#### 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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0
International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons license, and indicate if changes were made.
Springer (1974)
1. Andrea , S.A. : On homeomorphisms of the plane which have no fixed points . Abh. Math. Sem. Hamburg 30 , 61 - 74 ( 1967 )
2. Beck , A. : Continuous flows in the plane , Grundlehren der Mathematischen Wissenschaften 201 .
3. Béguin , F. , Le Roux , F. : Ensemble oscillant d'un homéomorphisme de Brouwer, homéomorphismes de Reeb. Bull. Soc. Math. France 131 ( 2 ), 149 - 210 ( 2003 )
4. Bhatia , N.P. , Szegö , G.P. : Stability Theory of Dynamical Systems . Springer, Berlin ( 1970 )
5. Brouwer , L.E.J.: Beweis des ebenen Translationssatzes . Math. Ann. 72 , 37 - 54 ( 1912 )
6. Dumortier , F. , Llibre , J. , Artés , J.C. : Qualitative Theory of Planar Differential Systems . Springer, Berlin ( 2006 )
7. Hájek , O. : Dynamical Systems in the Plane . Academic Press, London ( 1968 )
8. Homma , T. , Terasaka , H.: On the structure of the plane translation of Brouwer . Osaka Math. J. 5 , 233 - 266 ( 1953 )
9. Irwin , M.C. : Smooth Dynamical Systems . Academic Press, London ( 1980 )
10. Kaplan , W. : Regular curve-families filling the plane I . Duke Math. J. 7 , 154 - 185 ( 1940 )
11. Les´niak, Z. : On a decomposition of the plane for a flow free mappings . Publ. Math. Debrecen 75 ( 1-2 ), 191 - 202 ( 2009 )
12. Les´niak, Z. : On boundaries of parallelizable regions of flows of free mappings , Abstr. Appl. Anal . 2007 ( 2007 ), Article ID 31693
13. Les´niak, Z. : On strongly irregular points of a Brouwer homeomorphism embeddable in a flow , Abstr. Appl. Anal . 2014 ( 2014 ), Article ID 638784
14. Les´niak, Z. : On the first prolongational limit set of flows of free mappings . Tamkang J. Math. 39 , 263 - 269 ( 2008 )
15. Les´niak, Z. : On the structure of Brouwer homeomorphisms embeddable in a flow , Abstr. Appl. Anal . 2012 ( 2012 ), Article ID 248413
16. Les´niak, Z. : On the topological equivalence of flows of Brouwer homeomorphisms . J. Differ. Equ. Appl . 22 , 853 - 864 ( 2016 )
17. Le Roux , F. : Classes de conjugaison des flots du plan topologiquement équivalents au flot de Reeb. C. R. Acad. Sci. Paris 328 ( 1 ), 45 - 50 ( 1999 )
18. Matsumoto , S.: A characterization of the standard Reeb flow . Hokkaido Math. J . 42 ( 1 ), 69 - 80 ( 2013 )
19. McCann , R.C. : Planar dynamical systems without critical points . Funkcial. Ekvac . 13 , 67 - 95 ( 1970 )
20. Nakayama , H.: On dimensions of non-Hausdorff sets for plane homeomorphisms . J. Math. Soc. Jpn . 47 ( 4 ), 789 - 793 ( 1995 )