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

and Applied Analysis Volume 2014, Article ID 638784, 7 pages http://dx.doi.org/10.1155/2014/638784 2 Abstract and Applied Analysis regular. A point which is positively or negatively irregular is called irregular, otherwise it is regular. We say that a setU ⊂ R2 is invariant underf iff(U) = U. An invariant simply connected region U ⊂ R2 is said to be parallelizable if there exists a homeomorphism φ U mapping U onto R2 such that


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 {  : ∈ R} under the operation of composition which satisfies the following conditions: (1) the function  : R 2 × R → R 2 , (, ) =   () is continuous, (2)  0 () =  for  ∈ R 2 , (3)   (  ()) =  + () for  ∈ R 2 , ,  ∈ R.
We say that a Brouwer homeomorphism  is embeddable in a flow if there exists a flow {  :  ∈ R} such that  =  1 .Then for each  ∈ R \ {0},   is a Brouwer homeomorphism.
For any sequence of subsets (  ) ∈Z + of the plane we define limes superior lim sup  → ∞   as the set of all points  ∈ R 2 such that any neighbourhood of  has common points with infinitely many elements of the sequence (  ) ∈Z + .For any subset  of the plane we define the positive limit set   () as the limes superior of the sequence of its iterates (  ()) ∈Z + and negative limit set   () as the limes superior of the sequence ( − ()) ∈Z + .Under the assumption that  is compact, Nakayama [1] A point  is called positively irregular if   () ̸ = 0 for each Jordan domain  containing  in its interior and negatively irregular if   () ̸ = 0 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 R 2 \ ).A point which is not positively irregular is said to be positively regular.Similarly, a point which is not negatively irregular is called negatively 2 Abstract and Applied Analysis regular.A point which is positively or negatively irregular is called irregular, otherwise it is regular.
We say that a set  ⊂ R 2 is invariant under  if () = .An invariant simply connected region  ⊂ R 2 is said to be parallelizable if there exists a homeomorphism   mapping  onto R 2 such that The homeomorphism   occurring in this equality is called a parallelizing homeomorphism of |  .On account of the Brouwer Translation Theorem, for each  ∈ R 2 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  = Z + or  = {1, . . ., } for a positive integer , {   :  ∈ Z + }, and .The sets {  :  ∈ } and {   :  ∈ Z + } 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    ∩   (   ) = 0 for  ∈ Z \ {0}.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  + () ̸ = 0, otherwise it is weakly positively irregular.Similarly,  is strongly negatively irregular if  − () ̸ = 0, 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 ,  ∈ R 2 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.

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 {  :  ∈ R} such that  1 = .Let  ⊂ R 2 be a simply connected region such that   () =  for  ∈ R. We say that  is a parallelizable region of the flow if there exists a homeomorphism   mapping  onto R 2 such that Such a homeomorphism   will be called a parallelizing homeomorphism of the flow {  |  :  ∈ R}.It is known that for any simply connected region  which is invariant under the flow {  :  ∈ R} the existence of a parallelizing homeomorphism of |  is equivalent to the existence of a parallelizing homeomorphism of {  |  :  ∈ R} (see [6]).
By the trajectory of a point  ∈ R 2 we mean the set   := {  () :  ∈ R}.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 {  :  ∈ R} contained in  (see [7], page 49).Such a set  we will call a section in  (or a local section of {  :  ∈ R}).On account of the Whitney-Bebutov Theorem (see [7], page 52), for each  ∈ R 2 there exists a parallelizable region   containing .Without loss of generality we can assume that the parallelizing homeomorphism    satisfies the condition    () = (0, 0).
For a flow {  :  ∈ R} and a point  ∈ R 2 we define the first positive prolongational limit set and the first negative prolongational limit set of  by  + () := { ∈ R 2 : there exist sequences (  ) ∈Z + , (  ) ∈Z + such that   → ,   → +∞, The set () :=  + () ∪  − () is called the first prolongational limit set of .For a subset  ⊂ R 2 we define The set (R 2 ) will be called the first prolongational limit set of the flow {  :  ∈ R}.For all ,  ∈ R 2 we have In [5] it has been proven that for each point  ∈ R 2 the set  + () is contained in  + ().Now we prove the converse inclusion.

Abstract and Applied Analysis 3
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  0 and  0 be local sections of {  :  ∈ R} such that  ∈  0 and  ∈  0 .
Let  be a Jordan domain containing  in its interior.If  0 ∩ bd ̸ = 0, then by compactness of bd, there exists a  0 ∈  0 ∩   such that  0 is the only common point of bd with the subarc  of  0 having  and  0 as its endpoints.If  0 ∩bd = 0, then we put  :=  0 ∩(  ∪  ).Take an   > 0 such that (,   ) ⊂ int  and (,   ) ∩   is contained in the union of all trajectories having a common point with , where (,   ) denotes the ball with centre  and radius   .Fix a  > 0 and an   > 0. Without loss of generality we can assume that (,   ) ∩ (,   ) = 0. Now we take an  ∈ (0,   ) for which there exists a  ∈  0 ∩   such that dist(, ) > , where dist denotes the Euclidean metric on the plane.Then bd(, ) ∩  0 ∩   ̸ = 0.By compactness of bd(, ), there exists a  0 ∈  0 ∩   such that  0 is the only common point of bd(, ) with the subarc  of  0 having  and  0 as its endpoints.Denote by  the union of all trajectories having a common point with the arc .Since  0 ∈   , each trajectory contained in  is a subset of the component of cl   0 \  which contains   and   0 , where   0 denotes the strip between trajectories   and   0 of points  and  0 .
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 {  :  ∈ R} and let  ∈ R 2 .Then  + () =  + () and  − () =  − (), and consequently the set of all strongly irregular points of  is equal to the first prolongational limit set of the flow {  :  ∈ R}.

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 {  :  ∈ R} containing  each element   of the flow, for  ∈ R \ {0} or  > 0, respectively, can be treated as .
Corollary 5. Let  be a Brouwer homeomorphism which is embeddable in a flow {  :  ∈ R}.Then the set of all strongly irregular points of   is the same for all  ∈ R \ {0}.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  > 0.

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 () ∩  = 0. Hence for every parallelizable region  we have () ⊂ bd.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 () = bd 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.
For any distinct trajectories   1 ,   2 , and   3 of {  :  ∈ R} one of the following two possibilities must be satisfied: exactly one of the trajectories   1 ,   2 , and   3 is contained in the strip between the other two or each of the trajectories   1 ,   2 , and   3 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    |   |   (, ,  ∈ {1, 2, 3} and , ,  are different).In the second case we will write |   ,    ,    | (cf.[10]).
A tree  + ⊂ Z < + will be termed admissible if the following conditions hold: (i)  + contains the sequence 1 and no other one-element sequence; (ii) if ̂ is in  + and  > 1, then so also is ̂( − 1).
A tree  − ⊂ Z < − will be termed admissible if the following conditions hold: (iii)  − contains the sequence −1 and no other oneelement sequence; (iv) if ̂ is in  − and  < −1, then so also is ̂( + 1).
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 {  :  ∈ R} be a flow of Brouwer homeomorphisms.Then there exists a family of trajectories {  :  ∈ } and a family of maximal parallelizable regions {  :  ∈ }, where  =  + ∪  − is an admissible of finite sequences, such that  1 =  −1 ,  1 =  −1 , and Proposition 7 (see [11]).Let {  :  ∈ R} be a flow of Brouwer homeomorphisms.Then there exists a family of the parallelizing homeomorphisms {  :  ∈  + }, where   :   → R 2 ,   are those occurring in Theorem 6, and for each ̂ ∈ where given by the relation ℎ ̂ :=  ̂ ∘ ( |   ∩ ̂ ) −1 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 continuous functions  ̂ describe the time needed for the flow {  :  ∈ R} to move from the point with coordinates (0, ] ̂ ()) in the chart  ̂ until it reaches the point with coordinates (0, ) 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 in the case where |  ,   ̂ ,  ̂ |.
Let us assume that for each   ∈  \ {1, −1} there exists a homeomorphism    : cl    →  0 ∪   ∪   such that where {  :  ∈ R} is given by (21  • in case   +1 ⊂  + (   ), and by Now we can prove the following conjugacy result.[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  −1 ∈  − we extend Φ  from ⋃  =−1    to Φ −1 defined on ⋃  =−1    ∪   −1 in a similar way.