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We study the set of all strongly irregular points of a Brouwer homeomorphism

In this part we recall the requisite definitions and results concerning Brouwer homeomorphisms and flows of such homeomorphisms.

By a

the function

For any sequence of subsets

A point

We say that a set

Homma and Terasaka [

Let

For an irregular point

Homma and Terasaka [

A counterpart of Theorem

In this section we study the structure of the set of all irregular points for Brouwer homeomorphisms embeddable in a flow.

Let

By the trajectory of a point

For a flow

The set

In [

Let

Let

Let

Now we take an

By the assumption that

Fix any

Since an analogous reasoning can be applied to the set of strongly negatively irregular points and the first negative prolongational limit set, our considerations can be summarized in the following way.

Let

Let

After a reparametrization of the flow

Let

In this section we describe the form of any flow of Brouwer homeomorphisms. To give a sufficient condition for the topological conjugacy of flows of Brouwer homeomorphisms one can use covers of the plane by maximal parallelizable regions. We will study the functions which express the relations between parallelizing homeomorphisms of such regions.

It is known that a simply connected region

Let

For any distinct trajectories

Let

A tree

if

A tree

if

The set

Now we recall results describing the flows of Brouwer homeomorphisms.

Let

Let

The above proposition is formulated for

The homeomorphisms

The continuous functions

The functions

Let us consider the case where

Suppose, on the contrary, that there exists a sequence

By the fact that

The functions

In this section we consider the problem of topological conjugacy of a class of flows of Brouwer homeomorphisms. To prove our result we use the form of such flows.

We say that flows

In [

Let

Put

Put

For each flow

Consider a constant

Now we introduce a class of flows of Brouwer homeomorphisms. Put

Consider a flow

Fix an

Let us assume that for each

A standard generalized Reeb flow can have either a finite number of maximal parallelizable regions or an infinite number of such regions. The first case holds if the set of indices

A generalized Reeb flow with

A generalized Reeb flow with

Consider a flow

Now we can prove the following conjugacy result.

Let

Assume that one of the conditions (a) and (b) holds. First, let us note that there exists a topological conjugacy

Fix an

Fix any

The author declares that there is no conflict of interests regarding the publication of this paper.