On the Topological Conjugacy of Brouwer Flows

We study the problem of topological conjugacy of Brouwer flows. We give a sufficient and necessary condition for Brouwer flows to be topologically conjugate. To obtain this result we use a cover of the plane by maximal parallelizable regions and relations between parallelizing homeomorphisms of these regions. We show that for topologically equivalent Brouwer flows there exists a one-to-one correspondence between such covers of the plane.


Introduction
In this paper we deal with the problem of topological conjugacy of Brouwer flows. In [2,6,16] one can find necessary and sufficient conditions for such flows to be topologically equivalent. Topologically conjugate flows are topologically equivalent, but not conversely. In [3] it has been constructed an uncountable family of topologically equivalent Brouwer flows such that any two elements of the family are not topological conjugate. Our aim is to give a sufficient and necessary condition for topologically equivalent Brouwer flows to be topologically conjugate. After introducing basic notions concerning Brouwer flows, we recall the decomposition theorem for such flows used in the proof of the main result of this paper. We also present properties of the transition maps between parallelizing homeomorphisms and prove an extension result. In Sect. 2 we prove that for topologically equivalent Brouwer flows one can take the same admissible class of finite sequences occurring in the decomposition theorem. Section 3 contains the statement and the proof of the main result of this paper. It gives a sufficient condition for topologically equivalent Brouwer flows to be topologically conjugate. At the end of this paper we prove that this condition is also necessary. Now we briefly recall the notions that will be used in this paper. A more detailed introduction concerning Brouwer flows can be found in [15,16,20].
By a flow we mean a family { f t : t ∈ R} of homeomorphisms of a topological space X onto itself such that f t • f s = f t+s , t, s ∈ R, and the function F : X × R → X given by F(x, t) = f t (x) is continuous. Let us note that any flow { f t : t ∈ R} is a group under the operation of composition, where f 0 is the identity function and the inverse of f t is equal to f −t for t ∈ R. Let { f t : t ∈ R} and {g t : t ∈ R} be flows defined on topological spaces X and Y , respectively. Then they are said to be topologically equivalent if there exists a homeomorphism : X → Y mapping trajectories of { f t : t ∈ R} onto trajectories of {g t : t ∈ R} and preserving the orientations of the trajectories (see [9, p. 32]). We say that the flows are topologically conjugate if there exists a homeomorphism : X → Y such that homeomorphism we mean an orientation-preserving homeomorphism of the plane onto itself which has no fixed points (cf. [5,8]). It is known that if { f t : t ∈ R} is a Brouwer flow, then f t is a Brouwer homeomorphism for each t ∈ R\{0} (see [1]).
For each p ∈ R 2 let us denote by C p the trajectory of p, i.e., C p := { f t ( p) : t ∈ R}. Each trajectory of a Brouwer flow is an unbounded closed set and divides the plane into two invariant unbounded simply connected regions (see [1]). Thus, every Brouwer flow is a plane flow with no compact trajectories. Conversely, if a flow { f t : t ∈ R} has no fixed points and no periodic trajectories, then for each t ∈ R\{0} the homeomorphism f t has no fixed points. Thus, { f t : t ∈ R} is a Brouwer flow, since every element of a plane flow has to preserve orientation. Hence, every trajectory of such flow is a closed subset of the plane. This result can be also obtained from a generalization of the Poincaré-Bendixson theorem which can be found in [7].
In this paper we study plane flows with trajectories which are unbounded and closed sets. Therefore, any two different trajectories C p 1 , C p 2 determine the strip, i.e., the component of R 2 \(C p 1 ∪ C p 2 ) which contains C p 1 and C p 2 in its boundary. The strip between C p 1 and C p 2 will be denoted by D p 1 p 2 . It is an invariant simply connected region.
For any distinct trajectories C p 1 , C p 2 , C p 3 of { f t : t ∈ R} one of the following two possibilities must be satisfied: Exactly one of the trajectories C p 1 , C p 2 , C p 3 is contained in the strip between the other two, or each of the trajectories C p 1 , C p 2 , C p 3 is contained in the strip between the other two. In the first case if C p j is the trajectory which lies in the strip between C p i and C p k we will write C p i |C p j |C p k (i, j, k ∈ {1, 2, 3} and i, j, k are different). In the second case we will write |C p i , C p j , C p k | (cf. [10]).
Let { f t : t ∈ R} be a Brouwer flow and U ⊂ R 2 be a simply connected invariant region, i.e., f t (U ) = U for t ∈ R. We say that U is a parallelizable region if there exists a homeomorphism ϕ f, U mapping U onto R 2 such that Such a homeomorphism ϕ f, U will be called a parallelizing homeomorphism of the flow { f t | U : t ∈ R}. On account of the Whitney-Bebutov theorem (see [4, p. 52]), for each p ∈ R 2 there exists a parallelizable region U p such that p ∈ U p . We will consider a cover of the plane by maximal parallelizable regions, i.e., such parallelizable regions which are contained properly in no other parallelizable region. It is known that if U is a maximal parallelizable region, then J f (U ) = bd U (see [19]). As the set of indices of such a cover we take a class of finite sequences of integers described below.
Let X be a nonempty set. Denote by X <ω the set of all finite sequences of elements of X . By a tree on X we mean a subset T of X <ω which is closed under initial segments, i.e., for all positive integers m, n such that n > m, if (x 1 , . . . , x m , . . . , x n ) ∈ T , then (x 1 , . . . , x m ) ∈ T .
Let α = (x 1 , . . . , x n ) ∈ X <ω . Then, for any x ∈ X by α x we denote the sequence (x 1 , . . . , x n , x). If trees A + ⊂ Z <ω + and A − ⊂ Z <ω − satisfy the conditions (i) A + contains the sequence 1 and no other one-element sequence, (ii) if α k is in A + and k > 1, then so also is α (k − 1), (iii) A − contains the sequence −1 and no other one-element sequence, (iv) if α k is in A − and k < −1, then so also is α (k + 1), then the set A := A + ∪ A − will be said to be admissible class of finite sequences. Now we can recall the main results describing the structure of Brouwer flows. The first of them says that for any Brouwer flow there exists a cover of the plane consisting of maximal parallelizable regions which can be indexed in a convenient way by an admissible class of finite sequences. The second one describes relations between the parallelizing homeomorphisms of overlapping elements of the cover. Theorem 1.1 (See [11]) Let { f t : t ∈ R} be a Brouwer flow. Then there exist a family of trajectories {C α : α ∈ A} and a family of maximal parallelizable regions given by the relation has the form Similarly, there exists a family of the parallelizing homeomorphisms given by relation (7) satisfies (8).
The construction of the families occurring in Theorem 1.1 starts from a trajectory denoted by C 1 = C −1 and a maximal parallelizable region U 1 = U −1 . Moreover, we denote by H 1 and H −1 the components of R 2 \C 1 . Having constructed an index α ∈ A and the corresponding C α , U α and H α we denote, in case bd U α ∩ H α = ∅, by C α i the trajectories contained in bd U α ∩ H α starting from i = 1 and taking subsequent positive integers i if α ∈ A + and starting from i = −1 and taking subsequent negative integers i if α ∈ A − . Then for each trajectory C α i we take a maximal parallelizable region U α i containing it and denote by H α i the component of R 2 \C α i which does not contain U α .
In the next part of this section we prove that the homeomorphisms ν α i occurring in Theorem 1.2 can be extended to one of the endpoints of the interval (c α α i , d α α i ). This result will be used in the main part of this paper.
The homeomorphisms ν α i can be either increasing or decreasing. The extension of the homeomorphism ν α i will contain the number 0 in its image. If such an extension exists, then ν α i (c α α i ) = 0 in the case where ν α i is decreasing, and ν α i (d α α i ) = 0 in the case where ν α i is increasing. The necessary condition for the existence of such an extension is the correspondence by ϕ α between ν −1 α i (0) and a trajectory contained in U α . In the chart ϕ α i the number 0 corresponds to C α i , since the second coordinate in ϕ α i of every point of C α i is equal to 0, i.e., To find a trajectory which corresponds to ν −1 α i (0) we will use properties of the first prolongational limit set of boundary trajectories of a parallelizable region. [12]) Let U be a parallelizable region of { f t : t ∈ R}. Let p ∈ bd U and q 1 , q 2 ∈ U . Assume that q 1 , q 2 ∈ J f ( p). Then C q 1 = C q 2 .

Proposition 1.3 (See
Using the above proposition one can obtain the following corollary.

Corollary 1.4 Let U be a maximal parallelizable region of
For each α i ∈ A denote by C α α i the unique trajectory contained in U α ∩ J(C α i ) occurring in Corollary 1.4. We will show that the trajectory C α α i corresponds either to c α α i or to d α α i . This depends on how C α α i is situated in relation to C α and C α i . The following result shows that the relation between the trajectories C α α i , C α and C α i determines the kind of monotonicity of ν α i .
Now we can proceed to the extension result mentioned above.
i.e., ν α i (s) and s correspond to the same trajectory contained in The continuous functions μ α i occurring in Theorem 1.2 describe the time needed for the flow { f t : t ∈ R} to move from the point with coordinates (0, ν α i (s)) in the chart ϕ α i until it reaches the point with coordinates (0, s) in the chart ϕ α . In other words, μ α i describe the time needed for the flow to move from a point from the section The following result describes the limits of sequences (μ α i (s n )) n∈Z + analogous to sequences (ν α i (s n )) n∈Z + considered in the proof of Proposition 1.6. Proposition 1.7 (See [13]) The functions μ α i occurring in Theorem 1.2 satisfy the condition Proposition 1.7 describes the property that the trajectories C α i and C α α i cannot be contained in the same parallelizable region, whereas Proposition 1.6 corresponds to the fact that the trajectories C α i and C α α i have no disjoint invariant neighborhoods.

Topological Equivalence of Brouwer Flows
In this part we present results concerning the topological equivalence of Brouwer flows. In particular, we prove a result about relations between covers of the plane by families of maximal parallelizable regions for topologically equivalent Brouwer flows. Topologically equivalent Brouwer flows considered in this section will be denoted by { f t : t ∈ R} and {g t : t ∈ R}. Therefore, to distinguish between corresponding elements of the families occurring in Theorem 1.1 we will add the letters f and g to the subscripts. For instance, the maximal parallelizable regions will be denoted by U f,α and U g,α , respectively.
We start from a property of the topological equivalence of Brouwer flows concerning maximal parallelizable regions. Proof Let U be a maximal parallelizable region of the flow { f t : t ∈ R}. Then there exists a section K in U . Since is a homeomorphism, the set (K ) is a section in (U ). Hence, (U ) is a parallelizable region of the flow {g t : t ∈ R}. Suppose that (U ) is not a maximal parallelizable region of the flow {g t : t ∈ R}. Then there exists a parallelizable region U extending (U ). Using the same reasoning as above, we obtain that for −1 (U ) is a parallelizable region of the flow { f t : t ∈ R}, since −1 realizes the topological equivalence of {g t : t ∈ R} and { f t : t ∈ R}. Moreover, we have U ⊂ −1 (U ) and U = −1 (U ), which is a contradiction.
In [16] it has been proved that a homeomorphism which realizes the topological equivalence of Brouwer flows preserves the first prolongational limit set. Namely, we have the following result. [16]) Let { f t : t ∈ R} and {g t : t ∈ R} be topologically equivalent Brouwer flows and : R 2 → R 2 be a homeomorphism which realizes the equivalence.

Theorem 2.2 (See
In the main step of the proof of this result it has been showed that if q ∈ J f ( p), then (q) ∈ J g ( ( p)), where : R 2 → R 2 is a homeomorphism which realizes the topological equivalence of { f t : t ∈ R} and {g t : t ∈ R}. To prove this fact some properties of the first prolongational limit set given in [14] have been used. Therefore, we can state the following result.
For a given Brouwer flow { f t : t ∈ R} let us consider families {C f,α : α ∈ A} and {U f,α : α ∈ A} of trajectories and maximal parallelizable regions, respectively, occurring in Theorem 1.1. We will find corresponding trajectories C g,α and maximal parallelizable regions U g,α for any Brouwer flow {g t : t ∈ R} topologically equivalent to { f t : t ∈ R}. The admissible class of finite sequences occurring in Theorem 1.1 is not unique for a given flow, so we can usually choose a convenient A when solving a problem topological conjugacy.
By Corollary 1.4, for each α i ∈ A there exists a uniquely determined trajectory It plays an important role in Propositions 1.6 and 1.7 describing the properties of ν α i and μ α i . Similarly, for each α i ∈ A there exists a unique trajectory C α g,α i contained in J g (C g,α i ) ∩ U g,α . The main part of the proof of the next theorem is to show that any homeomorphism which realizes the topological equivalence maps C α f,α i onto C α g,α i .
where C α f,α i and C α g,α i are the unique trajectories contained in J f (C f,α i ) ∩U f,α and J g (C g,α i ) ∩ U g,α , respectively. Proof By Proposition 2.1, the set U g,α is a maximal parallelizable region of {g t : t ∈ R} for each α ∈ A.
From the assumption that realizes the topological equivalence we obtain that C g,α is a trajectory of the flow {g t : t ∈ R}. Since is a homeomorphism, conditions (1)- (6) are satisfied. In particular, conditions (5) and (6) hold, since every homeomorphism preserves the considered ternary relations.
Let us fix an α ∈ A. Put I α := {i ∈ Z : α i ∈ A}, and take any i ∈ I α . By Proposition 2.3 the trajectory Thus, on account of Corollary 1.4, we have For Brouwer flows { f t : t ∈ R} and {g t : t ∈ R} let us consider the homeomorphisms ν f,α i and ν g,α i , respectively, occurring in Theorem 1.2. By Proposition 1.5 the monotonicity of these homeomorphisms depends on the trajectories C f,α , C α f,α i , C f,α i and C g,α , C α g,α i , C {g t },α i , respectively. Therefore, from the above theorem we obtain the following result. Then Thus, by Proposition 1.5 the homeomorphisms ν f,α i and ν g,α i have the same kind of monotonicity, since preserves the considered ternary relations.

Topological Conjugacy of Brouwer Flows
In this section we deal with the problem of topological conjugacy of Brouwer flows. We prove a sufficient condition for Brouwer flows to be topologically conjugate. First let us recall the known result concerning the problem of topological conjugacy of flows defined on the plane.
Consider a subclass of the family of Brouwer flows which consists of all flows for which the first prolongational limit set contains exactly two trajectories. Such flows are called Reeb flows, and each element of this flow except the identity is said to be a Reeb homeomorphism. Let { f t : t ∈ R} be a Reeb flow. We take A = {−1, 1, 11} as the admissible class of finite sequences occurring in Theorem 2.4. Then we can have J f (R 2 ) = C f,1 ∪ C f,11 , C f,−1 = C f, 1 and C 1 f, 11 = C 1 f, 11 . The flow has two maximal parallelizable regions U f,1 = U f,−1 and U f, 11 .
Le Roux [17] has given a classification of conjugacy classes of such flows (see also [18]). The result can be stated in the following form. In [13] it has been given a sufficient condition for topological conjugacy of Brouwer flows for which the admissible class of finite sequences A contains only sequences with elements equal to 1 or − 1. Such flows have been called generalized Reeb flows. Now we consider the problem of topological conjugacy for any Brouwer flows.
Let { f t : t ∈ R} and {g t : t ∈ R} be topologically equivalent Brouwer flows and : R 2 → R 2 be a homeomorphism which realizes the equivalence. Let {C f,α : α ∈ A} be a family of trajectories and {U f,α : α ∈ A} a family of maximal parallelizable regions of { f t : t ∈ R} occurring in Theorem 1.1, where A is an admissible class of finite sequences. Let {ϕ f,α : α ∈ A}, where ϕ f,α : U f,α → R 2 , be a family of parallelizing homeomorphisms occurring in Theorem 1.2. Define C g,α := (C f,α ) and U g,α := (U f,α ) for α ∈ A. Then, by Theorem 2.4, {C g,α : α ∈ A} is a family of trajectories and {U g,α : α ∈ A} is a family of maximal parallelizable regions of {g t : t ∈ R} satisfying the assertion of Theorem 1.1. Let {ϕ g,α : α ∈ A}, where ϕ g,α : U g,α → R 2 , be a family of parallelizing homeomorphisms described in Theorem 1.2.
Let us fix α i ∈ A + and put where c α f,α i , d α f,α i are those occurring in Theorem 1.2. Then Then, Hence, since every homeomorphism preserves the considered ternary relations. Moreover, from Proposition 1.5 we get that b α f,α i ≥ 0 and b α g,α i ≥ 0 for all α i ∈ A + . In a similar way we can define b α f,α i and b α g,α i for α i ∈ A − . Namely, if we put then for the topologically equivalent flows { f t : t ∈ R} and {g t : t ∈ R} condition (12) also holds for all α i ∈ A − . Moreover, b α f,α i ≤ 0 and b α g,α i ≤ 0 for all α i ∈ A − . Now we can proceed to the proof of a sufficient condition for the topological conjugacy of Brouwer flows. To obtain this result we use the fact that for any Brouwer flow one can find a cover of the plane by maximal parallelizable regions. The trajectories of these regions are mapped onto horizontal straight lines by charts of the cover. After such a linearization we considered transition maps h α i for overlapping regions of this cover. A homeomorphism which conjugates the flows is constructed by induction. The crucial step is to extend the homeomorphism from the regions of the cover onto their boundary trajectories. satisfying are defined by (10) and (11). Moreover, we assume that for {C g,α : α ∈ A}, {U g,α : α ∈ A}, {ϕ g,α : α ∈ A} be the families described before the statement of this theorem. Then For every positive integer n let us put A + n := {α ∈ A + : |α| = n} and A − n := {α ∈ A − : |α| = n}, where |α| denotes the length of the sequence α. Then by the definition of an admissible class of finite sequences we get that if A + n = ∅ for some n, then A + m = ∅ for all m > n. Similarly, if A − n = ∅ for some n, then A − m = ∅ for all m > n. For each positive integer n such that A + n = ∅ we define U + f,n by taking U + f,n := G f,1 in case n = 1 and U + f,n : In the same way we define the regions U + g,n and U − g,n . Let : R 2 → R 2 be a homeomorphism which realizes the topological equivalence. Then for all α i, α j ∈ A, i = j, since for each α ∈ A the homeomorphism ϕ g,α • • ϕ −1 f,α is strictly increasing with respect to the second variable. Let us start the construction of a homeomorphism which conjugates the flows { f t : t ∈ R} and {g t : t ∈ R}. First, we consider the regions U f,1 : and U g,1 := U + g,1 ∪ U − g,1 . Let us note that there exists a topological conjugacy 1 : U f,1 → U g,1 of flows { f t | U f,1 : t ∈ R} and {g t | U g,1 : t ∈ R}, since U f,1 and U g,1 are parallelizable regions. Let us remind that C f,1 = C f,−1 , U f,1 = U f,−1 , ϕ f,1 = ϕ f,−1 and C g,1 = C g,−1 , U g,1 = U g,−1 , ϕ g,1 = ϕ g,−1 . Thus, we can take −1 = 1 . Moreover, we can choose 1 to satisfy the condition 1 (C f,1 ) = C g,1 and 1 (K f,1 ) = K g,1 . Without loss of generality we can assume that A + 2 = ∅ and A − 2 = ∅. Otherwise, we do need no other conditions on 1 on G f,1 or G f,−1 . Since −1 = 1 , we also will consider sequences −1 j ∈ A for 1 .
Let us fix any α i ∈ A + n+1 . We will define α i : G f,α i → G g,α i and use it to construct a homeomorphism n+1 : U + f,n+1 → U + g,n+1 which conjugates the flows : t ∈ R} and {g t | U + g,n+1 : t ∈ R}. The first step is to define α i on C f,α i . To do this we use the form of the function α : G f,α → G g,α used in the construction of n : U + f,n → U + g,n . Denote by (t, s) the coordinates of points belonging to U f,α in the chart ϕ f,α and by (t , s ) the coordinates of points belonging to U f,α i in the chart ϕ f,α i . Then by (8) we have t = μ f,α i (s) + t and s = ν f,α i (s) for t ∈ R and s ∈ (c α f,α i , d α f,α i ). Similarly, denote by (u, v) the coordinates of points belonging to U g,α in the chart ϕ g,α and by (u , v ) the coordinates of points belonging to U g,α i in the chart ϕ g,α i .
, since { f t : t ∈ R} and {g t : t ∈ R} are topologically equivalent. From the assumption we have that there exist a continuous function γ α i : I α f,α i → R and an increasing homeomorphism β α i : I α f,α i → I α g,α i such that Then for the function h f,α i given by (7) we . Thus, by the assumption, we have Since lim s →0 η α i (s ) = −a α g,α i + η α and lim s →0 ξ α i (s ) = 0, we can extend continuously the functions η α i and ξ α i onto (d f,α i , 0] by putting η α i (0) := −a α g,α i + η α and ξ α i (0) := 0. The value η α i (0) we will denote by η α i . Thus, we have obtained a continuous extension of α on C f,α i . Now define α i :G f,α i → G f,α i by the formula α i : g,α i j and θ α i restricted to I α i f,α i j is equal to β α i j for all α i j ∈ A + . Note that the restrictions of the functions α i and α to C f,α i are equal. Therefore, we can define a homeomorphism n+1 : Then n+1 conjugates the flows : t ∈ R} and {g t | U + g,n+1 : t ∈ R}.
In the same way we can extend n from U − f,n to n+1 defined on U − f,n+1 . Let U f,n := U + f,n ∪ U − f,n . To define a homeomorphism which realizes the topological conjugacy between the flows { f t : t ∈ R} and {g t : t ∈ R} we put The function is well defined, since for k > n we have Now we prove that the assumption of Theorem 3.2 concerning the relation between the functions μ f,α i and μ f,α i is a necessary condition for the topological conjugacy of Brouwer flows.
As in the proof of Theorem 3.2 we can find an and C g,ε α i := (C f,ε α i ). Since : R 2 → R 2 is a homeomorphism which realizes the topological conjugacy, we get that there exists a unique ε α g,α i > 0 such that interval (c α g,α i , c α g,α i + ε α g,α i ) does not contain any b α g,α j for j = i and C g,ε α i = ϕ −1 g,α (R × {c α g,α i + ε α g,α i }). Denote by φ α from the section K f,α i to the section K f,α for each s ∈ (c α f,α i , c α f,α i + ε α f,α i ). Put L g,α := (K f,α ) and L g,α i := (K f,α i ). Then L g,α is a section in U g,α and L g,α i is a section in U g,α i , since is a homeomorphism which maps trajectories of { f t : t ∈ R} onto trajectories of {g t : t ∈ R}. Moreover, by the assumption that : R 2 → R 2 realizes the topological conjugacy we get that μ f,α i (s) is equal to the time needed for the flow {g t : t ∈ R} to move along the trajectory ϕ −1 g,α (R × {β α i (s)}) from the section L g,α i to the section L g,α for each s ∈ (c α f,α i , c α f,α i + ε α f,α i ). Let K g,α := ϕ −1 g,α ({0} × R) and K g,α i := ϕ −1 g,α i ({0} × R). For every s ∈ (c α f,α i , c α f,α i + ε α f,α i ) denote by τ g,α (s) the time needed for the flow {g t : t ∈ R} to move along the trajectory ϕ −1 g,α (R × {β α i (s)}) from K g,α to L g,α and by τ g,α i (s) the time needed to move along this trajectory from K g,α i to L g,α i . Then for each s ∈ (c α f,α i , c α f,α i + ε α f,α i ) we have Since, by Proposition 1.6, v = c α g,α i corresponds to v = 0, we get that lim s→c α f,α i τ α i (s) is the time needed to move along trajectory C g,α i from K g,α i to L g,α i . Thus, lim s→c α f,α i γ α i (s) = a α g,α i for some a α g,α i ∈ R. Summing up, we proved that relation (13) is a sufficient and necessary condition for topologically equivalent Brouwer flows to be topologically conjugate.
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