On the Topological Conjugacy of Brouwer Flows

Bulletin of the Malaysian Mathematical Sciences Society, Nov 2017

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.

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On the Topological Conjugacy of Brouwer Flows

On the Topological Conjugacy of Brouwer Flows Zbigniew Les´niak 0 1 0 Department of Mathematics, Pedagogical University , Podchora ̧z ̇ych 2, 30-084 Kraków , Poland 1 Mathematics Subject Classification Primary 39B12; Secondary 54H20 , 37E30 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. Communicated by Rosihan M. Ali. Brouwer homeomorphism; Brouwer flow; First prolongational limit set; Topological equivalence; Topological conjugacy 1 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 {gt : 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 {gt : 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 ( f t (x )) = gt ( (x )), x ∈ X, t ∈ R. For a flow { f t : t ∈ R} defined on a Hausdorff space X and a point p ∈ X one can define the first positive prolongational limit set and the first negative prolongational limit set of p by J +f( p) := {q ∈ X : there exist sequences ( pn)n∈Z+ and (tn)n∈Z+ such that pn → p, tn → +∞, f tn ( pn) → q as n → +∞}, J −f( p) := {q ∈ X : there exist sequences ( pn)n∈Z+ and (tn)n∈Z+ such that pn → p, tn → −∞, f tn ( pn) → q as n → +∞} (cf. [4, p. 25]). Moreover, let us put for every p ∈ X and J f ( p) := J +f( p) ∪ J −f( p) J f (H ) := J f ( p) p∈H for all H ⊂ X . Now let us restrict to a class of flows defined on R2. A flow { f t : t ∈ R} will be called a Brouwer flow if it contains a Brouwer homeomorphism, where by a Brouwer homeomorphism we mean an orientation-preserving homeomorphism of the plane t t ∈ R} is a onto itself which has no fixed points (cf. [5,8]). It is known that if { f : Brouwer flow, then f t is a Brouwer homeomorphism for each t ∈ R\{0} (see [1]). For each p ∈ R2 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 p1 , C p2 determine the strip, i.e., the component of R2\(C p1 ∪ C p2 ) which contains C p1 and C p2 in its boundary. The strip between C p1 and C p2 will be denoted by D p1 p2 . It is an invariant simply connected region. For any distinct trajectories C p1 , C p2 , C p3 of { f t : t ∈ R} one of the following two possibilities must be satisfied: Exactly one of the trajectories C p1 , C p2 , C p3 is contained in the strip between the other two, or each of the trajectories C p1 , C p2 , C p3 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 pi and C pk we will write C pi |C p j |C pk (i, j, k ∈ {1, 2, 3} and i, j, k are different). In the second case we will write |C pi , C p j , C pk | (cf. [10]). t t ∈ R} be a Brouwer flow and U ⊂ R2 be a simply connected invariant Let { f : 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 R2 such that f t (x ) = ϕ −f,1U (ϕ f, U (x ) + (t, 0)), x ∈ U, t ∈ R. 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 ∈ R2 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 (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 ). 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. t t ∈ R} be a Brouwer flow. Then there exist a family of Theorem 1.1 (See [11]) Let { f : 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 Cα ⊂ Uα for α ∈ A, Uα = R2, α∈A Uα ∩ Uα i = ∅ for α i ∈ A, Cα i ⊂ bd Uα for α i ∈ A, |Cα, Cα i1 , Cα i2 | for α i1, α i2 ∈ A, i1 = i2, Cα|Cα i |Cα i j for α i j ∈ A. t t ∈ R} be a Brouwer flow. Then there exists a family Theorem 1.2 (See [11]) Let { f : of the parallelizing homeomorphisms {ϕα : α ∈ A+}, where ϕα : Uα → R2, Uα are those occurring in Theorem 1.1, such that for each α i ∈ A+ ϕα i (Uα ∩ Uα i ) = R × (cα i , 0), where cαα i ∈ R ∪ {−∞}, dαα i ∈ R ∪ {+∞} and cα i ∈ [−∞, 0) are some constants such that cαα i < dαα i and at least one of the constants cαα i , dαα i is finite. Moreover, there exist a continuous function μα i : (cαα i , dαα i ) → R and a homeomorphism να i : (cαα i , dαα i ) → (cα i , 0) such that the homeomorphism given by the relation has the form hα i : R × cαα i , dαα i → R × (cα i , 0) hα i := ϕα i ◦ ϕα|Uα∩Uα i −1 hα i (t, s) = (μα i (s) + t, να i (s)), t ∈ R, s ∈ cαα i , dαα i . ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) Similarly, there exists a family of the parallelizing homeomorphisms {ϕα : α ∈ A−}, where ϕα : Uα → R2, Uα are those occurring in Theorem 1.1, ϕ−1 = ϕ1, such that for each α i ∈ A− ϕα i (Uα ∩ Uα i ) = R × (0, cα i ), where cαα i ∈ R ∪ {−∞}, dαα i ∈ R ∪ {+∞} and cα i ∈ (0, +∞) are some constants such that cαα i < dαα i and at least one of the constants cαα i , dαα i is finite. Moreover, there exist a continuous function μα i : (cαα i , dαα i ) → R and a homeomorphism να i : (cαα i , dαα i ) → (0, cα i ) such that the homeomorphism The construction of the families occurring in Theorem 1.1 starts from a trajectory denoted by C1 = C−1 and a maximal parallelizable region U1 = U−1. Moreover, we denote by H1 and H−1 the components of R2\C1. Having constructed an index α ∈ A and the corresponding Cα, Uα and Hα we denote, in case bd Uα ∩ Hα = ∅, by Cα i the trajectories contained in bd Uα ∩ Hα 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 Cα i we take a maximal parallelizable region Uα i containing it and denote by Hα i the component of R2\Cα i which does not contain Uα. The parallelizing homeomorphisms ϕα : Uα → R2 occurring in Theorem 1.2 satisfy the conditions ϕα(Cα) = R × {0} and ϕα(Uα ∩ Hα) = R × (0, +∞). Each of these homeomorphisms maps trajectories contained in Uα onto horizontal straight lines. The parallelizing homeomorphisms will be also called charts. In the next part of this section we prove that the homeomorphisms να i occurring in Theorem 1.2 can be extended to one of the endpoints of the interval (cαα i , dαα i ). This result will be used in the main part of this paper. The homeomorphisms να i can be either increasing or decreasing. The extension of the homeomorphism να i will contain the number 0 in its image. If such an extension exists, then να i (cαα i ) = 0 in the case where να i is decreasing, and να i (dαα i ) = 0 in the case where να i is increasing. The necessary condition for the existence of such an extension is the correspondence by ϕα between να−1i(0) and a trajectory contained in Uα. In the chart ϕα i the number 0 corresponds to Cα i , since the second coordinate in ϕα i of every point of Cα i is equal to 0, i.e., ϕα i (Cα i ) = R × {0}. To find a trajectory which corresponds to να−1i(0) we will use properties of the first prolongational limit set of boundary trajectories of a parallelizable region. Proposition 1.3 (See [12]) Let U be a parallelizable region of { f t : t ∈ R}. Let p ∈ bd U and q1, q2 ∈ U . Assume that q1, q2 ∈ J f ( p). Then Cq1 = Cq2 . Using the above proposition one can obtain the following corollary. Corollary 1.4 Let U be a maximal parallelizable region of { f t : t ∈ R}. Let p ∈ bd U . Then the set J f ( p) ∩ U consists of exactly one trajectory. Proof Since U is a maximal parallelizable region of the flow { f t : t ∈ R}, we have bd U = J f (U ). Hence, J f ( p) ∩ U = ∅. By Proposition 1.3 each element of the set J f ( p) ∩ U belongs to the same trajectory. For each α i ∈ A denote by Cαα i the unique trajectory contained in Uα ∩ J(Cα i ) occurring in Corollary 1.4. We will show that the trajectory C α α i corresponds either to cα i or to dαα i . This depends on how Cαα i is situated in relation to Cα and Cα i . α The following result shows that the relation between the trajectories Cαα i , Cα and Cα i determines the kind of monotonicity of να i . Proposition 1.5 (See [11]) If Cα|Cαα i |Cα i or Cα = Cαα i , then the homeomorphism να i is decreasing and cαα i > 0 or cαα i = 0, respectively. However, in case |Cα, Cαα i , Cα i |, the homeomorphism να i is increasing and dαα i > 0. Now we can proceed to the extension result mentioned above. Proposition 1.6 If Cα|Cαα i |Cα i or Cα = Cαα i , then ϕα(Cαα i ) = R × {cα i } and να i : α (cαα i , dαα i ) → (cα i , 0) can be extended to a homeomorphism defined on [cαα i , dαα i ) by putting να i (cαα i ) = 0. However, if |Cα, Cαα i , Cα i |, then ϕα(Cαα i ) = R×{dαα i } and να i can be extended to a homeomorphism defined on (cαα i , dαα i ] by putting να i (dαα i ) = 0. ϕα−1(R × {dαα i }) = Cαα i . Proof By the definition of να i we have ϕα−1i(R × {να i (s)}) = ϕα−1(R × {s}) for s ∈ (cαα i , dαα i ), i.e., να i (s) and s correspond to the same trajectory contained in Uα ∩ Uα i . First, let us consider the case where Cα|Cαα i |Cα i or Cα = Cαα i . Then να i is a decreasing homeomorphism. Hence, for any sequence (sn)n∈Z+ such that sn ∈ (cαα i , dαα i ) for n ∈ Z+, we have limn→∞ sn = cαα i if and only if limn→∞ να i (sn) = 0, since C α α i ⊂ J f (Cα i ). To prove that ϕα−1(R × {cαα i }) = Cαα i , we take a sequence ( pn)n∈Z+ of points from Uα ∩ Uα i which tends to a p ∈ Cα i . Then there exists the corresponding sequence (sn)n∈Z+ such that sn ∈ (cαα i , dαα i ), pn ∈ ϕα−1i(R×{να i (sn)})sauncdhlitmhant→f ∞tn (νpαni)(s→n) =q 0. Since Cαα i ⊂ J f (Cα i ), there exists a sequence (tn)n∈Z+ as n → +∞ for some q ∈ Cαα i . Moreover, f tn ( pn) ∈ ϕα−1i(R × {να i (sn)}) = ϕα−1(R × {sn}). Thus, Cαα i has to correspond to cα i by ϕα, since limn→∞ sn = cα i α α and the sequence ( f tn ( pn))n∈Z+ tends to a point of Cαα i . The case where |Cα, Cαα i , Cα i | is similar. Then να i is an increasing homeomorphism. Replacing in the above consideration cα i by dαα i , we obtain that α The continuous functions μα 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, 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, where Kϕα i := ϕα−1i({0} × R) and Kϕα := ϕα−1({0} × R). The following result describes the limits of sequences (μα i (sn))n∈Z+ analogous to sequences (να i (sn))n∈Z+ considered in the proof of Proposition 1.6. Proposition 1.7 (See [13]) The functions μα i occurring in Theorem 1.2 satisfy the condition lims→cαα i μα i (s) = −∞ +∞ if Cα i ⊂ J +f(Cαα i ), if Cα i ⊂ J −f(Cα i ) in the case where Cα|Cαα i |Cα i or Cα = Cαα i , or the condition lims→dαα i μα i (s) = −∞ +∞ if Cα i ⊂ J +f(Cαα i ), if Cα i ⊂ J −f(Cα i ) in the case where |Cα, Cαα i , Cα i |. Proposition 1.7 describes the property that the trajectories Cα i and Cαα i cannot be contained in the same parallelizable region, whereas Proposition 1.6 corresponds to the fact that the trajectories Cα i and Cαα i have no disjoint invariant neighborhoods. 2 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 {gt : 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 Ug,α, respectively. We start from a property of the topological equivalence of Brouwer flows concerning maximal parallelizable regions. Proposition 2.1 Let { f t : t ∈ R} and {g : t t ∈ R} be topologically equivalent Brouwer flows and : R2 → R2 be a homeomorphism which realizes the equivalence. Then if U is a maximal parallelizable region of the flow { f t : t ∈ R}, then (U ) is a maximal parallelizable region of the flow {gt : t ∈ R}. 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 {gt : t ∈ R}. Suppose that (U ) is not a maximal parallelizable region of the flow {gt : 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 {gt : 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. Theorem 2.2 (See [16]) Let { f : t t ∈ R} be topologically equivalent t t ∈ R} and {g : Brouwer flows and : R2 → R2 be a homeomorphism which realizes the equivalence. Then (J f (R2)) = Jg(R2), where J f (R2) and Jg(R2) denote the first prolongational limit set of { f t : t ∈ R} and {gt : t ∈ R}, respectively. In the main step of the proof of this result it has been showed that if q ∈ J f ( p), then (q) ∈ Jg( ( p)), where : R2 → R2 is a homeomorphism which realizes the topological equivalence of { f : t t ∈ R}. To prove this fact some t t ∈ R} and {g : properties of the first prolongational limit set given in [14] have been used. Therefore, we can state the following result. Proposition 2.3 Let { f t : t ∈ R} and {g : t t ∈ R} be topologically equivalent Brouwer flows and : R2 → R2 be a homeomorphism which realizes the equivalence. Then for all p, q ∈ R2 if q ∈ J f ( p), then (q) ∈ Jg( ( p)). 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 Cg,α and maximal parallelizable regions Ug,α for any Brouwer flow {gt : 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 C αf,α i contained in J f (C f,α i ) ∩ U f,α. It plays an important role in Propositions 1.6 and 1.7 describing the properties of να i and μα i . Similarly, for each α i ∈ A there exists a unique trajectory Cgα,α i contained in Jg(Cg,α i ) ∩ Ug,α. The main part of the proof of the next theorem is to show that any homeomorphism which realizes the topological equivalence maps C αf,α i onto Cgα,α i . Theorem 2.4 Let { f t : t ∈ R} and {g : t t ∈ R} be topologically equivalent Brouwer flows and : R2 → R2 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}, where A = A+ ∪ A− is an admissible class of finite sequences, such that U f,1 = U f,−1, C f,1 = C f,−1 and conditions ( 1 )–( 6 ) are satisfied. Define Cg,α := (C f,α) and Ug,α := (U f,α) for α ∈ A. Then {Cg,α : α ∈ A} is a family of trajectories and {Ug,α : α ∈ A} is a family of maximal parallelizable regions of {gt : t ∈ R} such that conditions ( 1 )–( 6 ) hold. Moreover, we have (C αf,α i ) = Cgα,α i , α i ∈ A, ( 9 ) where C αf,α i and Cgα,α i are the unique trajectories contained in J f (C f,α i ) ∩ U f,α and Jg(Cg,α i ) ∩ Ug,α, respectively. (C αf,α i ) = Cgα,α i . Proof By Proposition 2.1, the set Ug,α is a maximal parallelizable region of {gt : t ∈ R} for each α ∈ A. From the assumption that realizes the topological equivalence we obtain that Cg,α is a trajectory of the flow {gt : 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 Iα := {i ∈ Z : α i ∈ A}, and take any i ∈ Iα. By Proposition 2.3 the trajectory (C αf,α i ) is contained in Jg(Cg,α i ). Moreover, (C αf,α i ) ⊂ Ug,α, since C αf,α i ⊂ U f,α. Thus, on account of Corollary 1.4, we have For Brouwer flows { f : t t ∈ R} let us consider the homeomorphisms t t ∈ R} and {g : ν f,α i and νg,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 αf,α i , C f,α i and Cg,α, Cgα,α i , C{gt },α i , respectively. Therefore, from the above theorem we obtain the following result. Corollary 2.5 Let { f t : t ∈ R} and {g : t t ∈ R} be topologically equivalent Brouwer flows. Then for each α i ∈ A the homeomorphisms ν f,α i and νg,α i occurring in Theorem 1.2 are either both increasing or both decreasing. Proof Let : R2 → R2 be the homeomorphism occurring in Theorem 2.4. Then (C f,α) = Cg,α, (C αf,α i ) = Cgα,α i , (C f,α i ) = C{gt },α i . Thus, by Proposition 1.5 the homeomorphisms ν f,α i and νg,α i have the same kind of monotonicity, since preserves the considered ternary relations. Let us remark that Brouwer flows having the same admissible class of finite sequences A (in the decomposition described in Theorem 2.4) may not be topologically equivalent. Consider the case A = {−1, 1, 11, 12} and C 1f,11 = C 1f,12 = C f,1, Cg1,11 = Cg,1, Cg1,12 = Cg,1. Then J f (R2) = C f,1 ∪ C f,11 ∪ C f,12 and Jg(R2) = Cg,1 ∪ Cg,11 ∪ Cg,12 ∪ Cg1,12. Thus, by Theorem 2.2, the flows { f t : t ∈ R} and {gt : t ∈ R} are not topologically equivalent. 3 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 (R2) = C f,1 ∪ C f,11, C f,−1 = C f,1 and C 1f,11 = C 1f,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. Theorem 3.1 (See [17]) Let { f : t t ∈ R} be Reeb flows which have t t ∈ R} and {g : the same trajectories. Then they are topologically conjugate if and only if there exists a homeomorphism β : [0, +∞) → [0, +∞) and a continuous function γ : [0, +∞) → R such that μ f,11 = μg,11 ◦ β + γ , where μ f,11, μg,11 are continuous mappings occurring in Theorem 1.2. 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 {gt : t ∈ R} be topologically equivalent Brouwer flows and : R2 → R2 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 → R2, be a family of parallelizing homeomorphisms occurring in Theorem 1.2. Define Cg,α := (C f,α) and Ug,α := (U f,α) for α ∈ A. Then, by Theorem 2.4, {Cg,α : α ∈ A} is a family of trajectories and {Ug,α : α ∈ A} is a family of maximal parallelizable regions of {gt : t ∈ R} satisfying the assertion of Theorem 1.1. Let {ϕg,α : α ∈ A}, where ϕg,α : Ug,α → R2, be a family of parallelizing homeomorphisms described in Theorem 1.2. Let us fix α i ∈ A+ and put α b f,α i := d αf,α i if |C f,α, C αf,α i , C f,α i |, c f,α i if C f,α|C αf,α i |C f,α i or C f,α = C αf,α i , α where cαf,α i , d αf,α i are those occurring in Theorem 1.2. Then ϕ f,α C αf,α i = R × α b f,α i . Similarly, let Then, Hence, dgα,α i if |Cg,α, Cgα,α i , Cg,α i |. cg,α i if Cg,α|Cgα,α i |Cg,α i or Cg,α = Cgα,α i , α ϕg,α Cgα,α i = R × α bg,α i . ϕg,α ◦ ( 10 ) ( 11 ) since, by Theorem 2.4, (C αf,α i ) = Cgα,α i . Thus, for the topologically equivalent flows { f t : t ∈ R} and {gt : t ∈ R} we have α α α α b f,α i = c f,α i iff bg,α i = cg,α i , ( 12 ) since every homeomorphism preserves the considered ternary relations. Moreover, from Proposition 1.5 we get that bαf,α i ≥ 0 and bgα,α i ≥ 0 for all α i ∈ A+. In a similar way we can define bαf,α i and bgα,α i for α i ∈ A−. Namely, if we put and bαf,α i := bgα,α i := d αf,α i if C f,α|C αf,α i |C f,α i or C f,α = C αf,α i , c f,α i if |C f,α, C αf,α i , C f,α i |, α dgα,α i if Cg,α|Cgα,α i |Cg,α i or Cg,α = Cgα,α i , cg,α i if |Cg,α, Cgα,α i , Cg,α i |, α then for the topologically equivalent flows { f t : t ∈ R} and {gt : t ∈ R} condition ( 12 ) also holds for all α i ∈ A−. Moreover, bαf,α i ≤ 0 and bgα,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 hα 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. Theorem 3.2 Let { f : t t ∈ R} be topologically equivalent Brouwer t t ∈ R} and {g : flows. Assume that for each α i ∈ A there exists a continuous function γα i : I αf,α i → R and an increasing homeomorphism βα i : I αf,α i → Igα,α i such that satisfying lims→cαf,α i βα i (s) = cg,α i , lims→cαf,α i γα i (s) = agα,α i for some agα,α i ∈ α R, where μ f,α i , μg,α i , cαf,α i , d αf,α i , cgα,α i , dgα,α i are those occurring in Theorem 1.2, and I αf,α i := (cαf,α i , cαf,α i + εαf,α i ), Igα,α i := (cgα,α i , cgα,α i + εαf,α i ) in case bgα,α i = cgα,α i or I αf,α i := (d αf,α i − εαf,α i , d αf,α i ), Igα,α i := (dgα,α i − εgα,α i , dgα,α i ) in case bgα,α i = dgα,α i for some εαf,α i > 0 and εgα,α i > 0, where bαf,α i and bgα,α i are defined by ( 10 ) and ( 11 ). Moreover, we assume that for α i, α j ∈ A, i = j we have βα i = βα j in the case where C αf,α i = C αf,α j and C f,α i , C f,α j are contained in the same component of R\C αf,α i . Then the flows { f t : t ∈ R} and {gt : t ∈ R} are topologically conjugate. Proof On account of Theorem 2.4 we can take the same set of indices A for the flows { f t : t ∈ R} and {gt : t ∈ R}. Let {C f,α : α ∈ A}, {U f,α : α ∈ A}, {ϕ f,α : α ∈ A} and {Cg,α : α ∈ A}, {Ug,α : α ∈ A}, {ϕg,α : α ∈ A} be the families described before the statement of this theorem. Then C f,α = ϕ −f,1α(R × {0}) and Cg,α = ϕg−,1α(R × {0}). Let K f,α := ϕ −f,1α({0} × R) and Kg,α := ϕg−,1α({0} × R) for α ∈ A. For all α ∈ A put G f,α := ϕ −f,1α(R × [0, +∞)) if α ∈ A+ and G f,α := ϕ −f,1α(R × (−∞, 0]) if α ∈ A−. By H f,α denote the component of R2\C f,α which contains G f,α. In particular, we have C f,α i ⊂ H f,α for all α i ∈ A. Similarly, let Gg,α := ϕg−,1α(R × [0, +∞)) if α ∈ A+ and Gg,α := ϕg−,1α(R × (−∞, 0]) if α ∈ A−. By Hg,α denote the component of R2\Cg,α which contains Gg,α. Then Cg,α i ⊂ Hg,α for all α i ∈ A. For every positive integer n let us put An+ := {α ∈ A+ : |α| = n} and An− := {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+ = ∅ for some n, then n Am+ = ∅ for all m > n. Similarly, if An− = ∅ for some n, then Am− = ∅ for all m > n. For each positive integer n such that An+ = ∅ we define U +f,n by taking U +f,n := G f,1 in case n = 1 and U +f,n := U +f,n−1 ∪ α∈An+ G f,α in case n > 1. Similarly, if An− = ∅, then we put U −f,n := G f,−1 in case n = 1 and U −f,n := U −f,n−1 ∪ α∈An− G f,α in case n > 1. In the same way we define the regions Ug+,n and Ug−,n. Let : R2 → R2 be a homeomorphism which realizes the topological equivalence. Then α α α α b f,α i < b f,α j iff bg,α i < bg,α j ( 14 ) for all α i, α j ∈ A, i = j , since for each α ∈ A the homeomorphism ϕg,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 {gt : t ∈ R}. First, we consider the regions U f,1 := U +f,1 ∪ U −f,1 and Ug,1 := Ug+,1 ∪ Ug−,1. Let us note that there exists a topological conjugacy 1 : U f,1 → Ug,1 of flows { f t |U f,1 : t ∈ R} and {gt |Ug,1 : t ∈ R}, since U f,1 and Ug,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 Cg,1 = Cg,−1, Ug,1 = Ug,−1, ϕg,1 = ϕg,−1. Thus, we can take −1 = 1. Moreover, we can choose 1 to satisfy the condition 1(C f,1) = Cg,1 and 1(K f,1) = Kg,1. Without loss of generality we can assume that A2+ = ∅ and A2− = ∅. 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. We can choose 1 in such a way that 1(C 1f,1 i ) = Cg1,1 i and 1(C −f,1−1 j ) = Cg−,1−1 j , i.e., condition ( 9 ) holds for 1 and 1 i, −1 j ∈ A. (This condition is required for the existence of an extension of 1 to a topological conjugacy of flows { f t : t ∈ R} and {gt : t ∈ R}.) More precisely, since ϕ f,1 : U f,1 → R2 and ϕg,1 : Ug,1 → R2 are parallelizing homeomorphisms for { f t |U f,1 : t ∈ R} and {gt |Ug,1 : t ∈ R}, respectively, we can put 1( p) := (ϕg−,11 ◦ ψ1 ◦ ϕ f,1)( p) for p ∈ U f,1, where ψ1 : θR12(0→)=R02,,θψ11(b(t1f,,1s)i ):== (btg1,,1θ1i(,sθ)1)(ba −nf,d1−1θ1j ): R= →bg−,1−R1 ijsaandhoθm1ereosmtroicrptehdistmo Isu1f,c1hi tahnadt I −f,−11 j is equal to β1 i and β−1 j , respectively, for all 1 i, −1 j ∈ A. The existence n(C f,α) = Cg,α and of such θ1 is guaranteed by condition ( 14 ). Let us note that 1(K f,1) = Kg,1, since ψ1(0, s) = (0, θ1(s)). Fix a positive integer n such that An++1 = ∅. Assume that we have defined a homeomorphism n : U +f,n → Ug+,n which conjugates the flows { f t |U +f,n : t ∈ R} and {gt |Ug+,n : t ∈ R} in such a way such that for each α i ∈ A+ with |α| ≤ n we have n C αf,α i = Cgα,α i , α := ϕg−,1α ◦ ψα ◦ (ϕ f,α|G f,α}), where i.e., condition ( 9 ) holds for n. The homeomorphism n is obtained by means of 1 defined on U f,1 and α defined on G f,α i . The function 1 has been defined in the first step. For α ∈ A+ with |α| ≤ n we have α : G f,α → G f,α given by the formula ψα(t, s) = (ηα + t, θα(s)), (t, s) ∈ R × [0, +∞), and θα : [0, +∞) → [0, +∞) is an increasing homeomorphism. In case An++1 = ∅ we have θα(bαf,α i ) = bgα,α i and θα restricted to I αf,α i is equal to βα i . By the construction of 1 we have η1 = 0, since 1(K f,1) = Kg,1. The numbers ηα are constructed in the subsequent steps. Let us fix any α i ∈ An++1. We will define α i : G f,α i → Gg,α i and use it to construct a homeomorphism n+1 : U +f,n+1 → Ug+,n+1 which conjugates the flows { f t |U +f,n+1 : t ∈ R} and {gt |Ug+,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 → Gg,α used in the construction Let us recall that ψα = ϕg,α ◦ α ◦ (ϕ f,α|G f,α )−1 and of n : U +f,n → Ug+,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 ∈ (cαf,α i , d αf,α i ). Similarly, denote by (u, v) the coordinates of points belonging to Ug,α in the chart ϕg,α and by (u , v ) the coordinates of points belonging to Ug,α i in the chart ϕg,α i . Then u = μg,α i (v) + u and v = νg,α i (v) for u ∈ R and v ∈ (cgα,α i , dgα,α i ). ψα(t, s) = (ηα + t, θα(s)), (t, s) ∈ R × [0, +∞), where ηα ∈ R and θα : [0, +∞) → [0, +∞) is an increasing homeomorphism such that θα(bαf,α i ) = bgα,α i and θα restricted to I αf,α i is equal to βα i . Passing to coordinates we have u = ηα + t and v = θα(s) for t ∈ R and s ∈ [0, +∞) if (u, v) = ψα(t, s). Consider the case where bαf,α i = c f,α i . (The case bαf,α i = d αf,α i is similar.) Then α bgα,α i = cgα,α i , since { f t : t ∈ R} and {gt : t ∈ R} are topologically equivalent. From the assumption we have that there exist a continuous function γα i : I αf,α i → R and an increasing homeomorphism βα i : I αf,α i → Igα,α i such that u = −γα i (ν −f,1α i (s )) + ηα + t . Put and lims→cαf,α i βα i (s) = cg,α i , lims→cαf,α i γα i (s) = agα,α i for some agα,α i ∈ R. α By ( 8 ) we have t = μ f,α i (s) + t and s = ν f,α i (s) for t ∈ R and s ∈ (cαf,α i , d αf,α i ). Hence, t = t − μ f,α i (ν −f,1α i (s )) and s = ν −f,1α i (s ). By the definition of the ψα, if (u, v) = ψα(t, s), then u = ηα + t and v = βα i (s) for t ∈ R and s ∈ (cαf,α i , cαf,α i + ε f,α i ). (In case bαf,α i = d αf,α i we would have α s ∈ (d αf,α i − ε f,α i , d αf,α i ).) α α α Put d f,α i := ν f,α i (c f,α i + ε f,α i ). Then for the function h f,α i given by ( 7 ) we have h f,α i (R × I αf,α i ) = R × (d f,α i , 0)). Put W f,α i := ϕ −f,1α i (R × (d f,α i , 0)). Let φαα i (t , s ) := (ϕg,α i ◦ α ◦ ϕ −f,1α i )(t , s ) for (t , s ) ∈ R × (d f,α i , 0). Since u = μg,α i (v) + u and v = νg,α i (v), we have u = μg,α i (βα i (s)) + ηα + t and v = νg,α i (βα i (s)). Hence, u = μg,α i (βα i (ν −f,1α i (s ))) + ηα + t − μ f,α i (ν −f,1α i (s )) and v = νg,α i (βα i (ν −f,1α i (s ))). Thus, by the assumption, we have and for s ∈ (d f,α i , 0). Then ηα i (s ) := −γα i (ν −f,1α i (s )) + ηα ξα i (s ) := νg,α i (βα i (ν −f,1α i (s ))) φαα i (t , s ) := (ηα i (s ) + t , ξα i (s )), (t , s ) ∈ R × (d f,α i , 0). α i :G f,α i → (ϕ f,α i |G f,α i ), where Since lims →0 ηα i (s ) = −agα,α i + ηα and lims →0 ξα i (s ) = 0, we can extend continuously the functions ηα i and ξα i onto (d f,α i , 0] by putting ηα i (0) := −agα,α i + ηα and ξα i (0) := 0. The value ηα i (0) we will denote by ηα i . Thus, we have obtained a continuous extension of α on C f,α i . Now define G f,α i by the formula α i := ϕg−,1α i ◦ ψα i ◦ ψα i (t, s) = (ηα i + t, θα i (s)), (t, s) ∈ R × [0, +∞), and θα i :[0, +∞) → [0, +∞) is an increasing homeomorphism such that θα i (bαf,αi i j ) = bgα,αi i j and θα i restricted to I αf,αi i j is equal to βα 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 : U +f,n+1 → Ug+,n+1 which conjugates the flows { f t |U +f,n+1 : t ∈ R} and {gt |Ug+,n+1 : t ∈ R} by n+1( p) := n( p) if p ∈ U +f,n, α i ( p) if p ∈ G f,α i , α i ∈ An++1. Then n+1 conjugates the flows { f t |U +f,n+1 : t ∈ R} and {gt |Ug+,n+1 : t ∈ R}. ( p) := n( p), p ∈ U f,n, n ∈ Z+. The function n+1|U +f,n . is well defined, since for k > n we have U +f,n ⊂ U +f,k and k |U +f,n = 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. Proposition 3.3 Let { f t : t ∈ R} and {g : t t ∈ R} be topologically conjugate Brouwer flows. Then for each α i ∈ A there exist a continuous function γα i : I αf,α i → R and an increasing homeomorphism βα i : I αf,α i → Igα,α i such that relation ( 13 ) holds, i.e., where μ f,α i , μg,α i , cαf,α i , d αf,α i , cgα,α i , dgα,α i are those occurring in Theorem 1.2, cagαn,dα Ii αfo,αr iI:αf=,α (ic:αf=,α i(,dcαfα,fα,αi i−+εεαff,,αα ii,),dIαf,αα i ), Igα,α i := (dgα,α i − εgα,α i , dgα,α i ) in case α g,α i := (cgα,α i , cgα,α i + εαf,α i ) in case bgα,α i = bgα,α i = dgα,α i for some εαf,α i > 0 and εgα,α i > 0, where bαf,α i and bgα,α i are defined by ( 10 ) and ( 11 ). Moreover, we have lims→bαf,α i βα i (s) = bgα,α i , lims→bαf,α i γα i (s) = agα,α i for some agα,α i ∈ R. Proof Let {C f,α : α ∈ A}, {U f,α : α ∈ A}, {ϕ f,α : α ∈ A} and {Cg,α : α ∈ A}, {Ug,α : α ∈ A}, {ϕg,α : α ∈ A} be the families described before the statement of Theorem 3.2. Let : R2 → R2 be a homeomorphism which realizes the topological conjugacy. Let us fix an arbitrary α i ∈ A. We will consider the case where bgα,α i = α cg,α i . (The second case is similar.) By Theorem 1.2 we have U f,α ∩ U f,α i = ϕ −f,1α(R × (cαf,α i , d αf,α i )) and Ug,α ∩ Ug,α i = ϕg−,1α(R × (cgα,α i , dgα,α i )). As in the proof of Theorem 3.2 we can find an ε f,α i > 0 such that the interval (cαf,α i , cαf,α i + ε f,α i ) does not contain any bαf,α j α α for j = i . Put C f,εα i := ϕ −f,1α(R × {cαf,α i + εαf,α i }) and Cg,εα i := (C f,εα i ). Since : R2 → R2 is a homeomorphism which realizes the topological conjαugacy, we get that there exists a unique εgα,α i > 0 such that interval (cgα,α i , cgα,α i + εg,α i ) does not contain any bgα,α j for j = i and Cg,εα i = ϕg−,1α(R × {cg,α i + εg,α i }). Denote by φαα i α α the restriction of the function ϕg,α ◦ ◦ ϕ −f,1α to the set R × (cαf,α i , cαf,α i + εαf,α i ). Then φαα i : R × (cαf,α i , cαf,α i + εαf,α i ) → R × (c{αgt },α i , cgα,α i + εgα,α i ). Thus, if (u, v) = ψαα i (t, s), then v = βα i (s), where βα i : (cαf,α i , cαf,α i + εαf,α i ) → (cgα,α i , cgα,α i + εg,α i ) is an increasing homeomorphism. α Let K f,α := ϕ −f,1α({0} × R) and K f,α i := ϕ −f,1α i ({0} × R). Then μ f,α i (s) describes the time needed for the flow { f t : t ∈ R} to move along the trajectory ϕ −f,1α(R × {s}) from the section K f,α i to the section K f,α for each s ∈ (cαf,α i , cαf,α i + ε f,α i ). α Put L g,α := (K f,α ) and L g,α i := (K f,α i ). Then L g,α is a section in Ug,α and L g,α i is a section in Ug,α i , since is a homeomorphism which maps trajectories of { f t : t ∈ R} onto trajectories of {gt : t ∈ R}. Moreover, by the assumption that : R2 → R2 realizes the topological conjugacy we get that μ f,α i (s) is equal to the time needed for the flow {gt : t ∈ R} to move along the trajectory ϕg−,1α (R × {βα i (s)}) from the section L g,α i to the section L g,α for each s ∈ (cαf,α i , cαf,α i + ε f,α i ). α Let Kg,α := ϕg−,1α ({0} × R) and Kg,α i := ϕg−,1α i ({0} × R). For every s ∈ (cαf,α i , cαf,α i + εαf,α i ) denote by τg,α (s) the time needed for the flow {gt : t ∈ R} to move along the trajectory ϕg−,1α (R × {βα i (s)}) from Kg,α to L g,α and by τg,α i (s) the time needed to move along this trajectory from Kg,α i to L g,α i . Then for each s ∈ (cαf,α i , cαf,α i + ε f,α i ) we have α μg,α i (βα i (s)) = μ f,α i (s) + τα i (s) − τα (s). Define γα i : (cαf,α i , cαf,α i + ε f,α i ) → α R by putting γα i (s) := τα (s) − τα i (s), s ∈ (cαf,α i , cαf,α i + ε f,α i ). α Then μ f,α i (s) = (μg,α i ◦ βα i )(s) + γα i (s) for each s ∈ (cαf,α i , cαf,α i + ε f,α i ). α Since C αf,α i = ϕ −f,1α (R × {cαf,α i }), Cgα,α i = ϕg−,1α (R × {cgα,α i }) and (C αf,α i ) = Cgα,α i , we have lims→cαf,α i βα i (s) = cgα,α i . Moreover, by the definition of γα i and τα we have lims→cαf,α i γα i (s) = τα (cαf,α i ) − lims→cαf,α i τα i (s). Since, by Proposition 1.6, v = cgα,α i corresponds to v = 0, we get that lims→cαf,α i τα i (s) is the time needed to move along trajectory Cg,α i from Kg,α i to L g,α i . Thus, lims→cαf,α i γα i (s) = agα,α i for some agα,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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Zbigniew Leśniak. On the Topological Conjugacy of Brouwer Flows, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 1-17, DOI: 10.1007/s40840-017-0567-8