AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/638784 638784 Research Article On Strongly Irregular Points of a Brouwer Homeomorphism Embeddable in a Flow Leśniak Zbigniew Anderson Douglas R. 1 Institute of Mathematics Pedagogical University Podchorążych 2 30-084 Kraków Poland up.krakow.pl 2014 16102014 2014 25 02 2014 25 07 2014 16 10 2014 2014 Copyright © 2014 Zbigniew Leśniak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the set of all strongly irregular points of a Brouwer homeomorphism f 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.

1. Introduction

In this part we recall the requisite definitions and results concerning Brouwer homeomorphisms and flows of such homeomorphisms.

By a Brouwer homeomorphism we mean an orientation preserving homeomorphism of the plane onto itself which has no fixed points. By a flow we mean a group of homeomorphisms of the plane onto itself {ft:tR} under the operation of composition which satisfies the following conditions:

the function F:R2×RR2,F(x,t)=ft(x) is continuous,

f0(x)=x for xR2,

ft(fs(x))=ft+s(x) for xR2, t,sR.

We say that a Brouwer homeomorphism f is embeddable in a flow if there exists a flow {ft:tR} such that f=f1. Then for each tR{0}, ft is a Brouwer homeomorphism.

For any sequence of subsets (An)nZ+ of the plane we define limes superior limsupnAn as the set of all points pR2 such that any neighbourhood of p has common points with infinitely many elements of the sequence (An)nZ+. For any subset B of the plane we define the positive limit set ωf(B) as the limes superior of the sequence of its iterates (fn(B))nZ+ and negative limit set αf(B) as the limes superior of the sequence (f-n(B))nZ+. Under the assumption that B is compact, Nakayama  proved that(1)ωf(B)={qR2:there  exist  sequences  (pj)jZ+,  (nj)jZ+  such  that  pjB,njZ+,nj+,  fnj(pj)q  as  j+qR2:there  exist  sequences  (pj)jZ+},αf(B)={qR2:there  exist  sequences  (pj)jZ+,  (nj)jZ+  such  that  pjB,njZ+,nj+,    f-nj(pj)q  as  j+qR2:there  exist  sequences  (pj)jZ+}.

A point p is called positively irregular if ωf(B) for each Jordan domain B containing p in its interior and negatively 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 R2J). A point which is not positively irregular is said to be positively regular. Similarly, a point which is not negatively irregular is called negatively regular. A point which is positively or negatively irregular is called irregular, otherwise it is regular.

We say that a set UR2 is invariant under f if f(U)=U. An invariant simply connected region UR2 is said to be parallelizable if there exists a homeomorphism φU mapping U onto R2 such that (2)f(x)=φU-1(φU(x)+(1,0))forxU. The homeomorphism φU occurring in this equality is called a parallelizing homeomorphism of f|U. On account of the Brouwer Translation Theorem, for each pR2 there exists a parallelizable region U containing p (see ).

Homma and Terasaka  proved a theorem describing the structure of an arbitrary Brouwer homeomorphism. The theorem can be formulated in the following way.

Theorem 1 (see [<xref ref-type="bibr" rid="B3">3</xref>], First Structure Theorem).

Let f be a Brouwer homeomorphism. Then the plane is divided into at most three kinds of pairwise disjoint sets: {Oi:iI}, where I=Z+ or I={1,,n} for a positive integer n, {Oi:iZ+}, and F. The sets {Oi:iI} and {Oi:iZ+} 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 Oifn(Oi)= for nZ{0}. The set F is invariant and closed and consists of all irregular points.

For an irregular point p 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 is strongly positively irregular if P+(p), otherwise it is weakly positively irregular. Similarly, p is strongly negatively irregular if P-(p), otherwise it is weakly negatively irregular. We say that p is strongly irregular if it is strongly positively irregular or strongly negatively irregular. Otherwise, an irregular point p is said to be weakly irregular.

Homma and Terasaka  proved that for all p,qR2(3)qP+(p)pP-(q). Nakayama  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 ).

A counterpart of Theorem 1 for a Brouwer homeomorphism embeddable in a flow has been given in . 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.

Let f be a Brouwer homeomorphism. Assume that there exists a flow {ft:tR} such that f1=f. Let UR2 be a simply connected region such that ft(U)=U for tR. We say that U is a parallelizable 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))forxU,tR. Such a homeomorphism φU will be called a parallelizing homeomorphism of the flow {ft|U:tR}. It is known that for any simply connected region U which is invariant under the flow {ft:tR} the existence of a parallelizing homeomorphism of f|U is equivalent to the existence of a parallelizing homeomorphism of {ft|U:tR} (see ).

By the trajectory of a point pR2 we mean the set Cp:={ft(p):tR}. It is known that a region U is parallelizable if and only if there exists a topological line K 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:tR} contained in U (see , page 49). Such a set K we will call a section in U (or a local section of {ft:tR}). On account of the Whitney-Bebutov Theorem (see , page 52), for each pR2 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:tR} and a point pR2 we define the first positive prolongational limit set and the first negative prolongational limit set of p by (5)J+(p){qR2:there  exist  sequences  (pn)nZ+,(tn)nZ+  such  that  pnp,tn+,ftn(pn)q  as  n+qR2:    there  exist  sequences  (pn)nZ+},J-(p){qR2:there  exist  sequences  (pn)nZ+,(tn)nZ+  such  that  pnp,tn-,ftn(pn)q  as  n+qR2  :  there  exist  sequences  (pn)nZ+}.

The set J(p):=J+(p)J-(p) is called the first prolongational limit set of p. For a subset HR2 we define (6)J(H):=pHJ(p). The set J(R2) will be called the first prolongational limit set of the flow {ft:tR}. For all p,qR2 we have (7)qJ+(p)pJ-(q).

In  it has been proven that for each point pR2 the set P+(p) is contained in J+(p). Now we prove the converse inclusion.

Theorem 2.

Let f be a Brouwer homeomorphism which is embeddable in a flow {ft:tR} and let pR2. Then J+(p)P+(p).

Proof.

Let qJ+(p). Denote by Spq the strip between trajectories Cp and Cq of points p and q, respectively. Then for each zSpq 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 SqzCp (see ). Let K0 and L0 be local sections of {ft:tR} such that pK0 and qL0.

Let B be a Jordan domain containing p in its interior. If K0bdB, then by compactness of bdB, there exists a p0K0Spq such that p0 is the only common point of bdB with the subarc K of K0 having p and p0 as its endpoints. If K0bdB=, then we put K:=K0(SpqCp). Take an rB>0 such that B(p,rB)intB 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 an r(0,rq) for which there exists a yL0Spq such that dist(q,y)>r, where dist denotes the Euclidean metric on the plane. Then bdB(q,r)L0Spq. By compactness of bdB(q,r), there exists a q0L0Spq 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 q0Spq, each trajectory contained in W is a subset of the component of clSqq0Cp which contains Cq and Cq0, where Sqq0 denotes the strip between trajectories Cq and Cq0 of points q and q0.

By the assumption that qJ+(p), there exist sequences (pn)nZ+ and (tn)nZ+ such that pnp, tn+, ftn(pn)q as n+. Thus there exists an n0Z+ such that for all n>n0 we have tn>T, pnB(p,rB) and ftn(pn)B(q,r)W. Then, for every n>n0 there exists αnR such that ftn+αn(pn)L. Moreover, by the definition of rB, for every n>n0 there exists xnK and βnR such that fβn(xn)=pn. Thus ftn+αn+βn(xn)L for n>n0.

Fix any n>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 WL, since L is a section in W. By continuity of fkn at p there exists a ynfkn(K) such that xn and yn belong to the same component of WL, since any neighbourhood of fkn(p) must contain a point from fkn(K). Thus fkn(K) has a common point wn with L. Then wnB(q,r) and hence wnB(q,rq). Taking zn=f-kn(wn) we have znB, since KB. 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 qωf(B). Consequently qP+(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.

Let f be a Brouwer homeomorphism which is embeddable in a flow {ft:tR} and let pR2. 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:tR}.

Corollary 4.

Let f 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:tR} containing f each element ft of the flow, for tR{0} or t>0, respectively, can be treated as f.

Corollary 5.

Let f be a Brouwer homeomorphism which is embeddable in a flow {ft:tR}. Then the set of all strongly irregular points of ft is the same for all tR{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 region U 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 . For the convenience of the reader, we outline the essential ideas in that proof.

Let U be a parallelizable region. Assume that there exists a point pbdU such that pJ(U). Denote by D1 the component of R2Cp which has a common point with U and by D2 the other component of R2Cp. Let V be a parallelizable region which contains p and put V1VD2. Let U1UCpV1. We show that J(q)U1= for each qU1, which means that U1 is a parallelizable region. To see this we consider three cases. First, let us consider the case where qU. Then J(q)clD1, since qD1. Hence by parallelizability of U, we have J(q)U= and by the assumption that pJ(U), we get J(q)Cp=. Thus J(q)U1=. Now, let qV1. Then J(q)clD2. Hence J(q)U1=, since by parallelizability of V we have J(q)(CpV1)=. Finally, let qCp. Then, as in the previous case, J(q)(CpV1)=, and by the assumption that pJ(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 trajectories Cp1, Cp2, and Cp3 of {ft:tR} 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. ).

Let X 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 a tree 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 xX by α    ^x we denote the sequence (x1,,xn,x). A node α=(x1,,xn)T of a tree T is said to be terminal if there is no node of properly extending it; that is, there is no element xX such that α    ^xT.

A tree A+Z+<ω will be termed admissible if the following conditions hold:

A+ contains the sequence 1 and no other one-element sequence;

if α    ^k is in A+ and k>1, then so also is α    ^(k-1).

A tree A-Z-<ω will be termed admissible if the following conditions hold:

A- contains the sequence -1 and no other one-element sequence;

if α    ^k is in A- and k<-1, then so also is α    ^(k+1).

The set AA+A- will be said to be admissible 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 [<xref ref-type="bibr" rid="B11">11</xref>]).

Let {ft:tR} be a flow of Brouwer homeomorphisms. Then there exists a family of trajectories {Cα:αA} and a family of maximal parallelizable regions {Uα:αA}, where A=A+A- is an admissible class of finite sequences, such that U1=U-1, C1=C-1, and (8)CαUαforαA,αAUα=R2,UαUα    ^iforα    ^iA,Cα    ^ibdUαforα    ^iA,|Cα,Cα    ^i1,Cα    ^i2|forα    ^i1,α    ^i2A,i1i2,Cα|Cα    ^i|Cα    ^i    ^jforα    ^i    ^jA.

Proposition 7 (see [<xref ref-type="bibr" rid="B11">11</xref>]).

Let {ft:tR} be a flow of Brouwer homeomorphisms. Then there exists a family of the parallelizing homeomorphisms {φα:αA+}, where φα:UαR2, Uα are those occurring in Theorem 6, and for each α    ^iA+(9)φα    ^i(UαUα    ^i)=R×(cα    ^i,0),φα(UαUα    ^i)=R×(cα,dα), where cαR{-}, dαR{+}, 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|UαUα    ^i)-1 has the form (11)hα    ^i(t,s)=(μα    ^i(s)+t,να    ^i(s)),tR,s(cα,dα).

The above proposition is formulated for αA+, but the analogous result holds for αA-. 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 homeomorphisms να    ^i occurring in Proposition 7 can be either increasing or decreasing. For each α    ^iA denote by Cα    ^iα the unique trajectory contained in UαJ(Cα    ^i) (the uniqueness has been proven in ). From the construction of the families {Cα:αA} and {Uα:αA} 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 ).

The continuous functions μα    ^i describe the time needed for the flow {ft:tR} 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 functions μα    ^i occurring in Proposition 7 satisfy the condition (12)limscαμα    ^i(s)={-ifCα    ^iJ+(Cα    ^iα),+ifCα    ^iJ-(Cα    ^i) in the case where Cα|Cα    ^iα|Cα    ^i or Cα=Cα    ^iα or the condition (13)limsdαμα    ^i(s)={-ifCα    ^iJ+(Cα    ^iα),+ifCα    ^iJ-(Cα    ^i) in the case where |Cα,Cα    ^iα,Cα    ^i|.

Proof.

Let us consider the case where Cα|Cα    ^iα|Cα    ^i or Cα=Cα    ^iα and assume that Cα    ^iJ+(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, pKφαCα    ^iα and qKφα    ^iCα    ^i. Then qJ+(p). Thus there exist sequences (pn)nZ+ and (tn)nZ+ such that pnp, tn+, and ftn(pn)q as n+. This means that there exist sequences (un)nZ+, (sn)nZ+ such that un0, sncα, 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+un0 as n+, since ftn(pn)q as n+. Hence μα    ^i(sn)-, since tn+ and un0. Consequently, liminfscαμα    ^i(s)=-.

Suppose, on the contrary, that there exists a sequence (sn)nZ+ such that sncα and μα    ^i(sn)c for some cR. Consider the sequence (pn)nZ+ such that φα(pn)=(0,sn). Then each element of the sequence (pn)nZ+ belongs to Kφα. Moreover, the sequence (pn)nZ+ tends to the point p such that φα(p)=(0,cα). Hence pKφαCα    ^iα. On the other hand, by (11) (15)φα    ^i(pn)=hα    ^i(0,sn)=(μα    ^i(sn),να    ^i(sn)). Hence limn+φα    ^i(pn)=(c,0). Consequently limn+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    ^iCα    ^iα=.

By the fact that να    ^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    ^iνα    ^i-1)(v)+t,v),tR,v(cα    ^i,0) is continuous. Moreover, putting sνα    ^i-1(v) in Proposition 8 we obtain the following result.

Corollary 9.

The functions ηα    ^i:(cα    ^i,0)R given by (17)ηα    ^iμα    ^iνα    ^i-1, where μα    ^i and να    ^i are those occurring in Proposition 7, satisfy the condition (18)limv0ηα    ^i(v)={-ifCα    ^iJ+(Cα    ^iα),+ifCα    ^iJ-(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:tR} and {gt:tR}, where ft,gt:R2R2, are topologically conjugate if there exists a homeomorphism Φ of the plane onto itself such that (19)gt=Φ-1ftΦ,tR.

In  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:tR} and {gt:tR} be topologically conjugate flows of Brouwer homeomorphisms and let Φ:R2R2 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:tR} and {gt:tR}, 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:=P0P1P2LxLy. Consider the flow {gt:tR}, where for each tR the homeomorphism gt:UU is defined by (21)gt(x,y){(2tx,2-ty)if(x,y)P0LxLy,(x,2-ty)if(x,y)P1,(2tx,y)if(x,y)P2. Then J+(U)=Lx and J-(U)=Ly.

Put A+{1,(1,1)}, U1P1LyP0, U(1,1)P0LxP2, C1Ly, C(1,1)Lx and A-{-1}, U-1U1, C-1C1. Then C(1,1)1C1 and C(1,1)J+(C1). Let (22)Kφ1{(s,1):sR},Kφ(1,1){(1,s):sR}. Note that the trajectories of {gt:tR} contained in P0 are given by the equation xy=s for s(0,+). Hence (23)μ(1,1):(0,+)R,μ(1,1)(s)=log2s, since glog2s(1,s)=(2log2s,2-log2ss)=(s,1). Moreover, (24)ν(1,1):(0,+)(-,0),ν(1,1)(s)=-s.

For each flow {ft:tR}, where ft:UU for tR, having the same trajectories (including the orientation) as the flow {gt:tR} 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 pP0Kφ(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)lims0μ{ft},(1,1)(s)=-.

Consider a constant σ(μ{ft},(1,1))[0,+] defined by (26)σ(μ{ft},(1,1))limsupv0μ{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:tR} is topologically conjugate to the flow {gt:tR} given by (21) if and only if σ(μ{ft},(1,1))=0 (see ). 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. Put α11 and αn+1αn    ^1 for nZ+. For any positive integer k we define Ak{αn:1nk} and A+{αn:nZ+}. Similarly, put α-1-1, αn-1αn    ^-1 for nZ- and for any negative integer k let Ak{αn:kn-1} and A-:={αn:nZ-}.

Consider a flow {ht:tR} of Brouwer homeomorphisms ht:R2R2 such that A=A+A- can be given in one of the following forms:

A-={-1} and A+=Ak for some kZ+,

A-={-1} and A+=A+,

A-=A- and A+=A+.

We assume that UαnJ(R2)=Cαn for each αnA, where {Uαn:αnA} and {Cαn:αnA} are whose occurring in Theorem 6. Then Cαn+1αn=Cαn, since Cαn+1αnUαnJ(R2) for every αn+1A+. Similarly, Cαn-1αn=Cαn for every αn-1A-.

Fix an αn+1A+. Denote by Vαn+1 the strip between Cαn and Cαn+1. Then Vαn+1Uαn and |C,Cαn,Cαn+1| for every trajectory CVαn+1 (see ). In particular, if Cαn is equal to the vertical line {(n-1,y):yR} for each αnA+, then Vαn+1 is a vertical strip for each αn+1A+. In a similar way we define the strip Vαn-1 for αn-1A-.

Let us assume that for each αnA{1,-1} there exists a homeomorphism ψαn:clVαnP0LxLy such that (28)ht=ψαn-1gtψαn,tR, where {gt:tR} is given by (21). If CαnJ+(Cαn-1), then ψαn(Cαn-1)=Ly and ψαn(Cαn)=Lx. In case CαnJ-(Cαn-1) we have ψαn(Cαn-1)=Lx and ψαn(Cαn)=Ly. The flow {ht:tR} described above will be called a standard 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 indices A 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.

A generalized Reeb flow with A-={-1} and A+=A+.

A generalized Reeb flow with A-=A- and A+=A+.

Consider a flow {ft:tR} of Brouwer homeomorphisms which has the same trajectories as a standard generalized Reeb flow. For αnA{1,-1} and s(0,+) denote by Csαn the image of the trajectory {(x,y)P0:xy=s} of {gt:tR} under ψαn-1. For each αnA{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 Csαnψαn-1(Kφ(1,1)) to the unique point of Csαnψαn-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+1J+(Cαn), and by (30)μ{ft},αn*(v)max{μ{ft},αn(s):s[v,1]}-μ{ft},αn(v) in case Cαn+1J-(Cαn). Put (31)σ(μ{ft},αn)limsupv0μ{ft},αn*(v).

Now we can prove the following conjugacy result.

Theorem 11.

Let {ht:tR} 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:tR} is a flow of Brouwer homeomorphisms having the same trajectories including orientation as {ht:tR}. If σ(μ{ft},αn)=0 for all αnA{1,-1}, then the flows {ft:tR} and {ht:tR} are topologically conjugate.

Proof.

Assume that one of the conditions (a) and (b) holds. First, let us note that there exists a topological conjugacy Φ1:U1U1 of flows {ft|U1:tR} and {ht|U1:tR}, since U1 is a parallelizable region of each of these flows. More precisely, if φ{ft},1:U1R2 and φ{ht},1:U1R2 are parallelizing homeomorphisms for {ft|U1:tR} and {ht|U1:tR}, respectively, then for every pU1 we put Φ1(p):=(φ{ht},1-1φ{ft},1)(p).

Fix an αn+1A+. Assume that we have defined a homeomorphism Φn which conjugates {ft:tR} and {ht:tR} on the set i=1nUαi. Define Ft:P0LxLyP0LxLy by Ft:=ψαn+1ftψαn+1-1 for tR, 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:tR} and {gt|P0LxLy:tR} are topologically conjugate. Consequently {ft|Wαn+1:tR} and {ht|Wαn+1:tR} are topologically conjugate, where Wαn+1clVαn+1. Denote by ϕαn+1 the homeomorphism which conjugates these flows.

Fix any p0Cαn and put q1Φn(p0), q2ϕαn+1(p0). Take t0R such that ht0(q2)=q1 and define Φαn+1ht0|Wαn+1ϕαn+1. Then Φαn+1 conjugates the flows {ft|Wαn+1:tR} and {ht|Wαn+1:tR}, since (32)Φαn+1ft|Wαn+1=ht0|Wαn+1ϕαn+1ft|Wαn+1=ht0|Wαn+1ht|Wαn+1ϕαn+1=ht|Wαn+1ht0|Wαn+1ϕαn+1=ht|Wαn+1Φαn+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),pi=1nUαiVαn+1,Φαn+1(p),pVαn+1. Then Φn+1 conjugates {ft:tR} and {ht:tR} on the set i=1nUαiCαn+1. Since {ft:tR} and {ht:tR} are parallelizable on Uαn+1 we can extend the topological conjugacy Φn+1 on the component of Uαn+1Cαn+1 which do not contain Cαn (see ). 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-1A- we extend Φn from i=-1nUαi to Φn-1 defined on i=-1nUαiCαn-1 in a similar way.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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