# On Strongly Irregular Points of a Brouwer Homeomorphism Embeddable in a Flow

Authors: Zbigniew Leśniak
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the set of all strongly irregular points of a Brouwer homeomorphismf which is embeddable in a flow. We prove that this set is equal to the first prolongational limit set of any flow containing f. We also give a sufficient condition for a class of flows of Brouwer homeomorphisms to be topologically conjugate.


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## Body

## 1. Introduction

In this part we recall the requisite definitions and results concerning Brouwer homeomorphisms and flows of such homeomorphisms.By aBrouwer homeomorphism we mean an orientation preserving homeomorphism of the plane onto itself which has no fixed points. By aflow we mean a group of homeomorphisms of the plane onto itself {ft:t∈R} under the operation of composition which satisfies the following conditions:(1)
the functionF:R2×R→R2,F(x,t)=ft(x) is continuous,(2)
f0(x)=x for x∈R2,(3)
ft(fs(x))=ft+s(x) for x∈R2, t,s∈R. We say that a Brouwer homeomorphism f isembeddable in a flow if there exists a flow {ft:t∈R} such that f=f1. Then for each t∈R∖{0}, ft is a Brouwer homeomorphism.For any sequence of subsets(An)n∈Z+ of the plane we definelimes superiorlimsup⁡n→∞An as the set of all points p∈R2 such that any neighbourhood of p has common points with infinitely many elements of the sequence (An)n∈Z+. For any subset B of the plane we define thepositive limit setωf(B) as the limes superior of the sequence of its iterates (fn(B))n∈Z+ andnegative limit setαf(B) as the limes superior of the sequence (f-n(B))n∈Z+. Under the assumption that B is compact, Nakayama [1] proved that(1)ωf(B)={q∈R2:thereexistsequences(pj)j∈Z+,(nj)j∈Z+suchthatpj∈B,nj∈Z+,nj⟶+∞,fnj(pj)⟶qasj⟶+∞q∈R2:thereexistsequences(pj)j∈Z+},αf(B)={q∈R2:thereexistsequences(pj)j∈Z+,(nj)j∈Z+suchthatpj∈B,nj∈Z+,nj⟶+∞,f-nj(pj)⟶qasj⟶+∞q∈R2:thereexistsequences(pj)j∈Z+}.A pointp is calledpositively irregular if ωf(B)≠∅ for each Jordan domain B containing p in its interior andnegatively irregular if αf(B)≠∅ for each Jordan domain B containing p in its interior, where by a Jordan domain we mean the union of a Jordan curve J and the Jordan region determined by J (i.e., the bounded component of R2∖J). A point which is not positively irregular is said to bepositively regular. Similarly, a point which is not negatively irregular is callednegatively regular. A point which is positively or negatively irregular is calledirregular, otherwise it isregular.We say that a setU⊂R2 isinvariant under f if f(U)=U. An invariant simply connected region U⊂R2 is said to beparallelizable if there exists a homeomorphism φU mapping U onto R2 such that
(2)f(x)=φU-1(φU(x)+(1,0))forx∈U.
The homeomorphism φU occurring in this equality is called aparallelizing homeomorphism of f|U. On account of the Brouwer Translation Theorem, for each p∈R2 there exists a parallelizable region U containing p (see [2]).Homma and Terasaka [3] proved a theorem describing the structure of an arbitrary Brouwer homeomorphism. The theorem can be formulated in the following way.Theorem 1 (see [3], First Structure Theorem).
Letf be a Brouwer homeomorphism. Then the plane is divided into at most three kinds of pairwise disjoint sets: {Oi:i∈I}, where I=Z+ or I={1,…,n} for a positive integer n, {Oi′:i∈Z+}, and F. The sets {Oi:i∈I} and {Oi′:i∈Z+} are the components of the set of all regular points such that each Oi is a parallelizable unbounded simply connected region and each Oi′ is a simply connected region satisfying the condition Oi′∩fn(Oi′)=∅ for n∈Z∖{0}. The set F is invariant and closed and consists of all irregular points.For an irregular pointp of a Brouwer homeomorphism f the set P+(p) is defined as the intersection of all ωf(B) and the set P-(p) as the intersection of all αf(B), where B is a Jordan domain containing p in its interior. An irregular point p isstrongly positively irregular if P+(p)≠∅, otherwise it isweakly positively irregular. Similarly, p isstrongly negatively irregular if P-(p)≠∅, otherwise it isweakly negatively irregular. We say that p isstrongly irregular if it is strongly positively irregular or strongly negatively irregular. Otherwise, an irregular point p is said to beweakly irregular.Homma and Terasaka [3] proved that for all p,q∈R2(3)q∈P+(p)⟺p∈P-(q).
Nakayama [4] showed that for any Brouwer homeomorphism the set of strongly irregular points has no interior points. The set of weakly irregular points consists of all cluster points of the set of strongly irregular points which are not strongly irregular points (see [3]).A counterpart of Theorem1 for a Brouwer homeomorphism embeddable in a flow has been given in [5]. Namely, if a Brouwer homeomorphism is embeddable in a flow, then the set of regular points is a union of pairwise disjoint parallelizable unbounded simply connected regions.


## 2. Strongly Irregular Points

In this section we study the structure of the set of all irregular points for Brouwer homeomorphisms embeddable in a flow.Letf be a Brouwer homeomorphism. Assume that there exists a flow {ft:t∈R} such that f1=f. Let U⊂R2 be a simply connected region such that ft(U)=U for t∈R. We say that U is aparallelizable region of the flow if there exists a homeomorphism φU mapping U onto R2 such that
(4)ft(x)=φU-1(φU(x)+(t,0))forx∈U,t∈R.
Such a homeomorphism φU will be called aparallelizing homeomorphism of the flow {ft|U:t∈R}. It is known that for any simply connected region U which is invariant under the flow {ft:t∈R} the existence of a parallelizing homeomorphism of f|U is equivalent to the existence of a parallelizing homeomorphism of {ft|U:t∈R} (see [6]).By the trajectory of a pointp∈R2 we mean the set Cp:={ft(p):t∈R}. It is known that a region U is parallelizable if and only if there exists atopological lineK in U (i.e., a homeomorphic image of a straight line that is a closed set in U) such that K has exactly one common point with every trajectory of {ft:t∈R} contained in U (see [7], page 49). Such a set K we will call asection in U (or alocal section of {ft:t∈R}). On account of the Whitney-Bebutov Theorem (see [7], page 52), for each p∈R2 there exists a parallelizable region Up containing p. Without loss of generality we can assume that the parallelizing homeomorphism φUp satisfies the condition φUp(p)=(0,0). Then KφUp:=φUp-1({0}×R) is a local section containing p.For a flow{ft:t∈R} and a point p∈R2 we define thefirst positive prolongational limit set and thefirst negative prolongational limit set of p by
(5)J+(p)≔{q∈R2:thereexistsequences(pn)n∈Z+,(tn)n∈Z+suchthatpn⟶p,tn⟶+∞,ftn(pn)⟶qasn⟶+∞q∈R2:thereexistsequences(pn)n∈Z+},J-(p)≔{q∈R2:thereexistsequences(pn)n∈Z+,(tn)n∈Z+suchthatpn⟶p,tn⟶-∞,ftn(pn)⟶qasn⟶+∞q∈R2:thereexistsequences(pn)n∈Z+}.The setJ(p):=J+(p)∪J-(p) is called thefirst prolongational limit set of p. For a subset H⊂R2 we define
(6)J(H):=⋃p∈H‍J(p).
The set J(R2) will be called thefirst prolongational limit set of the flow {ft:t∈R}. For all p,q∈R2 we have
(7)q∈J+(p)⟺p∈J-(q).In [5] it has been proven that for each point p∈R2 the set P+(p) is contained in J+(p). Now we prove the converse inclusion.Theorem 2.
Letf be a Brouwer homeomorphism which is embeddable in a flow {ft:t∈R} and let p∈R2. Then J+(p)⊂P+(p).Proof.
Letq∈J+(p). Denote by Spq the strip between trajectories Cp and Cq of points p and q, respectively. Then for each z∈Spq the trajectory Cp is contained in the strip Sqz between trajectories Cq and Cz of points q and z, respectively, and the trajectories Cq and Cz are subsets of the same component of Sqz∖Cp (see [8]). Let K0 and L0 be local sections of {ft:t∈R} such that p∈K0 and q∈L0.
LetB be a Jordan domain containing p in its interior. If K0∩bdB≠∅, then by compactness of bdB, there exists a p0∈K0∩Spq such that p0 is the only common point of bdB with the subarc K of K0 having p and p0 as its endpoints. If K0∩bdB=∅, then we put K:=K0∩(Spq∪Cp). Take an rB>0 such that B(p,rB)⊂int⁡B and B(p,rB)∩Spqis contained in the union of all trajectories having a common point with K, where B(p,rB) denotes the ball with centre p and radius rB. Fix a T>0 and an rq>0. Without loss of generality we can assume that B(q,rq)∩B(p,rB)=∅.
Now we take anr∈(0,rq) for which there exists a y∈L0∩Spq such that dist⁡(q,y)>r, where dist⁡ denotes the Euclidean metric on the plane. Then bdB(q,r)∩L0∩Spq≠∅. By compactness of bdB(q,r), there exists a q0∈L0∩Spq such that q0 is the only common point of bdB(q,r) with the subarc L of L0 having q and q0 as its endpoints. Denote by W the union of all trajectories having a common point with the arc L. Since q0∈Spq, each trajectory contained in W is a subset of the component of cl⁡Sqq0∖Cp which contains Cq and Cq0, where Sqq0 denotes the strip between trajectories Cq and Cq0 of points q and q0.
By the assumption thatq∈J+(p), there exist sequences (pn)n∈Z+ and (tn)n∈Z+ such that pn→p, tn→+∞, ftn(pn)→q as n→+∞. Thus there exists an n0∈Z+ such that for all n>n0 we have tn>T, pn∈B(p,rB) and ftn(pn)∈B(q,r)∩W. Then, for every n>n0 there exists αn∈R such that ftn+αn(pn)∈L. Moreover, by the definition of rB, for every n>n0 there exists xn∈K and βn∈R such that fβn(xn)=pn. Thus ftn+αn+βn(xn)∈L for n>n0.
Fix anyn>n0 and take a positive integer kn such that kn>tn+αn+βn and kn>T. Then xn and fkn(xn) belong to different components of W∖L, since L is a section in W. By continuity of fkn at p there exists a yn∈fkn(K) such that xn and yn belong to the same component of W∖L, since any neighbourhood of fkn(p) must contain a point from fkn(K). Thus fkn(K) has a common point wn with L. Then wn∈B(q,r) and hence wn∈B(q,rq). Taking zn=f-kn(wn) we have zn∈B, since K⊂B. Consequently, for each n>n0 we have kn>T and fkn(zn)∈B(q,r). Hence kn→+∞ and fkn(zn)→q as n→+∞, which implies that q∈ωf(B). Consequently q∈P+(p).Since an analogous reasoning can be applied to the set of strongly negatively irregular points and the first negative prolongational limit set, our considerations can be summarized in the following way.Corollary 3.
Letf be a Brouwer homeomorphism which is embeddable in a flow {ft:t∈R} and let p∈R2. Then P+(p)=J+(p) and P-(p)=J-(p), and consequently the set of all strongly irregular points of f is equal to the first prolongational limit set of the flow {ft:t∈R}.Corollary 4.
Letf be a Brouwer homeomorphism which is embeddable in a flow. Then, for each flow containing f, the first prolongational limit set is the same.After a reparametrization of the flow{ft:t∈R} containing f each element ft of the flow, for t∈R∖{0} or t>0, respectively, can be treated as f.Corollary 5.
Letf be a Brouwer homeomorphism which is embeddable in a flow {ft:t∈R}. Then the set of all strongly irregular points of ft is the same for all t∈R∖{0}. Moreover, the set of all strongly positive irregular points of ft and the set of all strongly negative irregular points of ft are the same for all t>0.


## 3. Flows of Brouwer Homeomorphisms

In this section we describe the form of any flow of Brouwer homeomorphisms. To give a sufficient condition for the topological conjugacy of flows of Brouwer homeomorphisms one can use covers of the plane by maximal parallelizable regions. We will study the functions which express the relations between parallelizing homeomorphisms of such regions.It is known that a simply connected regionU is parallelizable if and only if J(U)∩U=∅. Hence for every parallelizable region U we have J(U)⊂bdU. In the case where U is a maximal parallelizable region (i.e., U is not contained properly in any parallelizable region), the boundary of U consists of strongly irregular points. It follows from the fact that for each maximal parallelizable region U the equality J(U)=bdU holds. The proof of this fact can be found in [9]. For the convenience of the reader, we outline the essential ideas in that proof.LetU be a parallelizable region. Assume that there exists a point p∈bdU such that p∉J(U). Denote by D1 the component of R2∖Cp which has a common point with U and by D2 the other component of R2∖Cp. Let V be a parallelizable region which contains p and put V1≔V∩D2. Let U1≔U∪Cp∪V1. We show that J(q)∩U1=∅ for each q∈U1, which means that U1 is a parallelizable region. To see this we consider three cases. First, let us consider the case where q∈U. Then J(q)⊂cl⁡D1, since q∈D1. Hence by parallelizability of U, we have J(q)∩U=∅ and by the assumption that p∉J(U), we get J(q)∩Cp=∅. Thus J(q)∩U1=∅. Now, let q∈V1. Then J(q)⊂cl⁡D2. Hence J(q)∩U1=∅, since by parallelizability of V we have J(q)∩(Cp∪V1)=∅. Finally, let q∈Cp. Then, as in the previous case, J(q)∩(Cp∪V1)=∅, and by the assumption that p∉J(U), we get J(q)∩U=∅. Thus we proved that J(U1)∩U1=∅, which means that U1 is parallelizable. Since U is contained properly in U1, we obtain that U cannot be a maximal parallelizable region.For any distinct trajectoriesCp1, Cp2, and Cp3 of {ft:t∈R} one of the following two possibilities must be satisfied: exactly one of the trajectories Cp1, Cp2, and Cp3 is contained in the strip between the other two or each of the trajectories Cp1, Cp2, and Cp3 is contained in the strip between the other two. In the first case if Cpj is the trajectory which lies in the strip between Cpi and Cpk we will write Cpi|Cpj|Cpk (i, j, k∈{1,2,3} and i, j, k are different). In the second case we will write |Cpi,Cpj,Cpk| (cf. [10]).LetX be a nonempty set. Denote by X<ω the set of all finite sequences of elements of X. A subset T of X<ω is called atree on X if it is closed under initial segments; that is, for all positive integers m, n such that n>m if (x1,…,xm,…,xn)∈T, then (x1,…,xm)∈T. Let α=(x1,…,xn)∈X<ω. Then, for any x∈X by α^x we denote the sequence (x1,…,xn,x). A node α=(x1,…,xn)∈T of a tree T is said to beterminal if there is no node of properly extending it; that is, there is no element x∈X such that α^x∈T.A treeA+⊂Z+<ω will be termedadmissible if the following conditions hold:(i)
A+ contains the sequence 1 and no other one-element sequence;(ii)
ifα^k is in A+ and k>1, then so also is α^(k-1).A treeA-⊂Z-<ω will be termedadmissible if the following conditions hold:(iii)
A- contains the sequence -1 and no other one-element sequence;(iv)
ifα^k is in A- and k<-1, then so also is α^(k+1).The setA≔A+∪A- will be said to beadmissible class of finite sequences, where A+ and A- are some admissible classes of finite sequences of positive and negative integers, respectively.Now we recall results describing the flows of Brouwer homeomorphisms.Theorem 6 (see [11]).
Let{ft:t∈R} be a flow of Brouwer homeomorphisms. Then there exists a family of trajectories {Cα:α∈A} and a family of maximal parallelizable regions {Uα:α∈A}, where A=A+∪A- is an admissible class of finite sequences, such that U1=U-1, C1=C-1, and
(8)Cα⊂Uαforα∈A,⋃α∈A‍Uα=R2,Uα∩Uα^i≠∅forα^i∈A,Cα^i⊂bdUαforα^i∈A,|Cα,Cα^i1,Cα^i2|forα^i1,α^i2∈A,i1≠i2,Cα|Cα^i|Cα^i^jforα^i^j∈A.Proposition 7 (see [11]).
Let{ft:t∈R} be a flow of Brouwer homeomorphisms. Then there exists a family of the parallelizing homeomorphisms {φα:α∈A+}, where φα:Uα→R2, Uα are those occurring in Theorem 6, and for each α^i∈A+(9)φα^i(Uα∩Uα^i)=R×(cα^i,0),φα(Uα∩Uα^i)=R×(cα,dα),
where cα∈R∪{-∞}, dα∈R∪{+∞}, and cα^i∈[-∞,0) are some constants such that cα<dα and at least one of the constants cα, dα is finite. Moreover, there exists a continuous function μα^i:(cα,dα)→R and a homeomorphism να^i:(cα,dα)→(cα^i,0) such that the homeomorphism
(10)hα^i:R×(cα,dα)⟶R×(cα^i,0)
given by the relation hα^i:=φα^i∘(φα|Uα∩Uα^i)-1 has the form
(11)hα^i(t,s)=(μα^i(s)+t,να^i(s)),t∈R,s∈(cα,dα).The above proposition is formulated forα∈A+, but the analogous result holds for α∈A-. The admissible class of finite sequences occurring in Theorem 6 is not unique for a given flow, so we can usually choose a convenient A when solving a problem of topological conjugacy.The homeomorphismsνα^i occurring in Proposition 7 can be either increasing or decreasing. For each α^i∈A denote by Cα^iα the unique trajectory contained in Uα∩J(Cα^i) (the uniqueness has been proven in [8]). From the construction of the families {Cα:α∈A} and {Uα:α∈A} occurring in Theorem 6 we obtain that, in case Cα|Cα^iα|Cα^i or Cα=Cα^iα, the homeomorphism να^i is decreasing and cα>0 or cα=0, respectively. However, in case |Cα,Cα^iα,Cα^i|, the homeomorphism να^i is increasing and dα>0 (see [11]).The continuous functionsμα^i describe the time needed for the flow {ft:t∈R} to move from the point with coordinates (0,να^i(s)) in the chart φα^i until it reaches the point with coordinates (0,s) in the chart φα. In other words, μα^i describe the time needed for the flow to move from a point from the section Kφα^i in Uα^i to a point from the section Kφα in Uα.Proposition 8.
The functionsμα^i occurring in Proposition 7 satisfy the condition
(12)lim⁡s→cαμα^i(s)={-∞ifCα^i⊂J+(Cα^iα),+∞ifCα^i⊂J-(Cα^i)
in the case where Cα|Cα^iα|Cα^i or Cα=Cα^iα or the condition
(13)lim⁡s→dαμα^i(s)={-∞ifCα^i⊂J+(Cα^iα),+∞ifCα^i⊂J-(Cα^i)
in the case where |Cα,Cα^iα,Cα^i|.Proof.
Let us consider the case whereCα|Cα^iα|Cα^i or Cα=Cα^iα and assume that Cα^i⊂J+(Cα^iα). The other cases are similar. Denote by p and q the points for which φα(p)=(0,cα) and φα^i(q)=(0,0); that is, p∈Kφα∩Cα^iα and q∈Kφα^i∩Cα^i. Then q∈J+(p). Thus there exist sequences (pn)n∈Z+ and (tn)n∈Z+ such that pn→p, tn→+∞, and ftn(pn)→q as n→+∞. This means that there exist sequences (un)n∈Z+, (sn)n∈Z+ such that un→0, sn→cα, where φα(pn)=(un,sn). Hence φα(ftn(pn))=φα(pn)+(tn,0)=(tn+un,sn) and by (11)
(14)hα^i(tn+un,sn)=(μα^i(sn)+tn+un,να^i(sn)).
Thus μα^i(sn)+tn+un→0 as n→+∞, since ftn(pn)→q as n→+∞. Hence μα^i(sn)→-∞, since tn→+∞ and un→0. Consequently, liminf⁡s→cαμα^i(s)=-∞.
Suppose, on the contrary, that there exists a sequence(sn)n∈Z+ such that sn→cα and μα^i(sn)→c for some c∈R. Consider the sequence (pn)n∈Z+ such that φα(pn)=(0,sn). Then each element of the sequence (pn)n∈Z+ belongs to Kφα. Moreover, the sequence (pn)n∈Z+ tends to the point p such that φα(p)=(0,cα). Hence p∈Kφα∩Cα^iα. On the other hand, by (11)
(15)φα^i(pn)=hα^i(0,sn)=(μα^i(sn),να^i(sn)).
Hence lim⁡n→+∞φα^i(pn)=(c,0). Consequently lim⁡n→+∞pn=q~, where q~ is a point such that φα^i(q~)=(c,0); that is, q~∈Cα^i. But this is impossible, since Cα^i∩Cα^iα=∅.By the fact thatνα^i:(cα,dα)→(cα^i,0) is a homeomorphism, the function h~α^i:R×(cα^i,0)→R×(cα^i,0) defined by
(16)h~α^i(t,s)≔((μα^i∘να^i-1)(v)+t,v),t∈R,v∈(cα^i,0)
is continuous. Moreover, putting s≔να^i-1(v) in Proposition 8 we obtain the following result.Corollary 9.
The functionsηα^i:(cα^i,0)→R given by
(17)ηα^i≔μα^i∘να^i-1,
where μα^i and να^i are those occurring in Proposition 7, satisfy the condition
(18)lim⁡v→0ηα^i(v)={-∞ifCα^i⊂J+(Cα^iα),+∞ifCα^i⊂J-(Cα^i).


## 4. Topological Conjugacy of Generalized Reeb Flows

In this section we consider the problem of topological conjugacy of a class of flows of Brouwer homeomorphisms. To prove our result we use the form of such flows.We say that flows{ft:t∈R} and {gt:t∈R}, where ft,gt:R2→R2, aretopologically conjugate if there exists a homeomorphism Φ of the plane onto itself such that
(19)gt=Φ-1∘ft∘Φ,t∈R.In [12] a lemma can be found which says that the set of strongly irregular points (called the set of singular pairs there) is invariant with respect to topological conjugacy of flows. Thus, by Corollary 3, we have the following result.Proposition 10.
Let{ft:t∈R} and {gt:t∈R} be topologically conjugate flows of Brouwer homeomorphisms and let Φ:R2→R2 be a homeomorphism which conjugates the flows. Then Φ(J{ft}(R2))=J{gt}(R2), where J{ft}(R2) and J{gt}(R2) denote the first prolongational limit set of {ft:t∈R} and {gt:t∈R}, respectively.Put(20)P0≔{(x,y)∈R2:x>0,y>0},P1≔{(x,y)∈R2:x<0,y>0},P2≔{(x,y)∈R2:x>0,y<0},Lx≔{(x,0)∈R2:x>0},Ly≔{(0,y)∈R2:y>0}
and U:=P0∪P1∪P2∪Lx∪Ly. Consider the flow {gt:t∈R}, where for each t∈R the homeomorphism gt:U→U is defined by
(21)gt(x,y)≔{(2tx,2-ty)if(x,y)∈P0∪Lx∪Ly,(x,2-ty)if(x,y)∈P1,(2tx,y)if(x,y)∈P2.
Then J+(U)=Lx and J-(U)=Ly.PutA+≔{1,(1,1)}, U1≔P1∪Ly∪P0, U(1,1)≔P0∪Lx∪P2, C1≔Ly, C(1,1)≔Lx and A-≔{-1}, U-1≔U1, C-1≔C1. Then C(1,1)1≔C1 and C(1,1)⊂J+(C1). Let
(22)Kφ1≔{(s,1):s∈R},Kφ(1,1)≔{(1,s):s∈R}.
Note that the trajectories of {gt:t∈R} contained in P0 are given by the equation xy=s for s∈(0,+∞). Hence
(23)μ(1,1):(0,+∞)⟶R,μ(1,1)(s)=log⁡2s,
since glog⁡2s(1,s)=(2log⁡2s,2-log⁡2ss)=(s,1). Moreover,
(24)ν(1,1):(0,+∞)⟶(-∞,0),ν(1,1)(s)=-s.For each flow{ft:t∈R}, where ft:U→U for t∈R, having the same trajectories (including the orientation) as the flow {gt:t∈R} given by (21), one can consider the function μ{ft},(1,1):(0,+∞)→R occurring in Proposition 7 which describes the time needed to move from each point p∈P0∩Kφ(1,1) to the point of Kφ1 belonging to the trajectory of p, that is, from the point of the form (1,s) to the point of the form (s,1) for some s∈(0,+∞). Then by Proposition 8(25)lim⁡s→0μ{ft},(1,1)(s)=-∞.Consider a constantσ(μ{ft},(1,1))∈[0,+∞] defined by
(26)σ(μ{ft},(1,1))≔limsup⁡v→0μ{ft},(1,1)*(v),
where μ{ft},(1,1)*:(0,1]→[0,+∞) is given by
(27)μ{ft},(1,1)*(v)≔μ{ft},(1,1)(v)-min⁡{μ{ft},(1,1)(s):s∈[v,1]}
(cf. [13, 14]). Then the flow {ft:t∈R} is topologically conjugate to the flow {gt:t∈R} given by (21) if and only if σ(μ{ft},(1,1))=0 (see [13]). In particular, this condition holds in the case where μ{ft},(1,1) is increasing.Now we introduce a class of flows of Brouwer homeomorphisms. Putα1≔1 and αn+1≔αn^1 for n∈Z+. For any positive integer k we define Ak≔{αn:1≤n≤k} and A+∞≔{αn:n∈Z+}. Similarly, put α-1≔-1, αn-1≔αn^-1 for n∈Z- and for any negative integer k let Ak≔{αn:k≤n≤-1} and A-∞:={αn:n∈Z-}.Consider a flow{ht:t∈R} of Brouwer homeomorphisms ht:R2→R2 such that A=A+∪A- can be given in one of the following forms:(a)
A-={-1} and A+=Ak for some k∈Z+,(b)
A-={-1} and A+=A+∞,(c)
A-=A-∞ and A+=A+∞. We assume that Uαn∩J(R2)=Cαn for each αn∈A, where {Uαn:αn∈A} and {Cαn:αn∈A} are whose occurring in Theorem 6. Then Cαn+1αn=Cαn, since Cαn+1αn⊂Uαn∩J(R2) for every αn+1∈A+. Similarly, Cαn-1αn=Cαn for every αn-1∈A-.Fix anαn+1∈A+. Denote by Vαn+1 the strip between Cαn and Cαn+1. Then Vαn+1⊂Uαn and |C,Cαn,Cαn+1| for every trajectory C⊂Vαn+1 (see [8]). In particular, if Cαn is equal to the vertical line {(n-1,y):y∈R} for each αn∈A+, then Vαn+1 is a vertical strip for each αn+1∈A+. In a similar way we define the strip Vαn-1 for αn-1∈A-.Let us assume that for eachαn∈A∖{1,-1} there exists a homeomorphism ψαn:cl⁡Vαn→P0∪Lx∪Ly such that
(28)ht=ψαn-1∘gt∘ψαn,t∈R,
where {gt:t∈R} is given by (21). If Cαn⊂J+(Cαn-1), then ψαn(Cαn-1)=Ly and ψαn(Cαn)=Lx. In case Cαn⊂J-(Cαn-1) we have ψαn(Cαn-1)=Lx and ψαn(Cαn)=Ly. The flow {ht:t∈R} described above will be called astandard generalized Reeb flow.A standard generalized Reeb flow can have either a finite number of maximal parallelizable regions or an infinite number of such regions. The first case holds if the set of indicesA of the flow is of the form (a). However, the second case holds if this set is of the form (b) or (c). The trajectories of a standard generalized Reeb flow with an infinite number of maximal parallelizable regions are shown in Figures 1 and 2 for the set A of the forms (b) and (c), respectively.Figure 1
A generalized Reeb flow withA-={-1} and A+=A+∞.Figure 2
A generalized Reeb flow withA-=A-∞ and A+=A+∞.Consider a flow{ft:t∈R} of Brouwer homeomorphisms which has the same trajectories as a standard generalized Reeb flow. For αn∈A∖{1,-1} and s∈(0,+∞) denote by Csαn the image of the trajectory {(x,y)∈P0:xy=s} of {gt:t∈R} under ψαn-1. For each αn∈A∖{1,-1} consider the function μ{ft},αn:(0,+∞)→R taking as μ{ft},αn(s) the time needed to move from the unique point of the set Csαn∩ψαn-1(Kφ(1,1)) to the unique point of Csαn∩ψαn-1(Kφ1). Define μ{ft},αn*:(0,1]→[0,+∞) by
(29)μ{ft},αn*(v)≔μ{ft},αn(v)-min⁡{μ{ft},αn(s):s∈[v,1]}
in case Cαn+1⊂J+(Cαn), and by
(30)μ{ft},αn*(v)≔max⁡{μ{ft},αn(s):s∈[v,1]}-μ{ft},αn(v)
in case Cαn+1⊂J-(Cαn). Put
(31)σ(μ{ft},αn)≔limsup⁡v→0⁡μ{ft},αn*(v).Now we can prove the following conjugacy result.Theorem 11.
Let{ht:t∈R} be a standard generalized Reeb flow. Let A be an admissible class of finite sequences satisfying one of the conditions (a)–(c). Assume that {ft:t∈R} is a flow of Brouwer homeomorphisms having the same trajectories including orientation as {ht:t∈R}. If σ(μ{ft},αn)=0 for all αn∈A∖{1,-1}, then the flows {ft:t∈R} and {ht:t∈R} are topologically conjugate.Proof.
Assume that one of the conditions (a) and (b) holds. First, let us note that there exists a topological conjugacyΦ1:U1→U1 of flows {ft|U1:t∈R} and {ht|U1:t∈R}, since U1 is a parallelizable region of each of these flows. More precisely, if φ{ft},1:U1→R2 and φ{ht},1:U1→R2 are parallelizing homeomorphisms for {ft|U1:t∈R} and {ht|U1:t∈R}, respectively, then for every p∈U1 we put Φ1(p):=(φ{ht},1-1∘φ{ft},1)(p).
Fix anαn+1∈A+. Assume that we have defined a homeomorphism Φn which conjugates {ft:t∈R} and {ht:t∈R} on the set ⋃i=1n‍Uαi. Define Ft:P0∪Lx∪Ly→P0∪Lx∪Ly by Ft:=ψαn+1∘ft∘ψαn+1-1 for t∈R, where ψαn+1 satisfies (28). Then σ(μ{Ft},(1,1))=σ(μ{ft},αn). Hence σ(μ{Ft},(1,1))=0, since by the assumption σ(μ{ft},αn)=0. Thus {Ft:t∈R} and {gt|P0∪Lx∪Ly:t∈R} are topologically conjugate. Consequently {ft|Wαn+1:t∈R} and {ht|Wαn+1:t∈R} are topologically conjugate, where Wαn+1≔cl⁡Vαn+1. Denote by ϕαn+1 the homeomorphism which conjugates these flows.
Fix anyp0∈Cαn and put q1≔Φn(p0), q2≔ϕαn+1(p0). Take t0∈R such that ht0(q2)=q1 and define Φαn+1≔ht0|Wαn+1∘ϕαn+1. Then Φαn+1 conjugates the flows {ft|Wαn+1:t∈R} and {ht|Wαn+1:t∈R}, since
(32)Φαn+1∘ft|Wαn+1=ht0|Wαn+1∘ϕαn+1∘ft|Wαn+1=ht0|Wαn+1∘ht|Wαn+1∘ϕαn+1=ht|Wαn+1∘ht0|Wαn+1∘ϕαn+1=ht|Wαn+1∘Φαn+1.
Moreover Φαn+1(p0)=q1, since
(33)Φαn+1(p0)=ht0(ϕαn+1(p0))=ht0(q2)=q1.
Hence Φαn+1|Cαn=Φn|Cαn. Thus we can define Φn+1 by
(34)Φn+1(p)≔{Φn(p),p∈⋃i=1n‍Uαi∖Vαn+1,Φαn+1(p),p∈Vαn+1.
Then Φn+1 conjugates {ft:t∈R} and {ht:t∈R} on the set ⋃i=1n‍Uαi∪Cαn+1. Since {ft:t∈R} and {ht:t∈R} are parallelizable on Uαn+1 we can extend the topological conjugacy Φn+1 on the component of Uαn+1∖Cαn+1 which do not contain Cαn (see [12]). Such an extension is really needed in case of (a) to obtain the conjugacy on the whole plane. In case of (c), for any αn-1∈A- we extend Φn from ⋃i=-1n‍Uαi to Φn-1 defined on ⋃i=-1n‍Uαi∪Cαn-1 in a similar way.



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*Source: 638784-2014-10-16.xml*