Consider three possibilities: [a] dim Wuv = 0; [b] 0 < dim Wuv < n; [c] dim Wuv = n.

[a] If dim Wuv = 0 then yj E W^v for all j starting from some one. Hence, i = v and y is a homoclinic point, that contradicts to Statement 2. Thus case [a] is impossible.

[c] If dim W'Uv = n then W^v = pv and y = pv, that contradicts to the condition y E W^i. Thus case [c] is impossible.

[b] If 0 < dim W^v < n then v > i as f has no homoclinic points. According to Lemma 1 there is a subsequence {yjr}, a sequence mr ^ and a point z E W£v such

k

that the sequence {fmr(yjr)} converges to the point z. As Mn = (J W^. then z E W^.

i=i i w

Similarly to above arguments, v = j,v = i and 0 < dim Wuw < n. Due to finiteness of the set Rf the case [b] is also impossible.

Thus, Wu. is a topological submanifold of the manifold Mn homeomorphic to RXpi.

i □

3.3. Asymptotic behaviour of the invariant manifolds of chain recurrent points

Proof of the item (3) of Theorem 1

i -i

Proof. Now we prove that (cl(Wu) \ Wu) C (J Wu.

i i j=i j

If pi is a sink then the set cl(W ^) \ Wpi is empty and the statement is automatically true. In the other cases let us consider a point x E (cl(Wp.) \ Wpi) and prove that x E WU for some v < i.

Pv

Indeed, as x E (cl(Wp.) \ (W^. Upi)) then there is a sequence {xm} C Wpi such that d(xm, x) ^ 0 for m ^ By item (1) of Theorem 1, x E Wuv for some v E {1,... , k}. There are three possibilities: (a) pv is a sink, (b) pv is a saddle, (c) pv is a source.

In the case (c) xm E Wuv for all m large enough. But then pv = pi and x E W™ that contradicts the assumption.

In the case (a) Wuv = pv, x = pv and xm E W£v for all m large enough. Then WU n WS = 0 and v <i is true.

i v

In the case (b) by Lemma 1 there are a subsequence xmj and a sequence kj such that the sequence yj = f-kj (xmj) converges to a point y E (W£v \pv). By the item (1) of Theorem 1 there is a point pw ERf such that y E Wuw, that is pv — pw. If w = i then the statement is true. If not then arguing as above we get that the point pw cannot be a source. The point pw is evidently not a sink because pv is a saddle point. Thus, the point pw is a saddle different from pv. Repeating the process, taking into account the finiteness of Rf and the absence of cycles, we obtain the statement in a finite number of steps. □

References

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4. Grines V., Pochinka O. Morse-Smale cascades on 3-manifolds. Russian Mathematical Surveys 68. No.1, 117-173 (2013).

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Получена 11.11.2019