Abstract
We study the set of all strongly irregular points of a Brouwer homeomorphism which is embeddable in a flow. We prove that this set is equal to the first prolongational limit set of any flow containing . 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 under the operation of composition which satisfies the following conditions:(1)the function is continuous,(2) for ,(3) for , . We say that a Brouwer homeomorphism is embeddable in a flow if there exists a flow such that . Then for each , is a Brouwer homeomorphism.
For any sequence of subsets of the plane we define limes superior as the set of all points such that any neighbourhood of has common points with infinitely many elements of the sequence . For any subset of the plane we define the positive limit set as the limes superior of the sequence of its iterates and negative limit set as the limes superior of the sequence . Under the assumption that is compact, Nakayama [1] proved that
A point is called positively irregular if for each Jordan domain containing in its interior and negatively irregular if for each Jordan domain containing in its interior, where by a Jordan domain we mean the union of a Jordan curve and the Jordan region determined by (i.e., the bounded component of ). 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 is invariant under if . An invariant simply connected region is said to be parallelizable if there exists a homeomorphism mapping onto such that The homeomorphism occurring in this equality is called a parallelizing homeomorphism of . On account of the Brouwer Translation Theorem, for each there exists a parallelizable region containing (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). Let be a Brouwer homeomorphism. Then the plane is divided into at most three kinds of pairwise disjoint sets: , where or for a positive integer , , and . The sets and are the components of the set of all regular points such that each is a parallelizable unbounded simply connected region and each is a simply connected region satisfying the condition for . The set is invariant and closed and consists of all irregular points.
For an irregular point of a Brouwer homeomorphism the set is defined as the intersection of all and the set as the intersection of all , where is a Jordan domain containing in its interior. An irregular point is strongly positively irregular if , otherwise it is weakly positively irregular. Similarly, is strongly negatively irregular if , otherwise it is weakly negatively irregular. We say that is strongly irregular if it is strongly positively irregular or strongly negatively irregular. Otherwise, an irregular point is said to be weakly irregular.
Homma and Terasaka [3] proved that for all 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 Theorem 1 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.
Let be a Brouwer homeomorphism. Assume that there exists a flow such that . Let be a simply connected region such that for . We say that is a parallelizable region of the flow if there exists a homeomorphism mapping onto such that Such a homeomorphism will be called a parallelizing homeomorphism of the flow . It is known that for any simply connected region which is invariant under the flow the existence of a parallelizing homeomorphism of is equivalent to the existence of a parallelizing homeomorphism of (see [6]).
By the trajectory of a point we mean the set . It is known that a region is parallelizable if and only if there exists a topological line in (i.e., a homeomorphic image of a straight line that is a closed set in ) such that has exactly one common point with every trajectory of contained in (see [7], page 49). Such a set we will call a section in (or a local section of ). On account of the Whitney-Bebutov Theorem (see [7], page 52), for each there exists a parallelizable region containing . Without loss of generality we can assume that the parallelizing homeomorphism satisfies the condition . Then is a local section containing .
For a flow and a point we define the first positive prolongational limit set and the first negative prolongational limit set of by
The set is called the first prolongational limit set of . For a subset we define The set will be called the first prolongational limit set of the flow . For all we have
In [5] it has been proven that for each point the set is contained in . Now we prove the converse inclusion.
Theorem 2. Let be a Brouwer homeomorphism which is embeddable in a flow and let . Then .
Proof. Let . Denote by the strip between trajectories and of points and , respectively. Then for each the trajectory is contained in the strip between trajectories and of points and , respectively, and the trajectories and are subsets of the same component of (see [8]). Let and be local sections of such that and .
Let be a Jordan domain containing in its interior. If , then by compactness of , there exists a such that is the only common point of with the subarc of having and as its endpoints. If , then we put . Take an such that and is contained in the union of all trajectories having a common point with , where denotes the ball with centre and radius . Fix a and an . Without loss of generality we can assume that .
Now we take an for which there exists a such that , where denotes the Euclidean metric on the plane. Then . By compactness of , there exists a such that is the only common point of with the subarc of having and as its endpoints. Denote by the union of all trajectories having a common point with the arc . Since , each trajectory contained in is a subset of the component of which contains and , where denotes the strip between trajectories and of points and .
By the assumption that , there exist sequences and such that , , as . Thus there exists an such that for all we have , and . Then, for every there exists such that . Moreover, by the definition of , for every there exists and such that . Thus for .
Fix any and take a positive integer such that and . Then and belong to different components of , since is a section in . By continuity of at there exists a such that and belong to the same component of , since any neighbourhood of must contain a point from . Thus has a common point with . Then and hence . Taking we have , since . Consequently, for each we have and . Hence and as , which implies that . Consequently .
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 be a Brouwer homeomorphism which is embeddable in a flow and let . Then and , and consequently the set of all strongly irregular points of is equal to the first prolongational limit set of the flow .
Corollary 4. Let be a Brouwer homeomorphism which is embeddable in a flow. Then, for each flow containing , the first prolongational limit set is the same.
After a reparametrization of the flow containing each element of the flow, for or , respectively, can be treated as .
Corollary 5. Let be a Brouwer homeomorphism which is embeddable in a flow . Then the set of all strongly irregular points of is the same for all . Moreover, the set of all strongly positive irregular points of and the set of all strongly negative irregular points of are the same for all .
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 is parallelizable if and only if . Hence for every parallelizable region we have . In the case where is a maximal parallelizable region (i.e., is not contained properly in any parallelizable region), the boundary of consists of strongly irregular points. It follows from the fact that for each maximal parallelizable region the equality 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.
Let be a parallelizable region. Assume that there exists a point such that . Denote by the component of which has a common point with and by the other component of . Let be a parallelizable region which contains and put . Let . We show that for each , which means that is a parallelizable region. To see this we consider three cases. First, let us consider the case where . Then , since . Hence by parallelizability of , we have and by the assumption that , we get . Thus . Now, let . Then . Hence , since by parallelizability of we have . Finally, let . Then, as in the previous case, , and by the assumption that , we get . Thus we proved that , which means that is parallelizable. Since is contained properly in , we obtain that cannot be a maximal parallelizable region.
For any distinct trajectories , , and of one of the following two possibilities must be satisfied: exactly one of the trajectories , , and is contained in the strip between the other two or each of the trajectories , , and is contained in the strip between the other two. In the first case if is the trajectory which lies in the strip between and we will write (, , and , , are different). In the second case we will write (cf. [10]).
Let be a nonempty set. Denote by the set of all finite sequences of elements of . A subset of is called a tree on if it is closed under initial segments; that is, for all positive integers , such that if , then . Let . Then, for any by we denote the sequence . A node of a tree is said to be terminal if there is no node of properly extending it; that is, there is no element such that .
A tree will be termed admissible if the following conditions hold:(i) contains the sequence and no other one-element sequence;(ii)if is in and , then so also is . A tree will be termed admissible if the following conditions hold:(iii) contains the sequence and no other one-element sequence;(iv)if is in and , then so also is .
The set will be said to be admissible class of finite sequences, where and 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 be a flow of Brouwer homeomorphisms. Then there exists a family of trajectories and a family of maximal parallelizable regions , where is an admissible class of finite sequences, such that , , and
Proposition 7 (see [11]). Let be a flow of Brouwer homeomorphisms. Then there exists a family of the parallelizing homeomorphisms , where , are those occurring in Theorem 6, and for each where , , and are some constants such that and at least one of the constants , is finite. Moreover, there exists a continuous function and a homeomorphism such that the homeomorphism given by the relation has the form
The above proposition is formulated for , but the analogous result holds for . The admissible class of finite sequences occurring in Theorem 6 is not unique for a given flow, so we can usually choose a convenient when solving a problem of topological conjugacy.
The homeomorphisms occurring in Proposition 7 can be either increasing or decreasing. For each denote by the unique trajectory contained in (the uniqueness has been proven in [8]). From the construction of the families and occurring in Theorem 6 we obtain that, in case or , the homeomorphism is decreasing and or , respectively. However, in case , the homeomorphism is increasing and (see [11]).
The continuous functions describe the time needed for the flow to move from the point with coordinates in the chart until it reaches the point with coordinates in the chart . In other words, describe the time needed for the flow to move from a point from the section in to a point from the section in .
Proposition 8. The functions occurring in Proposition 7 satisfy the condition in the case where or or the condition in the case where .
Proof. Let us consider the case where or and assume that . The other cases are similar. Denote by and the points for which and ; that is, and . Then . Thus there exist sequences and such that , , and as . This means that there exist sequences , such that , , where . Hence and by (11)
Thus as , since as . Hence , since and . Consequently, .
Suppose, on the contrary, that there exists a sequence such that and for some . Consider the sequence such that . Then each element of the sequence belongs to . Moreover, the sequence tends to the point such that . Hence . On the other hand, by (11)
Hence . Consequently , where is a point such that ; that is, . But this is impossible, since .
By the fact that is a homeomorphism, the function defined by is continuous. Moreover, putting in Proposition 8 we obtain the following result.
Corollary 9. The functions given by where and are those occurring in Proposition 7, satisfy the condition
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 and , where , are topologically conjugate if there exists a homeomorphism of the plane onto itself such that
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 and be topologically conjugate flows of Brouwer homeomorphisms and let be a homeomorphism which conjugates the flows. Then , where and denote the first prolongational limit set of and , respectively.
Put and . Consider the flow , where for each the homeomorphism is defined by Then and .
Put , , , , and , , . Then and . Let Note that the trajectories of contained in are given by the equation for . Hence since . Moreover,
For each flow , where for , having the same trajectories (including the orientation) as the flow given by (21), one can consider the function occurring in Proposition 7 which describes the time needed to move from each point to the point of belonging to the trajectory of , that is, from the point of the form to the point of the form for some . Then by Proposition 8
Consider a constant defined by where is given by (cf. [13, 14]). Then the flow is topologically conjugate to the flow given by (21) if and only if (see [13]). In particular, this condition holds in the case where is increasing.
Now we introduce a class of flows of Brouwer homeomorphisms. Put and for . For any positive integer we define and . Similarly, put , for and for any negative integer let and .
Consider a flow of Brouwer homeomorphisms such that can be given in one of the following forms:(a) and for some ,(b) and ,(c) and . We assume that for each , where and are whose occurring in Theorem 6. Then , since for every . Similarly, for every .
Fix an . Denote by the strip between and . Then and for every trajectory (see [8]). In particular, if is equal to the vertical line for each , then is a vertical strip for each . In a similar way we define the strip for .
Let us assume that for each there exists a homeomorphism such that where is given by (21). If , then and . In case we have and . The flow 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 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 of the forms (b) and (c), respectively.
Consider a flow of Brouwer homeomorphisms which has the same trajectories as a standard generalized Reeb flow. For and denote by the image of the trajectory of under . For each consider the function taking as the time needed to move from the unique point of the set to the unique point of . Define by in case , and by in case . Put
Now we can prove the following conjugacy result.
Theorem 11. Let be a standard generalized Reeb flow. Let be an admissible class of finite sequences satisfying one of the conditions (a)–(c). Assume that is a flow of Brouwer homeomorphisms having the same trajectories including orientation as . If for all , then the flows and are topologically conjugate.
Proof. Assume that one of the conditions (a) and (b) holds. First, let us note that there exists a topological conjugacy of flows and , since is a parallelizable region of each of these flows. More precisely, if and are parallelizing homeomorphisms for and , respectively, then for every we put .
Fix an . Assume that we have defined a homeomorphism which conjugates and on the set . Define by for , where satisfies (28). Then . Hence , since by the assumption . Thus and are topologically conjugate. Consequently and are topologically conjugate, where . Denote by the homeomorphism which conjugates these flows.
Fix any and put , . Take such that and define . Then conjugates the flows and , since
Moreover , since
Hence . Thus we can define by
Then conjugates and on the set . Since and are parallelizable on we can extend the topological conjugacy on the component of which do not contain (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 we extend from to defined on in a similar way.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.