# Topological equivalence of submersion functions and topological equivalence
of their foliations on the plane: the linear-like case

Francisco Braun and Ingrid S. Meza-Sarmiento  Departamento de Matemática,
Universidade Federal de São Carlos, 13565–905 São Carlos, São Paulo, Brazil
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

Let $f,g:\mathbb{R}^{2}\to\mathbb{R}$ be two submersion functions and
$\mathscr{F}(f)$ and $\mathscr{F}(g)$ be the regular foliations of
$\mathbb{R}^{2}$ whose leaves are the connected components of the levels sets
of $f$ and $g$, respectively. The topological equivalence of $f$ and $g$
implies the topological equivalence of $\mathscr{F}(f)$ and $\mathscr{F}(g)$,
but the converse is not true, in general. In this paper, we introduce the
class of linear-like submersion functions, which is wide enough in order to
contain non-trivial behaviors, and provide conditions for the validity of the
converse implication for functions inside this class. Our results lead us to a
complete topological invariant for topological equivalence in a certain
subclass of linear-like submersion functions.

###### Key words and phrases:

Regular foliation, topological classification, submersions

###### 2010 Mathematics Subject Classification:

Primary: 37C10, 57R30, 58K65; Secondary: 34C99.

This paper is in final form and no version of it will be submitted for
publication elsewhere.

##  1\. Introduction

We say that two functions $f,g:\mathbb{R}^{2}\to\mathbb{R}$ are _topologically
equivalent_ (resp. o-_topologically equivalent_) if there exist homeomorphisms
(resp. orientation preserving homeomorphisms)
$h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ and $\ell:\mathbb{R}\to\mathbb{R}$ such
that

$\ell\circ f=g\circ h.$

On the other hand, if $\mathscr{F}$ and $\mathscr{G}$ are regular foliations
of open subsets $A$ and $B$ of $\mathbb{R}^{2}$, respectively, we say they are
_topologically equivalent_ (resp. o-_topologically equivalent_) if there
exists a homeomorphism (resp. orientation preserving homeomorphisms) $h:A\to
B$ carrying each leaf of $\mathscr{F}$ onto a leaf of $\mathscr{G}$. In this
case we call $h$ an _equivalence homeomorphism_ (resp. _o-equivalence
homeomorphism_) between $\mathscr{F}$ and $\mathscr{G}$.

Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be a $C^{\infty}$ submersion function. It
is well known that $f$ induces a regular foliation of $\mathbb{R}^{2}$ whose
leaves are the connected components of $f$. We denote this foliation by
$\mathscr{F}(f)$.

If $f$ and $g$ are topologically equivalent $C^{\infty}$ submersion functions
of $\mathbb{R}^{2}$, the homeomorphism $h$ of the definition carries level
sets of $f$ onto level sets of $g$, preserving the connected components. Hence
$h$ is an equivalence homeomorphism proving that $\mathscr{F}(f)$ and
$\mathscr{F}(g)$ are topologically equivalent. Clearly this implication is
also true for o-topologically equivalences. But the converse is not true, even
in the polynomial case. Indeed, take

(1) $\displaystyle p(x,y)$ $\displaystyle=x(7x-5)+(x+1)^{2}x^{2}(x-2)^{2}y,$
$\displaystyle q(x,y)$ $\displaystyle=2x(4x-5)+(x+1)^{2}x^{2}(x-2)^{2}y.$

In Example 4.1 below we prove that $\mathscr{F}(p)$ and $\mathscr{F}(q)$ are
o-topologically equivalent although $p$ and $q$ are not topologically
equivalent. The main objective of this paper is to study this converse, i.e.
to provide conditions in order that the topological equivalence of
$\mathscr{F}(p)$ and $\mathscr{F}(q)$ implies the topological equivalence of
$p$ and $q$, for two given submersion functions
$p,q:\mathbb{R}^{2}\to\mathbb{R}$. At least for a class of submersions to be
defined soon.

The question of characterizing the topological equivalence of functions has
long been studied, in different frameworks. In the theory of singularities the
local topological equivalence of smooth or analytic functions is a classical
subject since the works of M. Morse [12, 13] in the topological classification
of continuous functions on $2$-dimensional manifolds. In the study of
integrable systems on the plane, it is possible to relate the topological
equivalence of such systems to the topological equivalence of their first
integrals. For instance, J. Martínez-Alfaro et al. [9, 10] gave complete
topological invariants for the integrable Morse-Bott systems and for their
first integrals, the so called Morse-Bott functions. In a similar line, K.R.
Meyer [11] associated a function, called $\xi$-function, to each Morse-Smale
system (a $\xi$-functions is not a first integrals of the system, as its level
sets may be transversal to it), and provided conditions in order to obtain the
equivalence between topological equivalence of two Morse-Smale systems and the
topological equivalence of the related $\xi$-functions.

Although the topological equivalence of functions is a very studied subject
and there are local topological invariants for its study, global results and
complete invariants are not so common in the literature.

Without singularities and so more close to our subject, V.V. Sharko and Y.Y.
Soroka [14] proved that a continuous function $f:\mathbb{R}^{2}\to\mathbb{R}$,
without critical points, is topologically equivalent to a projection if and
only if the level sets of $f$ are connected and non-empty. In the $C^{\infty}$
higher dimensional case, by using different tools, M. Tibăr and A. Zaharia
[16, Prop. 2.7] proved that if $M\subseteq\mathbb{R}^{n}$ is a $C^{\infty}$
submanifold of dimension $m+1$ and $g:M\to\mathbb{R}^{m}$ is a $C^{\infty}$
function without critical values having level sets diffeomorphic to
$\mathbb{R}$ and closed in $\mathbb{R}^{n}$, then $g$ is a $C^{\infty}$
_trivial fibration_ (see the definition further in this work), in other words
$g$ is topologically equivalent to a projection. Moreover, $M$ is
diffeomorphic to $\mathbb{R}^{m+1}$.

The foliation $\mathscr{F}(f)$ is _trivial_ , i.e. without separatrices (see
the precise definition further in this paper), in Sharko and Soroka’s case, in
fact the very same result says that connectedness of level sets of $f$ is
equivalent to triviality of $\mathscr{F}(f)$. So, in particular, when two
surjective submersion functions have trivial foliation they are topologically
equivalent.

The level sets of a submersion $f:\mathbb{R}^{2}\to\mathbb{R}$ need not be
connected or, equivalently, the foliation $\mathscr{F}(f)$ need not be
trivial, as the examples in (1) above illustrate, and the right above-
mentioned equivalence does not hold. So what are the additional hypothesis we
have to assume in order to guarantee that two given submersion functions
$f,g:\mathbb{R}^{2}\to\mathbb{R}$ having topologically equivalent foliations
$\mathscr{F}(f)$ and $\mathscr{F}(g)$ are topologically equivalent?

In this paper we address this question for the class of submersions of the
form

(2) $p(x,y)=r(x)+s(x)y,$

where $r(x)$ and $s(x)$ are $C^{\infty}$ real functions. We say that $p$ as in
(2) is a _linear-like_ function. This class of functions is wide enough for
containing trivial submersions (when $s(x)\neq 0$ for all $x$) and more
interesting examples, as the known S.A. Broughton’s function [4] $x+x^{2}y$
and yet more complicated ones, for instance our examples above as well as
examples presenting limit-separatrices in the associated foliation, see
sections 3 and 4. We answer the above question for the subclass of _finite
linear-like submersions_ , defined to be the linear-like ones where the zero
set of $s$ is finite, in the results below. In order to state them, we denote
by $B(p)$ the bifurcation set of a function $p$, see Definition 2.5 below.

######  Theorem 1.1.

Two finite linear-like submersions $p,\,q:\mathbb{R}^{2}\to\mathbb{R}$ are
o-topologically equivalent if and only if $B(p)=B(q)=\emptyset$ or the
following conditions hold:

  1. (a)

$\mathscr{F}(p)$ and $\mathscr{F}(q)$ are o-topologically equivalent.

  2. (b)

There exists an increasing bijection $\sigma:B(p)\to B(q)$ such that

$p^{-1}(\sigma^{-1}(c))=h^{-1}(q^{-1}(c))$

for each $c\in B(q)$, where $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is an
o-equivalence homeomorphism between $\mathscr{F}(p)$ and $\mathscr{F}(q)$. If
$B(p)$ is a singleton, there further exist $m\notin B(p)$ and an extension of
$\sigma$ to an increasing bijection $\overline{\sigma}:B(p)\cup\\{m\\}\to
B(q)\cup\\{\overline{\sigma}(m)\\}$ such that
$p^{-1}(m)=h^{-1}(q^{-1}(\overline{\sigma}(m)))$.

The appearing of the bifurcation set in Theorem 1.1 becomes natural if one is
aware that A. Bodin and M. Tibăr [1] proved that two _complex_ polynomials $p$
and $q$ with $n$ variables, $n\neq 3$, are topologically equivalent if and
only if it is possible to deform $p$ into $q$ by means of a continuous family
of polynomials $h_{s}$, with $h_{0}=p$ and $h_{1}=q$, such that $\deg h_{s}$,
the Euler characteristic of the general fibre of $h_{s}$ and the cardinality
of $B(h_{s})$ are independent of $s$. Our approach is different as we do not
use deformations, instead we use the foliations associated to $p$ and $q$ and
the relation in statement (b).

Anyway, even in the polynomial linear-like case, only assuming that the
general fibers of $p$ and $q$ have the same Euler characteristic and the same
bifurcation sets is not sufficient for the topological equivalence of $p$ and
$q$: in Example 4.1 we will explain that the polynomials of degree $7$ $p$ and
$q$ of (1) have the same bifurcation set and generic fibers with the same
Euler characteristic (with $\mathscr{F}(p)$ and $\mathscr{F}(q)$ topologically
equivalent), but even so $p$ and $q$ are not topologically equivalent.

Condition (b) is equivalent to ask the relation of topologically equivalence
of $p$ and $q$, $\ell\circ p=q\circ h$, in the pre-image of $B(p)$ (or $B(p)$
plus a point) by $p$. Actually we just need to test this condition in a
smaller set of points, according to the more general Theorem 5.2 below, see
also Theorem 1.3.

When $B(p)$ and $B(q)$ are singletons, we need to add points to them in
condition (b) in order to guarantee o-topological equivalence: indeed $p$ and
$-p$, where $p(x,y)=x+x^{3}y$, are not o-topologically equivalent although $p$
and $-p$ are topologically equivalent and $\mathscr{F}(p)$ and
$\mathscr{F}(-p)$ are o-topologically equivalent, see Example 4.2 below. If we
are just concerned with topological equivalence, then we can rephrase Theorem
1.1 as

######  Corollary 1.2.

Two finite linear-like submersions $p,\,q:\mathbb{R}^{2}\to\mathbb{R}$ are
topologically equivalent if and only if $B(p)=B(q)=\emptyset$ or the following
conditions hold:

  1. (a)

$\mathscr{F}(p)$ and $\mathscr{F}(q)$ are topologically equivalent.

  2. (b)

There exists a monotone bijection $\sigma:B(p)\to B(q)$ such that
$p^{-1}(\sigma^{-1}(c))=h^{-1}(q^{-1}(c))$ for each $c\in B(q)$, where
$h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is an equivalence homeomorphism between
$\mathscr{F}(p)$ and $\mathscr{F}(q)$.

See also the more general Corollary 5.3 below. In condition (b), we can not
assure that $\sigma$ is increasing if $p$ and $q$ are only topologically
equivalent, see Example 4.3.

From our results we obtain _complete topological invariants_ for topological
and o-topological equivalence of finite linear-like submersions by putting
together the _separatrix configuration_ $S\mathfrak{S}_{\mathscr{F}(p)}$ of
the foliation $\mathscr{F}(p)$, according to Definition 2.2 below, and some
“order” in the image of the set of _separatrices_ inside the bifurcation set
$B(p)$ of $p$ (for precise definitions of these concepts, see Section 2):

######  Theorem 1.3.

Two finite linear-like submersions $p,q:\mathbb{R}^{2}\to\mathbb{R}$ are
o-topologically equivalent if and only if $B(p)=B(q)=\emptyset$ or the
following conditions hold:

  1. (a)

The separatrix configurations of $\mathscr{F}(p)$ and $\mathscr{F}(q)$ are
isomorphic.

  2. (b)

There exists an increasing bijection $\sigma:B(p)\to B(q)$ such that
$\sigma\circ p(\gamma)=q\circ g(\gamma)$ for every separatrix
$\gamma\in\mathfrak{S}_{\mathscr{F}(p)}$, where $g$ is an isomorphism between
$S\mathfrak{S}_{\mathscr{F}(p)}$ and $S\mathfrak{S}_{\mathscr{F}(q)}$. If
$B(p)$ is a singleton, there further exist an ordinary leaf $\xi$ in
$S\mathfrak{S}_{\mathscr{F}(p)}$ with $p(\xi)=m\notin B(p)$ and an extension
of $\sigma$ to an increasing bijection $\overline{\sigma}:B(p)\cup\\{m\\}\to
B(q)\cup\\{\overline{\sigma}(m)\\}$ such that
$\overline{\sigma}(m)=q(g(\xi))$.

Since there are not vanishing phenomenon in the linear-like submersion case,
according to Corollary 3.6 below, the image of the separatrices of
$\mathscr{F}(p)$ agrees with the set $B(p)$, so the above relations make
sense. For topological equivalence we have

######  Corollary 1.4.

Two finite linear-like submersions $p,q:\mathbb{R}^{2}\to\mathbb{R}$ are
topologically equivalent if and only if $B(p)=B(q)=\emptyset$ or the following
conditions hold:

  1. (a)

The separatrix configurations of $\mathscr{F}(p)$ and $\mathscr{F}(q)$ are
isomorphic or anti-isomorphic.

  2. (b)

There exists a monotone bijection $\sigma:B(p)\to B(q)$ such that $\sigma\circ
p(\gamma)=q\circ g(\gamma)$ for every
$\gamma\in\mathfrak{S}_{\mathscr{F}(p)}$, where $g$ is an isomorphism or anti-
isomorphism between $S\mathfrak{S}_{\mathscr{F}(p)}$ and
$S\mathfrak{S}_{\mathscr{F}(q)}$.

The organization of the paper is as follows: Section 2 is mainly devoted to
recall some results on regular foliations on the plane, as well as bifurcation
theory. In Section 3 we study the linear-like functions. In Section 4 we give
examples of linear-like submersions with different kind of behaviors. The main
results stated above are then proved in Section 5 as a consequence of the more
general Theorem 5.2.

In order to simplify notations, sometimes throughout the text a singleton will
be viewed as just its element.

##  2\. Regular foliations and bifurcation sets

###  2.1. Regular foliations, chordal relations and a complete topological
invariant

Given an open set $U\subset\mathbb{R}^{2}$, we say that a collection of curves
$\mathscr{F}$ contained in $U$ is a _regular foliation_ of $U$ if

$U=\bigcup_{\gamma\in\mathscr{F}}\gamma,$

and for any $P\in U$ there exists an open neighborhood of $P$, $N(P)$, and a
homeomorphism $h:N(P)\to\mathbb{R}^{2}$ such that each curve of
$\mathscr{F}|_{N(P)}\dot{=}\\{\gamma\cap N(P)\ |\ \gamma\in\mathscr{F}\\}$ is
carried by $h$ onto a line $x=\textnormal{const}$. A curve of $\mathscr{F}$
will be also called _leaf_.

For instance, let $f:\mathbb{R}^{2}\to\mathbb{R}$ be a $C^{\infty}$ submersion
function. From the implicit function Theorem, $f$ defines a regular foliation
of $\mathbb{R}^{2}$, denoted by $\mathscr{F}(f)$, whose leaves are the
connected components of the level sets of $f$

$L_{c}=L_{c}(f)=f^{-1}(c),$

$c\in f(\mathbb{R}^{2})$. Foliations $\mathscr{F}(f)$ of $\mathbb{R}^{2}$ will
play the main role of this paper.

To any foliation $\mathscr{F}$ of an open set $U\subset\mathbb{R}^{2}$ we
associate the _space of leaves_ $U/\mathscr{F}$, defined to be the quotient
space $U/{\mathord{\sim}}$, where $\sim$ is the equivalence relation in $U$
saying that two points are related if they belong to the same leaf of
$\mathscr{F}$. This topological space, with the quotient topology, may not be
Hausdorff.

Two leaves of $\mathscr{F}$ whose classes can not be separated in
$U/\mathscr{F}$ are said _inseparable leaves_. We can equivalently define them
as follows: two leaves $l_{1},l_{2}\in\mathscr{F}$ are inseparable if for any
cross sections $C_{1}$ and $C_{2}$ through $l_{1}$ and $l_{2}$ there exists
another leaf $l_{3}\in\mathscr{F}$ intersecting $C_{1}$ and $C_{3}$.

We say that a leaf $l\in\mathscr{F}$ is a _separatrix_ if it lives in the
closure of the set of the inseparable leaves. A separatrix which is not an
inseparable leaf is called a _limit separatrix_. The set of separatrices of
$\mathscr{F}$ will be denoted by $\mathfrak{S}_{\mathscr{F}}$. This is a
closed set by definition. Each connected component of the complement of
$\mathfrak{S}_{\mathscr{F}}$ will be called a _canonical region_.

From now on we consider regular foliations of the entire plane. Each curve
$\gamma$ of such a foliation tends to infinity in both directions and so
divides the plane into two open regions such that $\gamma$ is the common
boundary. Rephrasing W. Kaplan [5, 6], we can view the leaves of such a
foliation as “chords” laying on a disc. Kaplan managed to prove that the
relations between triples of leaves of the foliation, the _chordal relations_
, characterize it as we will recall in Theorem 2.1. First we recall the
chordal relations.

Given $l_{1}$, $l_{2}$ and $l_{3}$ three different leaves of $\mathscr{F}$, we
say that $l_{2}$ _separates_ $l_{1}$ and $l_{3}$, denoting this by
$l_{1}|l_{2}|l_{3}$ or $l_{3}|l_{2}|l_{1}$, if $l_{1}$ and $l_{3}$ are in
different connected components of $\mathbb{R}^{2}\setminus l_{2}$. If $l_{i}$
does not separate $l_{j}$ and $l_{k}$ for any different $i,j,k\in\\{1,2,3\\}$
then we say that $l_{1}$, $l_{2}$ and $l_{3}$ form a _cyclic triple_. In this
case, for any points $p_{i}\in l_{i}$, $i=1,2,3$, there exists a Jordan curve
$p_{1}p_{2}p_{3}p_{1}$ intersecting the leaves $l_{1}$, $l_{2}$, $l_{3}$ only
at the points $p_{i}$, $i=1,2,3$, respectively. If this curve, from $p_{1}$ to
$p_{2}$ to $p_{3}$ and returning to $p_{1}$ spins counterclockwise (resp.
clockwise) we say that this cycle is _positive_ (resp. _negative_) and denote
this by $|l_{1},l_{2},l_{3}|^{+}$ (resp. $|l_{1},l_{2},l_{3}|^{-}$). The
notation $|l_{1},l_{2},l_{3}|^{\pm}$ shall mean $|l_{1},l_{2},l_{3}|^{+}$ or
$|l_{1},l_{2},l_{3}|^{-}$. By a _chordal relation_ of a triple of three
different leaves $l_{1},l_{2},l_{3}\in\mathscr{F}$ we will mean
$l_{1}|l_{2}|l_{3}$, $l_{2}|l_{1}|l_{3}$, $l_{1}|l_{3}|l_{2}$,
$|l_{1},l_{2},l_{3}|^{+}$ or $|l_{1},l_{2},l_{3}|^{-}$. For a very detailed
and more general presentation of this, see [5, Section 2].

Given two subsets $U$ and $V$ of leaves of two regular foliations of the
plane, we say that $U$ and $V$ are _isomorphic_ if there exists a bijection
$g:U\to V$ such that for any triple $l_{1},l_{2},l_{3}\in U$, the chordal
relations of $l_{1},l_{2},l_{3}$ and $g(l_{1}),g(l_{2}),g(l_{3})$ are the
same. We say that $U$ and $V$ are _anti-isomorphic_ if there is a bijection
$g:U\to V$ as above preserving all the chordal relations of triples of leaves
but the signal of the cycles, that are reversed by $g$. The main result of
Kaplan in [6] says that two regular foliations are o-topologically equivalent
if and only if the set of leaves of them are isomorphic. Indeed, the result is
quite more precise, see [6, Theorem 26]:

######  Theorem 2.1 ([6]).

If $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ are o-topologically equivalent
regular foliations of the plane with o-equivalence homeomorphism $h$, then $h$
defines an isomorphism between $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$.
Conversely, if $g$ is an isomorphism between $\mathscr{F}_{1}$ and
$\mathscr{F}_{2}$, then there exists an o-equivalence homeomorphism $h$
providing the o-topological equivalence of $\mathscr{F}_{1}$ and
$\mathscr{F}_{2}$, and such that $h(l)=g(l)$ for all $l\in\mathscr{F}_{1}$.

L. Markus [7] proved that in order to attest the o-topological equivalence
between two foliations, it is only necessary to provide an isomorphism between
the sets of separatrices plus one leaf in each canonical region. In order to
state Markus’s result we first define

######  Definition 2.2.

A _separatrix configuration_ of a given regular foliation $\mathscr{F}$ is the
set of separatrices $\mathfrak{S}_{\mathscr{F}}$ together with one leaf chosen
in each canonical region. We denote the separatrix configuration of
$\mathscr{F}$ by $S\mathfrak{S}_{\mathscr{F}}$.

Two separatrix configurations of a given regular foliation are isomorphic, see
again [7], so we can say _the_ separatrix configuration of a foliation. Here
is Markus mentioned result:

######  Theorem 2.3 ([7]).

Two regular foliations $\mathscr{F}_{1}$ and $\mathscr{F}_{2}$ are
o-topologically equivalent if and only if the separatrix configurations
$S\mathfrak{S}_{\mathscr{F}_{1}}$ and $S\mathfrak{S}_{\mathscr{F}_{2}}$ are
isomorphic. They are topologically equivalent if and only if the separatrix
configurations are isomorphic or anti-isomorphic.

This theorem says that the separatrix configuration is a complete topological
invariant of a regular foliation. By the _trivial foliation_ we mean any
representative of the class of foliations without separatrices.

Before presenting the last result of this subsection, we introduce a notation:
given $l_{1}$ and $l_{2}$ two leaves of $\mathscr{F}$, we define by
$[l_{1},l_{2}]$ (resp. $(l_{1},l_{2})$) to be the closed (resp. open)
connected set whose boundary is $l_{1}\cup l_{2}$. Further
$(l_{1},l_{2}]=(l_{1},l_{2})\cup l_{2}=[l_{1},l_{2}]\setminus l_{1}$.

The following result is Theorem 1 of Markus’s paper [8] for a regular
foliation $\mathscr{F}$ given by the orbits of a noncritical vector field of
the plane. After a glance in its proof, one concludes that it is valid for any
regular foliation of $\mathbb{R}^{2}$, by possibly using [5, Theorem 30] to
conclude the necessity of statement 2.

######  Theorem 2.4.

Let $\mathscr{F}$ be a regular foliation of the plane.

  1. (1)

A leaf $l\in\mathscr{F}$ is a separatrix if and only if for any pair of leaves
$l_{1},l_{2}\in\mathscr{F}$ such that $l_{1}|l|l_{2}$, it follows that
$[l_{1},l_{2}]$ contains a cyclic triple of solutions.

  2. (2)

On the other hand a leaf $l\in\mathscr{F}$ is not a separatrix if and only if
there exist leaves $l_{1},l_{2}\in\mathscr{F}$ with $l_{1}|l|l_{2}$ such that
$[l_{1},l_{2}]$ is the topological image of the plane strip
$[0,1]\times\mathbb{R}$ with the curves of $\mathcal{F}|_{[l_{1},l_{2}]}$
corresponding to the vertical lines of the strip.

###  2.2. Bifurcation sets

The topology of the fiber $f^{-1}(y)$ of a given continuous function $f$ may
change when $y$ varies. As our examples in the introduction section, these
changes can occur even when $y$ do not cross a singular value. The set where
this changes happen is the _bifurcation set_. More precisely:

######  Definition 2.5.

We say that a continuous map $p:X\to Y$ between topological spaces $X$ and $Y$
is a trivial fibration if there exists a topological space $F$ and a
homeomorphism $\varphi:F\times Y\to X$ such that $p\circ\varphi$ is the
trivial projection. We say further that $p$ is a locally trivial fibration at
$y\in Y$ if there is a neighbourhood $U$ of $y$ such that the restriction
$p^{-1}(U)\xrightarrow{\phantom{.}p\phantom{.}}U$ is a topologically trivial
fibration. When $y\in Y$ does not satisfy this property, we say that $y$ is a
bifurcation value. We denote by $B(p)$ the set of bifurcation values of $p$.
The set $B(p)$ is also called the bifurcation set of $p$. In case
$B(p)=\emptyset$ we say that $p$ is a _locally trivial fibration_.

Notice that if $p$ is a locally trivial fibration, then its fibers are
pairwise homeomorphic. It is known that if $p:\mathbb{K}^{n}\to\mathbb{K}$ is
a polynomial function, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, then
$B(p)$ is a finite set (see [15, Cor. 1.2.14]).

It is easy to check that if $p,q:X\to Y$ are topologically equivalent, i.e.,
if $q=\ell\circ p\circ h^{-1}$, with $\ell:Y\to Y$ and $h:X\to X$
homeomorphisms, then $p$ is a locally trivial fibration at $y\in Y$ if and
only if $q$ is a locally trivial fibration at $\ell(y)$. So we have

######  Lemma 2.6.

Let $p,\,q:X\to Y$ satisfying $\ell\circ p=q\circ h$, with $\ell:Y\to Y$ and
$h:X\to X$ homeomorphisms. Then $B(q)=\ell(B(p))$. In particular, if
$p,q:\mathbb{R}^{2}\to\mathbb{R}$ are o-topologically equivalent functions
then $B(p)$ and $B(q)$ are homeomorphic with an increasing homeomorphism.

##  3\. Linear-like functions

In this section we study the linear like functions $p(x,y)=r(x)+s(x)y$ as
defined in (2). We pay attention to the submersion ones, studying the
foliation $\mathscr{F}(p)$. We first describe the connected components of the
level sets $L_{c}$ of $p$, i.e., the leaves of $\mathscr{F}(p)$. This provides
a subdivision of $\mathbb{R}^{2}$ in certain vertical strips where the
behavior of $\mathscr{F}(p)$ is described. We study the inseparable leaves of
$\mathscr{F}(p)$ in Proposition 3.2 and the limit separatrices in Corollary
3.3. Then we systematically study the distinct behaviors of the foliation in
each of these mentioned strips. In particular we will see that the behavior of
$\mathscr{F}(p)$ in each of these strips depends only on the separatrices and
their “asymptotic” behaviour, as defined later.

We will end the section providing the bifurcation set of a finite linear-like
submersion in Theorem 3.5 and concluding that finite linear-like submersion
functions do not present the vanishing at infinity phenomenon.

It is straightforward to characterize when a linear-like function is a
submersion, as well as to describe its levels sets:

######  Lemma 3.1.

Let $p$ be a linear-like function as in (2). Then

  1. (a)

$p$ is a submersion if and only if $s(x)\neq 0$ $\forall x\in\mathbb{R}$ or
each zero $z$ of $s(x)$ is not simple and $r^{\prime}(z)\neq 0$.

  2. (b)

For each $c\in\mathbb{R}$, the level set $L_{c}=p^{-1}(c)$ of $p$ is given by

$L_{c}=\left\\{\big{(}x,N_{c}(x)\big{)}\in\mathbb{R}^{2}\ |\ s(x)\neq
0\right\\}\cup\left\\{(x,y)\in\mathbb{R}^{2}\ |\ s(x)=0,\,r(x)=c\right\\},$

where

$N_{c}(x)=\frac{c-r(x)}{s(x)}.$

From now on, $\nu_{a}$ will stand for the vertical straight line $x=a$,
$a\in\mathbb{R}$, i.e.,

$\nu_{a}=\\{a\\}\times\mathbb{R}.$

Also, we will denote

$Z_{p}=\\{x\in\mathbb{R}\ |\ s(x)=0\\}.$

Since this is a closed subset of $\mathbb{R}$, it follows that

$\mathbb{R}\setminus Z_{p}=\cup_{i=1}^{\infty}(a_{i},b_{i})$

is a disjoint countable many union of (some possible empty) open intervals
$(a_{i},b_{i})$, where $a_{i}$ could be $-\infty$ and $b_{i}$ could be
$\infty$. So the leaves of $\mathscr{F}(p)$ are completely described as being
of two types:

> In each non-empty strip $(a_{i},b_{i})\times\mathbb{R}$ there exists exactly
> one connected component of $L_{c}$ given by the graphic of the function
> $N_{c}$ restricted to $(a_{i},b_{i})$, for any $c\in\mathbb{R}$. For each
> $z\in Z_{p}$ the vertical straight line $\nu_{z}$ is a connected component
> of $L_{c}$, where $c=r(z)$.

######  Proposition 3.2.

Let $p$ be a linear-like submersion such that $Z_{p}$ is not empty. If
$a_{i}>-\infty$ (resp. $b_{i}<\infty$) then there exists exactly one leaf of
$\mathscr{F}(p)$ contained in the strip $(a_{i},b_{i})\times\mathbb{R}$ which
is inseparable to $\nu_{a_{i}}$ (resp. $\nu_{b_{i}}$). More precisely, letting
$c$ be the value assumed by $p$ in $\nu_{a_{i}}$ (resp. $\nu_{b_{i}}$), then
the graphic of the function $N_{c}$ defined in $(a_{i},b_{i})$ is inseparable
to $\nu_{a_{i}}$ (resp. $\nu_{b_{i}}$).

The collection of the above leaves are the inseparable leaves of
$\mathscr{F}(p)$.

###### Proof.

To conclude that $\nu_{a_{i}}$ (resp. $\nu_{b_{i}}$) and the leaf in
$(a_{i},b_{i})\times\mathbb{R}$ in the same level set of $\nu_{a_{i}}$ (resp.
$\nu_{b_{i}}$) are inseparable, we provide two proofs. The first one is a
direct application of [3, Corollary 3.3]: “Let $f:\mathbb{R}^{2}\to\mathbb{R}$
be a $C^{\infty}$ submersion. Let $\gamma_{1}$ and $\gamma_{2}$ be two
distinct connected components of a level set $f^{-1}(c)$. If $\lambda$ is an
injective $C^{\infty}$ curve contained in $[\gamma_{1},\gamma_{2}]$ connecting
$\gamma_{1}$ to $\gamma_{2}$, then there exists at least a pair of inseparable
leaves of $\mathscr{F}(f)$ contained in $[\gamma_{1},\gamma_{2}]$ intercepting
$\lambda$” and the fact that two leaves inseparable to each other are
connected components of the same level set of $p$.

The second one is as follows. Let $c\in\mathbb{R}$ be the value taken by $p$
in $\nu_{a_{i}}$ and denote by $\gamma_{c}$ the only connected component of
the level set $L_{c}$ contained in $(a_{i},b_{i})\times\mathbb{R}$, i.e., the
graphic of the function $N_{c}$ in $(a_{i},b_{i})$.

Let $S_{1}$ be a transversal section through $\gamma_{c}$ and $S_{2}$ be the
intersection of a transversal section through the straight line $\nu_{a_{i}}$
with the strip $[a_{i},b_{i})\times\mathbb{R}$.

The images $p(S_{1})$ and $p(S_{2})$ are intervals containing $c$ such that
for $\varepsilon>0$ small enough the first one contains the interval
$(c-\varepsilon,c+\varepsilon)$ and the second one the interval
$(c-\varepsilon,c)$ or $(c,c+\varepsilon)$. In any case, there exists $d\neq
c$ with $d\in p(S_{1})\cap p(S_{2})$.

The level set $L_{d}$ intersects $S_{1}$ and $S_{2}$ by construction. Since
there exists exactly one connected component of $L_{d}$ in the strip
$(a_{i},b_{i})\times\mathbb{R}$, this connected component must intersect
$S_{1}$ and $S_{2}$, proving that $\gamma_{c}$ and $\nu_{a_{i}}$ are
inseparable.

Now we prove that there are no other type of inseparable leaves. Since in any
strip $(a_{i},b_{i})\times\mathbb{R}$, there is only one connected component
of each level set of $p$, any leaf in such a strip can only be inseparable to
$\nu_{a_{i}}$ or to $\nu_{b_{i}}$.

So let $\nu_{z_{1}}$ and $\nu_{z_{2}}$, $z_{1},z_{2}\in Z_{p}$, be two leaves
inseparable to each other. There does not exist $z\in Z_{p}$ between $z_{1}$
and $z_{2}$ otherwise the leaf $\nu_{z}$ would separate $\nu_{z_{1}}$ from
$\nu_{z_{2}}$. So $z_{1},z_{2}\in\\{a_{i},b_{i}\\}$ for some $i$, finishing
the proof. ∎

The following is a direct consequence of Proposition 3.2.

######  Corollary 3.3.

Let $p$ be a linear-like submersion such that $Z_{p}\neq\emptyset$. The leaves
$\nu_{z}$ such that
$z\in\overline{\cup_{i=1}^{\infty}\\{a_{i},b_{i}\\}}\setminus\cup_{i=1}^{\infty}\\{a_{i},b_{i}\\}$
are the limit separatrices of $\mathscr{F}(p)$.

Next we describe the behavior of $\mathscr{F}(p)$ in each nonempty strip
$(a_{i},b_{i})\times\mathbb{R}$. We have the following possible configurations
of separatrices and canonical regions according to Proposition 3.2:

  1. (I)

If $a_{i}=-\infty$ (resp. $b_{i}=\infty$), we have one inseparable leaf and so
exactly two canonical regions of $\mathscr{F}(p)$ contained in
$(a_{i},b_{i})\times\mathbb{R}$.

  2. (II)

If $-\infty<a_{i}<b_{i}<\infty$ and the values that $p$ assumes in
$\nu_{a_{i}}$ and in $\nu_{b_{i}}$ are distinct, we have two inseparable
leaves and so three canonical regions of $\mathscr{F}(p)$ contained in
$(a_{i},b_{i})\times\mathbb{R}$

  3. (III)

If $-\infty<a_{i}<b_{i}<\infty$ and the values that $p$ assumes in
$\nu_{a_{i}}$ and in $\nu_{b_{i}}$ are equal, we have one inseparable leaf and
so two canonical regions of $\mathscr{F}(p)$ contained in
$(a_{i},b_{i})\times\mathbb{R}$.

In order to determine the foliation in each strip, we will act according to
Theorem 2.3, describing the separatrix configuration in each strip. This will
be done by determining the behavior of a leaf in each of the canonical regions
in configurations (I), (II) and (III). We first study the behavior of a leaf
in a canonical region having in its boundary at least two leaves inseparable
to each other. By Proposition 3.2 and the possible configurations (I), (II)
and (III) above, this boundary must contain a straight line $\nu_{a_{i}}$ or
$\nu_{b_{i}}$, and a connected component of the level set $L_{c}$, where $c$
is the value that $p$ takes in $\nu_{a_{i}}$ or in $\nu_{b_{i}}$,
respectively. We denote by $\gamma_{c}$ this connected component, which is
given by the graphic of the function $N_{c}$ restricted to $(a_{i},b_{i})$.
Observe that the straight lines $\nu_{a_{i}},\nu_{b_{i}}$ are vertical
asymptotes of $N_{c}$.

######  Lemma 3.4.

Any leaf $\gamma$ in the canonical region whose boundary contains
$\nu_{a_{i}}$ and $\gamma_{c}$ satisfies
$|\nu_{a_{i}},\gamma,\gamma_{c}|^{\pm}$ if
$\lim\limits_{x\to{a_{i}}^{+}}N_{c}(x)=\pm\infty$. Any leaf $\gamma$ in the
canonical region whose boundary contains $\nu_{b_{i}}$ and $\gamma_{c}$
satisfies $|\nu_{b_{i}},\gamma,\gamma_{c}|^{\mp}$ if $\lim\limits_{x\to
a^{-}_{i}}N_{c}(x)=\pm\infty$.

###### Proof.

We prove the case where $\gamma$ is in the canonical region whose boundary
contains $\nu_{a_{i}}$ and $\gamma_{c}$, and
$\lim\limits_{x\to{a_{i}}^{+}}N_{c}(x)=+\infty$. All the other situations have
a completely similar proof.

The leaf $\gamma$ is in particular contained in $(\gamma_{c},\nu_{a_{i}})$.
Moreover, $\gamma$ cannot separates $\gamma_{c}$ and $\nu_{a_{i}}$, otherwise
these two leaves will not be inseparable. So
$|\nu_{a_{i}},\gamma,\gamma_{c}|^{+}$ or
$|\nu_{a_{i}},\gamma,\gamma_{c}|^{-}$.

Let $p_{1}\in\nu_{a_{i}}$, $p_{2}\in\gamma$ and $p_{3}\in\gamma_{c}$ and
consider a Jordan closed curve $p_{1}p_{2}p_{3}p_{1}$ meeting these three
leaves only in $\\{p_{1},p_{2},p_{3}\\}$ and oriented from $p_{1}$ to $p_{2}$
to $p_{3}$ to $p_{1}$.

We know that the leaf $\gamma$ is also the graphic of a function $N_{d}$
defined in $(a_{i},a_{i+1})$ with $\lim\limits_{x\to
a_{i}^{+}}N_{d}(x)=\pm\infty$ and such that $N_{d}(x)\leq N_{c}(x)$ for all
$x\in(a_{i},a_{i+1})$. We must have $\lim\limits_{x\to
a_{i}^{+}}N_{d}(x)=-\infty$, otherwise $\nu_{a_{i}}$ and $\gamma_{c}$ will not
be inseparable. So the curve $p_{1}p_{2}p_{3}p_{1}$ is counterclockwise
oriented, otherwise $\gamma$ will cut this curve at least twice. So
$|\nu_{a_{i}},\gamma,\gamma_{c}|^{+}$. ∎

###  3.1. Configuration (I)

If $a_{i}=-\infty$, let $c_{b_{i}}$ be the value taken by $p$ in $\nu_{b_{i}}$
and $\gamma_{c_{b_{i}}}$ be the leaf given by the only connected component of
the level set $L_{c_{b_{i}}}$ in $(a_{i},b_{i})$. This curve separates the
plane into two open regions. One of them does not contain any separatrix and
so will determine one of the canonical regions. A given leaf $\gamma$ in this
canonical region has just one possible behavior. Precisely, by considering
$N_{d}(x)$ the function whose graphic is $\gamma$, $\lim\limits_{x\to
b_{i}^{-}}N_{d}(x)=\lim\limits_{x\to b_{i}^{-}}N_{c_{b_{i}}}(x)$. The other
canonical region is contained in the opposite open region. It has as boundary
$\gamma_{c_{b_{i}}}$ and $\nu_{b_{i}}$, so by Lemma 3.4 it follows that
$|\nu_{b_{i}},\gamma,\gamma_{c_{b_{i}}}|^{\mp}$ if
$\lim\limits_{x\to{b_{i}}^{-}}N_{c_{b_{i}}}(x)=\pm\infty$. See the separatrix
configuration of this strip in case $\lim\limits_{x\to
b_{i}^{-}}N_{c_{b_{i}}}(x)=+\infty$ in (a) of Fig. 1.

(a) $a_{i}=-\infty$ (b) $b_{i}=\infty$

Figure 1. Separatrix configurations on the strip
$(a_{i},b_{i}]\times\mathbb{R}$ when $a_{i}=-\infty$ and on the strip
$[a_{i},b_{i})\times\mathbb{R}$ when $b_{i}=+\infty$.

If $b_{i}=\infty$, by letting $c_{a_{i}}$ the value $p$ takes in
$\nu_{a_{i}}$, we have a completely analogous analysis: in the canonical
region having as boundary the curve $\gamma_{c_{a_{i}}}$ there is only one
possibility. While in the canonical region whose boundary is
$\nu_{a_{i}}\cup\gamma_{c_{a_{i}}}$ it follows from Lemma 3.4 that
$|\nu_{a_{i}},\gamma,\gamma_{c_{a_{i}}}|^{\pm}$ if
$\lim\limits_{x\to{a_{i}}^{+}}N_{c_{a_{i}}}(x)=\pm\infty$. See in (b) of Fig.
1 a depiction in case $\lim\limits_{x\to{a_{i}}^{+}}N_{c_{a_{i}}}(x)=+\infty$.

###  3.2. Configuration (II)

Let $c_{a_{i}}$ and $c_{b_{i}}$ be the values taken by $p$ in $\nu_{a_{i}}$
and $\nu_{b_{i}}$, and $\gamma_{c_{a_{i}}}$ and $\gamma_{c_{b_{i}}}$ be the
graphics of $N_{c_{a_{i}}}$ and $N_{c_{b_{i}}}$ in $(a_{i},b_{i})$,
respectively.

We assume $\lim\limits_{x\to a^{+}_{i}}N_{c_{a_{i}}}(x)=+\infty$, the other
possibility is completely analogous. We have eight possibilities a priori for
$\lim\limits_{x\to a_{i}^{+}}N_{c_{b_{i}}}(x)$, $\lim\limits_{x\to
b_{i}^{-}}N_{c_{a_{i}}}(x)$ and $\lim\limits_{x\to
b_{i}^{-}}N_{c_{b_{i}}}(x)$. But it is simple to conclude that, in order not
to break the inseparability of $\nu_{a_{i}}$ with $\gamma_{c_{a_{i}}}$ and of
$\nu_{b_{i}}$ with $\gamma_{c_{b_{i}}}$, as well as considering that different
leaves cannot intersect each other, the only possibilities are the following
four:
If $\lim\limits_{x\to a^{+}_{i}}N_{c_{b_{i}}}(x)=+\infty$,

  1. (a)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=-\infty$ and $\lim\limits_{x\to
b^{-}_{i}}N_{c_{b_{i}}}(x)=+\infty$,

  2. (b)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=-\infty$ and $\lim\limits_{x\to
b^{-}_{i}}N_{c_{b_{i}}}(x)=-\infty$.

And, if $\lim\limits_{x\to a^{+}_{i}}N_{c_{b_{i}}}(x)=-\infty$,

  1. (c)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=+\infty$ and $\lim\limits_{x\to
b^{-}_{i}}N_{c_{b_{i}}}(x)=+\infty$,

  2. (d)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=+\infty$ and $\lim\limits_{x\to
b^{-}_{i}}N_{c_{b_{i}}}(x)=-\infty$.

See Figure 2.

Figure 2. Separatrix configurations in $[a_{i},b_{i}]\times\mathbb{R}$ when
$c_{a_{i}}\neq c_{b_{i}}$ in case $\lim\limits N_{c_{a_{i}}}(x)=+\infty$. The
blue curve is $\gamma_{c_{a_{i}}}$ and the red one is $\gamma_{c_{b_{i}}}$.
Completely analogous situations will happen when $\lim\limits
N_{c_{a_{i}}}(x)=-\infty$.

Now we have to add one leaf in each of the three canonical regions for each of
these four cases.

For the canonical regions having in their boundaries at least two inseparable
leaves, the only possibility is given by Lemma 3.4 and the fact that the
leaves are graphics of functions: see regions labeled as $\Gamma_{1}$,
$\Gamma_{3}$ and $\Gamma_{5}$ in Fig. 2. Each of the other canonical regions
in (a), (b), (c) and (d) can be of two types: or it has only one leaf in its
boundary, labeled as $\Gamma_{2}$ in Fig. 2, or it has exactly two leaves that
are not inseparable to each other in its boundary, labeled as $\Gamma_{4}$ in
Fig. 2. In both cases, a leaf in such a region, being the graphic of a
function defined in $(a_{i},a_{i})$, must follows the behavior of
$N_{c_{a_{i}}}$ and $N_{c_{b_{i}}}$. So the only possibility is as given in
each case of Fig. 2.

###  3.3. Configuration (III)

Let $c_{a_{i}}$ be the common value $p$ attains in $\nu_{a_{i}}$ and in
$\nu_{b_{i}}$, and let $\gamma_{c_{a_{i}}}$ be the graphic of $N_{c_{a_{i}}}$
in $(a_{i},b_{i})$. We assume $\lim\limits_{x\to
a^{+}_{i}}N_{c_{a_{i}}}(x)=+\infty$. We distinguish two cases:

  1. (a)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=+\infty$, which is equivalent to
$|\nu_{a_{i}},\gamma_{c_{a_{i}}},\nu_{b_{i}}|^{-}$,

  2. (b)

$\lim\limits_{x\to b^{-}_{i}}N_{c_{a_{i}}}(x)=-\infty$, which is equivalent to
$\nu_{a_{i}}|\gamma_{c_{a_{i}}}|\nu_{b_{i}}$.

(With the assumption $\lim\limits_{x\to a^{+}_{i}}N_{c_{a_{i}}}(x)=-\infty$,
the $+$ and $-$ in (a) and in (b) must be exchanged and the reasoning is
similar.)

In case (a), Lemma 3.4 and the fact that leaves are graphics of functions
provide the behavior of any leaf in the canonical region bounded by
$\nu_{a_{i}}$, $\gamma_{c_{a_{i}}}$ and $\nu_{b_{i}}$. The other canonical
region has only $\gamma_{c_{a_{i}}}$ as its boundary, so any leaf there has
its behavior as $\gamma_{c_{a_{i}}}$. See (a) of Fig. 3.

Figure 3. Separatrix configurations in the strip
$[a_{i},b_{i}]\times\mathbb{R}$ when $p$ attains the same value $c_{a_{i}}$ in
$\nu_{a_{i}}$ and in $\nu_{b_{i}}$, in case $\lim\limits_{x\to
a_{i}^{+}}N_{c_{a_{i}}}(x)=+\infty$. Completely similar situations occur when
$\lim\limits_{x\to a_{i}^{+}}N_{c_{a_{i}}}(x)=-\infty$.

In case (b), both canonical regions will be determined by Lemma 3.4. See (b)
in Fig. 3.

So the separatrix configuration of $\mathscr{F}(p)$ in each strip
$[a_{i},b_{i}]\times\mathbb{R}$ is completely characterized by the
“asymptotic” behavior of the separatrices in $(a_{i},b_{i})\times\mathbb{R}$,
i.e., in order to determine it we only need to know whether these
separatrices, which are graphics of functions $N_{c}(x)$ defined in
$(a_{i},b_{i})$, where $c$ is the value $p$ takes in $\nu_{a_{i}}$ and/or
$\nu_{b_{i}}$, are such that the limit when $x$ tends to $a_{i}$ and/or
$b_{i}$ is $+\infty$ or $-\infty$. So in some sense the separatrix
configuration in a strip region is determined by the configuration of the
separatrices in it: the behavior of the ordinary leaves in each canonical
region are determined by the behavior of the separatrices according to the
configurations (I), (II) and (III). A similar behavior may not happen for
functions other than linear-like ones; indeed, in Example 4.4 we present a
submersion function where there are ordinary leaves “misbehaving”, i.e. they
do not respect Lemma 3.4.

Therefore, in order to determine the separatrix configuration of
$\mathscr{F}(p)$ for a given linear-like submersion $p$, we first find the
pairwise disjoint open intervals $(a_{i},b_{i})$ such that
$\mathbb{R}\setminus Z_{p}=\cup_{i}(a_{i},b_{i})$. Then we analyze the
asymptotic behavior of the functions $N_{c}(x)$ in the intervals
$(a_{i},b_{i})$ where $c$ is the value that $r$ takes in $a_{i}$ and/or
$b_{i}$. Since $\nu_{a_{i}}$ and $\nu_{b_{i}}$ are vertical asymptotes of
$N_{c}(x)$, it follows that the limit of $N_{c}(x)$ when $x\to a_{i}^{+}$
(resp. $x\to b_{i}^{-}$) is $\pm\infty$ if and only if the signal of
$N_{c}(x)$ with $x>a_{i}$ close to $a_{i}$ (resp. $x<b_{i}$ close to $b_{i}$)
is $\pm$. According to configurations (I), (II) and (III) this will determine
the separatrix configuration in each strip $[a_{i},b_{i}]\times\mathbb{R}$.

Further, we consider the straight lines $\nu_{z}$, where
$z\in\overline{\cup_{i}\\{a_{i},b_{i}\\}}\setminus\cup_{i}\\{a_{i},b_{i}\\}$,
which will be the limit separatrices. Finally in each connected component of
the complement of $\overline{\cup_{i}(a_{i},b_{i})}$, we take an $x$. It
follows that each canonical region away from
$\cup_{i}(a_{i},b_{i})\times\mathbb{R}$ contains a straight line $\nu_{x}$.
Hence our separatrix configuration is complete.

We end this section by determining the bifurcation set of a finite linear-like
submersion $p$. In this case, by writing
$Z_{p}=\\{a_{1},a_{2},\ldots,a_{k}\\}$, $a_{i}<a_{i+1}$, $i=1,\ldots,k-1$, we
have that the intervals that cover $\mathbb{R}\setminus\mathbb{Z}_{p}$ as
above are of the form $(a_{i},a_{i+1})$, $i=0,\ldots,k$, where $a_{0}=-\infty$
and $a_{k+1}=\infty$.

######  Theorem 3.5.

Let $p(x,y)=r(x)+s(x)y$ be a finite linear-like submersion. Then
$B(p)=r(Z_{p})$.

###### Proof.

Let $Z_{p}=\\{a_{1},a_{2},\ldots,a_{k}\\}$ as above and $c\notin r(Z_{p})$. By
Lemma 3.1 it follows that $p^{-1}(c)$ has $k+1$ connected components
$\gamma_{1},\ldots,\gamma_{k+1}$ such that
$\gamma_{i}\subset(a_{i-1},a_{i})\times\mathbb{R}$, $i=1,\ldots k+1$, where
$a_{0}=-\infty$ and $a_{k+1}=\infty$. Moreover, by Proposition 3.2 it follows
that none of the $\gamma_{i}$, $i=1,\ldots,k+1$, is a separatrix of
$\mathscr{F}(p)$. So it follows from Theorem 2.4 that for each
$i=1,\ldots,k+1$, there exist two leaves
$\beta_{i}^{1},\beta_{i}^{2}\in\mathscr{F}(p)$ and $\varepsilon_{i}>0$ such
that the closed region $[\beta_{i}^{1},\beta_{i}^{2}]$ is mapped by a
homeomorphism $h_{i}$ onto the strip
$[c-\varepsilon_{i},c+\varepsilon_{i}]\times\mathbb{R}$ such that the vertical
lines correspond to the leaves of $[\beta_{i}^{1},\beta_{i}^{2}]$,
$i=1,\ldots,k+1$. We can assume further that
$h_{i}^{-1}(\\{d\\}\times\mathbb{R})$ is the connected component of
$p^{-1}(d)$ in $[\beta_{i}^{1},\beta_{i}^{2}]$. Let $\varepsilon=\min_{1\leq
i\leq k+1}\\{\varepsilon_{i}\\}$ and define
$h:F\times(c-\varepsilon,c+\varepsilon)\to
p^{-1}(c-\varepsilon,c+\varepsilon)$, where
$F=\\{1,2,\ldots,k+1\\}\times\mathbb{R}$, by $h((i,x),y)=h_{i}^{-1}(y,x)$.
This is a homeomorphism and $p\circ h((i,x),y)=y$, proving by definition that
$c\notin B(p)$, hence $B(p)\subset r(Z_{p})$.

Now let $c\in r(Z_{p})$. By Lemma 3.1, $p^{-1}(c)$ has at least $k+2$
connected component whereas $p^{-1}(d)$ for any $d$ sufficiently close to $c$
has $k+1$ connected components. Therefore $c\in B(p)$ and hence $r(Z)=B(p)$. ∎

Next corollary will use the terminology of M. Tibăr and A. Zaharia paper [16].

######  Corollary 3.6.

A finite linear-like submersion does not have the vanishing at infinity
phenomenon.

###### Proof.

Let $c_{0}\in\mathbb{R}$. If $c_{0}\notin B(p)$, it follows by Theorem 3.5 and
Lemma 3.1 that $p^{-1}(c)$ has the same number of connected components, each
of them given by the graphic of the function $N_{c}(x)$ in each interval
$(a_{i},a_{i+1})$. If $c\to c_{0}^{+}$ or $c\to c_{0}^{-}$, it follows that
each connected component of $p^{-1}(c)$ will tend to the corresponding
connected component of $p^{-1}(c_{0})$.

Now if $c_{0}\in B(p)$, it follows by our analysis in each configuration (I),
(II) and (III) that each connected component of $p^{-1}(c)$ will split into
two or three connected component of $p^{-1}(c_{0})$ (one of them being
$\nu_{a_{i}}$, for some $a_{i}\in Z_{p}$) or will tend to a connected
component of $p^{-1}(c_{0})$, when $c\to c_{0}^{+}$ or $c\to c_{0}^{-}$. So
there are no connected components of $p^{-1}(c)$ vanishing at infinity when
$c\to c_{0}^{\pm}$. ∎

##  4\. Examples

We gather in this section the examples of the paper. We provide details of the
just mentioned examples throughout the paper, as well as we present some other
linear-like submersions $p$ and its foliation $\mathscr{F}(p)$. We include
examples of linear-like submersions $p$ where the set $Z_{p}$ is not discrete.
In this case, it can appear limit separatrices.

We begin by detailing the examples of the introduction section.

######  Example 4.1.

Let $p$ and $q$ be the functions defined in (1). They are submersions because
the derivatives of $r_{1}(x)=x(7x-5)$ and $r_{2}(x)=2x(4x-5)$ are not zero in
the set $Z_{p}=Z_{q}=\\{-1,0,2\\}$, according to Lemma 3.1.

We calculate in details the separatrix configuration of $\mathscr{F}(p)$ and
let to the reader the calculations on $\mathscr{F}(q)$. According to Section 3
we have to study the asymptotic behavior of the functions $N_{c}(x)$, defined
in Lemma 3.1 for $p$, for $c=r_{1}(-1)=12$, $r_{1}(0)=0$ and $r_{1}(2)=18$, on
the domains $(-\infty,-1)\cup(-1,0)$, $(-1,0)\cup(0,2)$ and
$(0,2)\cup(2,\infty)$, respectively. In order to do that, we only need to know
the signals of $N_{12}(x)$ close to $x=-1$ and close to $0$ (with $x<0$); of
$N_{0}(x)$ close to $x=-1$ (with $x>-1$), close to $x=0$ and close to $x=2$
(with $x<2$); and of $N_{18}(x)$ close to $x=0$ (with $x>0$), and close to
$x=2$. We conclude that the behaviors are like the ones in (a) of Fig. 4.

(a) $S\mathfrak{S}_{\mathscr{F}(p)}$ (b) $S\mathfrak{S}_{\mathscr{F}(q)}$

Figure 4. Separatrix configurations of $\mathscr{F}(p)$ and $\mathscr{F}(q)$.
In red we depict the separatrices contained in the level set $L_{12}$, in blue
the ones contained in $L_{0}$ and in green the ones in $L_{18}$.

According to Section 3 the ordinary leaves in each canonical region are
automatically determined and so we get the separatrix configuration of
$\mathscr{F}(p)$. Analogously, the reader can conclude that the separatrix
configuration of $\mathscr{F}(q)$ is as presented in (b) of Fig. 4.

After a glance in $S\mathfrak{S}_{\mathscr{F}(p)}$ and
$S\mathfrak{S}_{\mathscr{F}(q)}$ in Fig. 4, we readily conclude that they are
isomorphic and so, by Theorem 2.3, $\mathscr{F}(p)$ and $\mathscr{F}(q)$ are
o-topologically equivalent.

We now prove that $p$ and $q$ cannot be topologically equivalent.

By Theorem 3.5, we have that
$B(p)=B(q)=r_{1}(Z_{p})=r_{2}(Z_{q})=\\{0,12,18\\}$. If $p$ and $q$ were
topologically equivalent, then by Corollary 1.2 there would exist an
equivalence homeomorphism $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ and a monotone
bijection $\sigma:B(p)\to B(q)$ such that
$p^{-1}(\sigma^{-1}(c))=h^{-1}(q^{-1}(c))$, $c=0,12,18$. Since $h$ carries
leaves inseparable to each other onto leaves inseparable to each other and
preserves chordal relations of triples, it follows that the set $\\{\nu_{-1}$,
$\nu_{0},\nu_{2}\\}$ must be carried onto itself, keeping $\nu_{0}$ fixed.
Since $\sigma$ is monotone, it then follows that $h(\nu_{-1})=\nu_{2}$,
$h(\nu_{2})=\nu_{-1}$, and so $\sigma$ is the identity.

Now, $h$ must take the two separatrices in the region bounded by $\nu_{-1}$
and $\nu_{0}$ in (a) of Fig. 4 onto the two separatrices contained in the
region bounded by $\nu_{0}$ and $\nu_{2}$ in (b) of Fig. 4, keeping the
colors. But then the cyclic triple formed by $\nu_{-1}$, $\nu_{0}$ and the
blue separatrix between them in (a) of Fig. 4 would be taken to the non-cyclic
triple $\nu_{0}$, $\nu_{2}$ and the blue separatrix between them in (b) of
Fig. 4, a contradiction because $h$ must preserve chordal relations.
Therefore, $p$ and $q$ cannot be topologically equivalent.

A final comment in this example is that for any $c\in\mathbb{R}$ the number of
connected components of $p^{-1}(c)$ and $q^{-1}(c)$ is the same: for
$c=0,12,18$, this number is $5$, and for any different $c$ this number is $4$.
In particular the Euler characteristics of the generic fiber of $p$ and $q$
are the same.

######  Example 4.2.

Let $p:\mathbb{R}^{2}\to\mathbb{R}$ be the function given by

$p(x,y)=x+x^{3}y.$

It is simple to see that this is a submersion. Acting as in Example 4.1, we
obtain the separatrices of $\mathscr{F}(p)$ and $\mathscr{F}(-p)$, that are
the same, by taking $\nu_{0}$ and the graphic of the function
$N_{0}(x)=-x/x^{3}=-1/x^{2}$, with $x\neq 0$. The ordinary leaves are
automatically given, according to Section 3. It is clear that $\mathscr{F}(p)$
and $\mathscr{F}(-p)$ are o-topologically equivalent.

On the other hand, we _claim that $p$ and $-p$ cannot be topologically
equivalent_. Indeed, the difference between the functions is exactly in their
ordinary leaves, given by connected components of the level sets $L_{c}$ with
$c\neq 0$: the level set $L_{c}$ of $p$ is the level set $L_{-c}$ of $-p$, see
Fig. 5.

(a) $S\mathfrak{S}_{\mathscr{F}(p)}$ (b) $S\mathfrak{S}_{\mathscr{F}(-p)}$

Figure 5. Separatrix configurations of $\mathscr{F}(p)$ and $\mathscr{F}(-p)$.
The black lines are the connected components of the level set $L_{0}$, the
inseparable leaves; the red ones are the connected component of a level set
$L_{t}$, $t<0$, and the blue ones are connected components of a level set
$L_{t}$ for $t>0$.

If there exist preserving orientation homeomorphisms
$\ell:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such
that $\ell\circ(-p)=p\circ h$ it follows in particular that $\ell(0)=0$
(because $h$ sends separatrices onto separatrices) and so $h$ must take
positive (resp. negative) level sets of $p$ onto positive (resp. negative)
ones of $-p$. But since $h$ keeps chordal relations of triples, it must send
so the cyclic triple formed by $\nu_{0}$, the connected components of $L_{0}$
to the left of $\nu_{0}$ and a red curve between them in (a) of Fig. 5 onto
the cyclic triple formed by $\nu_{0}$, the connected component of $L_{0}$ to
the right of $\nu_{0}$ and a red curve between them in (b) of Fig. 5. But
these cycles have reversed orientation. So $p$ and $-p$ cannot be
o-topologically equivalent.

The following example shows that the “monotone” assumption cannot be
strengthened in corollaries 1.2, 1.4 and 5.3.

######  Example 4.3.

Let

$p(x,y)=x(3-2x)+(x-1)^{2}x^{2}y,\ \ \ \ q(x,y)=(x-1)(2x-1)-(x-1)^{2}x^{2}y.$

As above, these linear-like functions are submersions. Observe that

$\ell\circ p=q,$

with $\ell(t)=1-t$, hence $p$ and $q$ are topologically equivalent.

We have $Z_{p}=Z_{q}=\\{0,1\\}$, and $p(\nu_{0})=q(\nu_{1})=0$ and
$p(\nu_{1})=q(\nu_{0})=1$. For the function $p$, by analyzing the signal of
$N_{0}(x)$ near $0$ and in the interval $(0,1)$, as well as the signal of
$N_{1}(x)$ in the interval $(0,1)$ and near $1$, we conclude, according to
Section 3, that the separatrix configuration of $\mathscr{F}(p)$ is as shown
in (a) of Fig. 6.

(a) $S\mathfrak{S}_{\mathscr{F}(p)}$ (b) $S\mathfrak{S}_{\mathscr{F}(q)}$

Figure 6. Separatrix configurations of $\mathscr{F}(p)$ and of
$\mathscr{F}(q)$. In red there are connected components of level $0$ of $p$
and $q$. In blue, of level $1$.

Analogously, for the function $q$, after analyzing the signal of $N_{1}(x)$
near $0$ and in the interval $(0,1)$, and also the signal of $N_{0}(x)$ in the
interval $(0,1)$ and near $1$, we obtain the separatrix configuration of
$\mathscr{F}(q)$ as depicted in (b) of Fig. 6.

We claim that there are no _increasing_ bijection $\sigma:B(p)\to B(q)$ and no
isomorphism nor anti-isomorphism $g$ between $S\mathfrak{S}_{\mathscr{F}(p)}$
and $S\mathfrak{S}_{\mathscr{F}(q)}$ such that
$\sigma(p(\nu_{i}))=q(g(\nu_{i}))$, for $i=0,1$. So, in particular, condition
(b) of corollaries 1.2, 1.4 and 5.3 will not be satisfied with an increasing
$\sigma$.

Indeed, if $\sigma$ is increasing, then $\sigma(0)=0$ and $\sigma(1)=1$. In
particular, an existing $g$ should send $\nu_{0}$ onto $\nu_{1}$, because it
must preserve separability of leaves). Also, it should send $\alpha$, the
inseparable leaf to $\nu_{0}$ contained in $(\nu_{0},\nu_{1})$, onto the
inseparable leaf (related to $\mathscr{F}(q)$) to $\nu_{1}$ in
$(\nu_{0},\nu_{1})$, because it should also preserve inseparable leaves.
Analogously, it should send $\beta$, the inseparable leaf to $\nu_{1}$
contained in $(\nu_{0},\nu_{1})$ onto the inseparable leaf to $\nu_{0}$ in
$(\nu_{0},\nu_{1})$. But $|\nu_{0},\alpha,\beta|^{+}$ and
$g(\nu_{0})|g(\alpha)|g(\beta)$, hence $g$ cannot be an isomorphism or anti-
isomorphism.

Next example proves that away from the linear-like class only the behaviour of
the separatrices does not characterize the foliation.

######  Example 4.4.

Let

$f(x,y)=\left(x-\arctan y+\pi\right)(2x+\pi)\left(x-\arctan y\right)e^{2y}.$

First we show that this is a submersion. For each $y$, the partial derivatives
$f_{x}(x,y)$ and $f_{y}(x,y)$ are polynomials of degrees $2$ and $3$ in $x$,
respectively. The second one can be written as $(2x+\pi)g(x,y)$, where
$g(x,y)$ is a polynomial of degree $2$ in $x$ for each $y$. We have
$f_{x}(-\pi/2,y)\neq 0$ for all $y$. Further, the resultant, in $x$, between
$f_{x}(x,y)$ and $g(x,y)$ is given, up to a factor of a positive function of
$y$, and after substituting $y=\tan z$, by

$\pi^{2}+\cos^{4}z-4(z+\cos^{2}z)^{2},$

which is a strictly positive function in the interval $(-\pi/2,\pi/2)$, and so
there does not exists $y\in\mathbb{R}$ such that $f_{x}(x,y)$ and $g(x,y)$
have a common zero in $x$. Hence $f$ is a submersion.

Clearly the straight line $\nu_{-\pi/2}$ and the curves
$\gamma_{1}=\\{(\arctan y-\pi,y)\ |\ y\in\mathbb{R}\\}$ and
$\gamma_{2}=\\{(\arctan y,y)\ |\ y\in\mathbb{R}\\}$ correspond to the three
connected components of the level set $L_{0}$. Moreover, it is easy to see
that $f$ is negative in the region to the left of $\gamma_{1}$ and in the
region $(\nu_{-\pi/2},\gamma_{2})$, and it is positive in the other two
connected components of $\mathbb{R}^{2}\setminus L_{0}$. Further, $f$ assumes
all the negative values in the former two regions as well it assumes all the
positive values in the last two ones. Now for any given $c\in\mathbb{R}$,
$c\neq 0$, we observe that for fixed $y$ the discriminant of the polynomial of
degree $3$ $x\mapsto f(x,y)-c$ is positive (resp. negative) for $y>0$ (resp.
$y<0$) with $|y|$ big enough. It follows that for $y>0$ (resp. $y<0$) with
$|y|$ big enough, the equation $f(x,y)=c$ has three (resp. one) real
solutions. So, following the ideas of the proof of [2, Lemma 3.5.] (with the
signal of the discriminant being the opposite of $D$), we get that for any
$c\neq 0$, the level set $f^{-1}(c)$ has two connected components. In
particular, in the regions $(\gamma_{1},\nu_{-\pi/2})$ and
$(\nu_{-\pi/2},\gamma_{2})$ there is exactly one connected components of
$L_{c}$ for each $c>0$ and for each $c<0$, respectively. Hence acting
analogously as in the proof of Proposition 3.2, it follows that $\gamma_{1}$
is inseparable to $\nu_{-\pi/2}$, as well as $\nu_{-\pi/2}$ is inseparable to
$\gamma_{2}$.

Since, as observed right above, for any $c\neq 0$ the equation $f(x,y)=c$ for
fixed $y$ has only one real solution if $y$ is negative enough, it follows
that the ordinary leaves in $(\gamma_{1},\nu_{-1/2})$ and in
$(\nu_{-1/2},\gamma_{2})$ have its ends in the region $y>0$. Hence the
separatrix configuration of $\mathscr{F}(f)$ with level sets is as shown in
Fig. 7.

Figure 7. Separatrix configuration of $\mathscr{F}(f)$. The inseparable leaves
are drawn in black. In blue (resp. red), we draw ordinary leaves where $f$
assumes positive (resp. negative) values.

In particular, in the region $(\gamma_{1},\nu_{-1/2})$, the behavior of the
ordinary leaves are contrary to what it should be if we were in the linear-
like case, because it does not respect Lemma 3.4.

Next example justifies why we ask that both $\ell$ and $h$ preserve
orientation in our definition of o-topological equivalence of functions.

######  Example 4.5.

Let $p$ and $q$ be defined by

$p(x,y)=x+(x+1)^{2}x^{2}(x-1)^{2}y,\ \ \ \ q(x,y)=p(-x,y).$

Following the ideas of Section 3 as above, it is simple to see that $p$ and
$q$ are submersions such that the separatrix configurations of
${\mathscr{F}(p)}$ and of ${\mathscr{F}(q)}$ with level sets are as given in
Fig. 8.

(a) $S\mathfrak{S}_{\mathscr{F}(p)}$ (b) $S\mathfrak{S}_{\mathscr{F}(q)}$

Figure 8. Separatrix configurations of $\mathscr{F}(p)$ and of
$\mathscr{F}(q)$. Inseparable leaves where the functions assume the value $0$
are painted in black. In blue the inseparable leaves where the functions
assume $1$ and in red, the ones where the functions assume $-1$.

By taking $\ell(t)=t$ and $h(x,y)=(-x,y)$, we have

$\ell\circ q=p\circ h.$

Although, it is not difficult to verify that we can not define an isomorphism
between $\mathfrak{S}_{\mathscr{F}(p)}$ and $\mathfrak{S}_{\mathscr{F}(q)}$:
because if $g$ is such an isomorphism, we must have $g(\nu_{0})=\nu_{0}$, also
$g(\nu_{-1})=\nu_{-1}$ or $g(\nu_{-1})=\nu_{1}$; but for any of these two
possibilities, it follows that cycles will have their orientation reverted.
Therefore, the foliations $\mathscr{F}(p)$ and $\mathscr{F}(q)$ are not
o-topologically equivalent.

Since in our paper we are relating topological equivalence of functions with
topological equivalence of the foliations given by them, it would be awkward
to have $p$ and $q$ o-topologically equivalent together with $\mathscr{F}(p)$
and $\mathscr{F}(q)$ _not_ o-topologically equivalent. It is because of this
that we ask that both $\ell$ and $h$ must preserve orientation in our
definition.

Next example shows that all the possibilities presenting in configurations
(I), (II) and (III) can actually happen.

######  Example 4.6.

For the cases in configuration (I), all the examples above provide explicit
linear-like submersions.

For cases (a) and (d) of configuration (II), the third and the second strips
in (a) of Example 4.1 given by $p$ provide explicit submersions. For case (c),
let again $p$ of this example and define $q(x,y)=p(-x,y)$. The second strip of
$\mathscr{F}(q)$ will provide an example. Finally, for case (b), let

$p(x,y)=x(2-x^{2})+(x-1)^{2}(x+1)^{2}y.$

As above, it is simple to see that $p$ is a submersion such that the strip
$[-1,1]\times\mathbb{R}$ has the same separatrix configuration as given in (b)
of configuration (II).

For (a) and (b) of configuration (III), we consider the polynomials

$\displaystyle p(x,y)$ $\displaystyle=x(x-1)+x^{2}(x-1)^{2}y,$ $\displaystyle
q(x,y)$ $\displaystyle=x(1-2x)(x-1)+x^{2}(x-1)^{2}y,$

respectively. As above, it is simple to see that they are submersions and that
their separatrix configuration in $[0,1]\times\mathbb{R}$ agree with the ones
in (a) and (b) of configuration(III), respectively.

The following example exhibits linear-like submersions $p$ such that the set
$Z_{p}$ is not discrete.

######  Example 4.7.

Let $u:[0,\infty)\to\mathbb{R}$ be the $C^{\infty}$ function defined by
$u(0)=0$ and

$u(x)=e^{-\frac{1}{x}}\sin^{2}\frac{\pi}{x},\ x\neq 0,$

and let $s_{i}:\mathbb{R}\to\mathbb{R}$, $i=1,2,3$, be the $C^{\infty}$
functions given by

$s_{1}(x)=\left\\{\begin{array}[]{ll}u(x),&x\geq 0,\\\
e^{\frac{1}{x}},&x<0,\end{array}\right.\ \ \ \
s_{2}(x)=\left\\{\begin{array}[]{ll}u(x),&x\geq 0,\\\
u(-x),&x<0,\end{array}\right.\ \ \ \
s_{3}(x)=\left\\{\begin{array}[]{ll}u(x),&x\geq 0,\\\
0,&x<0.\end{array}\right.$

We define

$f_{i}(x,y)=x+s_{i}(x)y,\ i=1,2,3.$

Denoting $\mathbb{Z}^{*}=\mathbb{Z}\setminus\\{0\\}$ and
$\mathbb{Z}^{+}=\mathbb{Z}\cap\\{x\ |\ x>0\\}$, we have

$Z_{f_{1}}=\left\\{\frac{1}{n}\ |\ n\in\mathbb{Z}^{+}\right\\}\cup\\{0\\},\ \
\ \ Z_{f_{2}}=\left\\{\frac{1}{n}\ |\ n\in\mathbb{Z}^{*}\right\\}\cup\\{0\\},\
\ \ \ Z_{f_{3}}=Z_{f_{1}}\cup\\{x\ |\ x\leq 0\\}.$

In what follows we denote $1/0=\infty$. By Lemma 3.1 it follows that $f_{i}$,
$i=1,2,3$, are submersions. Moreover, by Proposition 3.2, the straight lines
$\nu_{1/n}$, $n\in Z^{+}$, together with the connected components of $L_{1/n}$
contained in $(1/(n+1),1/n)\cup(1/n,1/(n-1))$, $n\in Z^{+}$, are inseparable
leaves of $\mathscr{F}(f_{i})$, $i=1,2,3$. In the case of $f_{2}$, all the
straight lines $\nu_{-1/n}$, $n\in\mathbb{Z}^{+}$ are also inseparable leaves
with the connected components of $L_{-1/n}$ contained in
$(-1/(n-1),-1/n)\cup(-1/n,-1/(n+1))$, $n\in Z^{+}$.

By applying Corollary 3.3, it follows that $\nu_{0}$ is a limit separatrix in
cases of $f_{2}$ and $f_{3}$. In case of $f_{1}$, $\nu_{0}$ is inseparable
with a connected component of $L_{0}$ contained in $(-\infty,0)$. In the case
of $f_{2}$ there are inseparable leaves converging to $\nu_{0}$ from both
“sides” of it. In the cases of $f_{1}$ and $f_{3}$, there are inseparable
leaves converging from just one side. Further, in case of $f_{3}$, we observe
that the foliation is trivial in the region $x<0$: the leaves are the straight
lines $\nu_{x}$, $x<0$. Finally, we observe that in this last case, the limit-
separatrix $\nu_{0}$ is the only leaf in the boundary of a canonical region,
this does not happen in the other cases.

To complete the separatrix configurations of $\mathscr{F}(f_{i})$, $i=1,2,3$,
we apply the results of Section 3, obtaining that they are as depicted in Fig.
9.

(a) $S\mathfrak{S}_{\mathscr{F}(f_{1})}$ (b)
$S\mathfrak{S}_{\mathscr{F}(f_{2})}$

(c) $S\mathfrak{S}_{\mathscr{F}(f_{3})}$

Figure 9. Separatrix configurations of the foliations $\mathscr{F}(f_{i})$,
$i=1,2,3$. Inseparable leaves are painted with the same color. In (a) there is
exactly one curve inseparable with the straight line $\nu_{0}$ in the region
$x<0$. In (b) and (c), $\nu_{0}$ is a limit separatrix. In (c), the region
$x<0$ is trivial.

The last example provides a linear like submersion $f$ for any given closed
subset $Z\subset\mathbb{R}$, such that $Z_{f}$ matches $Z$.

######  Example 4.8.

Given $Z$ a closed set in $\mathbb{R}$, we know that $\mathbb{R}\setminus Z$
is the disjoint union of an enumerable quantity of disjoint open intervals of
the form $(a_{n},b_{n})$, where one of the $a_{n}$ might be $-\infty$ and one
of the $b_{n}$ might be $\infty$. In case $(a_{n},b_{n})$ is bounded, we
define the $C^{\infty}$ function
$s_{n}(x)=e^{1/\big{(}(x-a_{n})(x-b_{n})\big{)}}$ for $x\in(a_{n},b_{n})$ and
$s_{n}(x)=0$ for $x$ away from $(a_{n},b_{n})$. For an interval of the form
$(-\infty,b_{n})$ we put $s_{n}(x)=e^{1/(x-b_{n})}$ for $x\in(-\infty,b_{n})$
and $s_{n}(x)=0$ when $x$ is not in $(-\infty,b_{n})$. Analogously, when
$b_{n}=\infty$, we put $s_{n}(s)=e^{1/(a_{n}-x)}$ in $(a_{n},b_{n})$ and $0$
away from this interval. Then we consider $s(x)=\sum_{n=0}^{\infty}s_{n}(x)$.
This is a $C^{\infty}$ function with zero set being exactly $Z$.

By defining $f(x,y)=-x+s(x)y$ we will get a $C^{\infty}$ submersion by Lemma
3.1, because $s$ is flat in each $a_{n}$ and $b_{n}$. In each strip
$[a_{n},b_{n}]\times\mathbb{R}$ the foliation $\mathscr{F}(f)$ has separatrix
configuration as (d) of configuration (III) if $[a_{n},b_{n}]$ is bounded and
as (a) (with $+$ and $-$ exchanged) and (b) of configuration (I) if
$a_{n}=-\infty$ or $b_{n}=\infty$, respectively.

By Corollary 3.3 it follows that for any
$z\in\overline{\cup_{i}\\{a_{i},b_{i}\\}}\setminus\cup_{i}\\{a_{i},b_{i}\\}$,
the straight line $\nu_{z}$ is a limit separatrix of $\mathscr{F}(f)$. Then we
choose an $x$ in each connected component of the complement of
$\overline{\cup_{i}(a_{i},b_{i})}\setminus\cup_{i}(a_{i},b_{i})$, and the
curve $\nu_{x}$ will be an ordinary leaf completing the separatrix
configuration of $\mathscr{F}(f)$.

##  5\. Relation between topological equivalence of finite linear-like
submersions and topological equivalence of the foliations given by them

As discussed in the introduction section, the topological (resp.
o-topological) equivalence of submersion functions implies the topological
(resp. o-topological) equivalence of the foliations given by them. The
reciprocal result is not true even in the finite linear-like case, as examples
in Section 4 show. In this section, we add conditions in order to obtain a
reciprocal result in the finite linear-like class. The most general results we
obtain is Theorem 5.2 and its corollary. The results announced in the
introduction section will be consequences of them, and are proved below.

We need the following general lemma on regular foliations:

######  Lemma 5.1.

Let $\mathcal{A}$ be a canonical region of a regular foliation $\mathscr{F}$.
The following properties hold for leaves
$\gamma_{1},\gamma_{2},\alpha,\beta\in\mathscr{F}$ such that
$\gamma_{1},\gamma_{2}$ are contained in $\mathcal{A}$ and $\alpha,\beta$ are
not contained in $\mathcal{A}$:

  1. (a)

$\gamma_{1}|\gamma_{2}|\alpha$ or $\gamma_{2}|\gamma_{1}|\alpha$.

  2. (b)

The chordal relations of the triples $\gamma_{1}$, $\alpha$, $\beta$ and
$\gamma_{2}$, $\alpha$, $\beta$ are the same.

###### Proof.

If (a) is not true, then $|\gamma_{1},\gamma_{2},\alpha|^{\pm}$ and hence, in
particular, $\alpha$ is contained in $(\gamma_{1},\gamma_{2})$, which is an
open set contained in $\mathcal{A}$, a contradiction.

Now we prove (b). By using Section 2.3 of Kaplan’s paper [5], it is possible
to deliver a very technical proof of this statement by applying the axioms and
results on abstract chordal systems. We prefer to give a more self-contained
proof, by using only our definitions in Section 2.1. By (a) we can assume

(3) $\gamma_{1}|\gamma_{2}|\alpha.$

We have the following possibilities for $\gamma_{1}$, $\alpha$ e $\beta$: (i)
$\gamma_{1}|\alpha|\beta$, (ii) $\gamma_{1}|\beta|\alpha$, (iii)
$\alpha|\gamma_{1}|\beta$ and (iv) $|\gamma_{1},\alpha,\beta|^{\pm}$. Clearly
(3) and (i) is equivalent to (3) and $\gamma_{2}|\alpha|\beta$. In case (ii),
since $\beta$ cannot be contained in $[\gamma_{1},\gamma_{2}]$ it follows that
$\beta$ is contained in $(\gamma_{2},\alpha)$. If $\beta$ does not separate
$\gamma_{2}$ and $\alpha$, we let $B$ be the open connected set whose boundary
is $\gamma_{2}\cup\beta\cup\alpha$. The open connected region given by
$(\gamma_{1},\gamma_{2}]\cup B$ have as boundary
$\gamma_{1}\cup\beta\cup\alpha$, a contradiction with the assumption
$\gamma_{1}|\beta|\alpha$ of this case. In case (iii), since nor $\alpha$ nor
$\beta$ is contained in $[\gamma_{1},\gamma_{2}]$, it follows that $\alpha$
and $\beta$ are in distinct connected components of
$[\gamma_{1},\gamma_{2}]^{c}$, hence in particular they are in distinct
connected components of $\gamma_{2}^{c}$, i.e. $\alpha|\gamma_{2}|\beta$.
Finally, in case (iv), since nor $\alpha$ nor $\beta$ is contained in
$[\gamma_{1},\gamma_{2}]$, it follows by (3) that both $\alpha$ and $\beta$
are contained in the connected component of $\gamma_{2}^{c}$ not containing
$\gamma_{1}$. In particular $\alpha$ does not separate $\gamma_{2}$ and
$\beta$ as well as $\gamma_{2}$ does not separate $\alpha$ and $\beta$. If
$\beta$ separates $\gamma_{2}$ and $\alpha$, it follows in particular that the
region $(\gamma_{1},\gamma_{2}]\cup[\gamma_{2},\beta)$ is an open connected
region not containing $\alpha$, a contradiction with our assumptions in this
case. So $\gamma_{2},\alpha,\beta$ form a cyclic triple. If
$|\gamma_{1},\alpha,\beta|^{+}$, there exist $p_{1}\in\gamma_{1}$,
$p_{2}\in\alpha$ and $p_{3}\in\beta$ such that $p_{1}p_{2}p_{3}p_{1}$ is a
positive Jordan curve as in the definition of $|\gamma_{1},\alpha,\beta|^{+}$.
In particular, this curve cuts the leaf $\gamma_{2}$ twice in distinct points
$q_{1},q_{2}$. By sliding $q_{2}$ over $\gamma_{2}$ until it meets $q_{1}$ we
obtain a positively oriented Jordan curve $q_{1}p_{2}p_{3}q_{1}$ with the
properties guaranteeing that $|\gamma_{2},\alpha,\beta|^{+}$. Similarly
$|\gamma_{1},\alpha,\beta|^{-}$ implies $|\gamma_{2},\alpha,\beta|^{-}$. ∎

######  Theorem 5.2.

Let $p_{i}(x,y)=r_{i}(x)+s_{i}(x)y$, $i=1,2$, be two finite linear-like
submersions. Then $p_{1}$ is o-topologically equivalent to $p_{2}$ if and only
if $Z_{p_{1}}=Z_{p_{2}}=\emptyset$ or each of the following conditions hold:

  1. (a)

$\mathscr{F}(p_{1})$ and $\mathscr{F}(p_{2})$ are o-topologically equivalent.

  2. (b)

There exists an increasing bijection $\sigma:r_{1}(Z_{p_{1}})\to
r_{2}(Z_{p_{2}})$ such that

$\sigma(p_{1}(\nu_{a}))=p_{2}(h(\nu_{a}))$

for each $a\in Z_{p_{1}}$, where $h$ is an o-equivalence homeomorphism between
$\mathscr{F}(p_{1})$ and $\mathscr{F}(p_{2})$. If $r_{1}(Z_{p_{1}})$ is a
singleton, there further exist $m\notin r_{1}(Z_{p_{1}})$, an extension of
$\sigma$ to an increasing bijection
$\overline{\sigma}:r_{1}(Z_{p_{1}})\cup\\{m\\}\to
r_{2}(Z_{p_{2}})\cup\\{\overline{\sigma}(m)\\}$ and a connected component
$\xi$ of $p_{1}^{-1}(m)$ such that $\overline{\sigma}(m)=p_{2}(h(\xi))$.

###### Proof.

(Necessity) Assuming the existence of preserving orientation homeomorphisms
$\ell:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$ such
that $\ell\circ p_{1}=p_{2}\circ h$, it follows that $h$ is an o-equivalence
homeomorphism between $\mathscr{F}(p_{1})$ and $\mathscr{F}(p_{2})$, proving
(a). Moreover, by Lemma 2.6 and Theorem 3.5, it follows that
$\sigma=\ell|_{r_{1}(Z_{p_{1}})}$ (resp.
$\overline{\sigma}=\ell|_{r_{1}(Z_{p_{1}})\cup\\{m\\}}$ for any
$m\notin{r_{1}(Z_{p_{1}})}$) defines an increasing bijection between
$r_{1}\left(Z_{p_{1}}\right)$ and $r_{2}\left(Z_{p_{2}}\right)$ (resp. between
$r_{1}\left(Z_{p_{1}}\right)\cup\\{m\\}$ and
$r_{2}\left(Z_{p_{2}}\right)\cup\\{\overline{\sigma}(m)\\}$), and so the
formula of (b) follows from the identity $\ell\circ p_{1}=p_{2}\circ h$.

(Sufficiency) If $Z_{p_{1}}=Z_{p_{2}}=\emptyset$, we let $\tau_{i}$ be the
constant signal of $s_{i}(x)$, and $h_{i}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be
the orientation preserving homeomorphism defined by
$h_{i}(x,y)=\big{(}x,\tau_{i}r_{i}(x)+\tau_{i}s_{i}(x)y\big{)}$, with inverse
$h_{i}^{-1}(x,y)=\big{(}x,(y-\tau_{i}r_{i}(x))/(\tau_{i}s_{i}(x))\big{)}$, and
observe that $p_{i}=\tau_{i}q\circ h_{i}$, $i=1,2$, where $q$ is the
projection $q(x,y)=y$. Hence $p_{i}$ is o-topologically equivalent to the
function $\tau_{i}q$, $i=1,2$. As $q$ and $-q$ are o-topologically equivalent
it follows that $p_{1}$ and $p_{2}$ are o-topologically equivalent. (We can as
well invoke Sharko and Soroka [14] already mentioned result to guarantee that
$p_{1}$ and $p_{2}$ are both o-topologically equivalent to a projection.)

So we assume $Z_{p_{i}}\neq\emptyset$, $i=1,2$ (we can not have just one of
them empty, because of condition (b) and Theorem 3.5) and write

$Z_{p_{i}}=\\{a_{1}^{i},a_{2}^{i},\ldots,a_{k_{i}}^{i}\\},$

with $a_{j}^{i}<a_{j+1}^{i}$, $j=1,\ldots,k_{i}-1$, $i=1,2$. We first notice
that being an equivalence homeomorphism between $\mathscr{F}(p_{1})$ and
$\mathscr{F}(p_{2})$, $h$ must carry the region
$(-\infty,a_{1}^{1}]\times\mathbb{R}$ onto
$(-\infty,a_{1}^{2}]\times\mathbb{R}$ or onto
$[a_{k_{2}}^{2},\infty)\times\mathbb{R}$, in the light of configurations (I),
(II) and (III) of Section 3. So up to considering the function
$\widetilde{p}_{2}(x,y)=p_{2}(-x,-y)$, which is o-topologically equivalent to
$p_{2}$, in the place of $p_{2}$, we can assume that $h$ carries
$(-\infty,a_{1}^{1}]\times\mathbb{R}$ onto
$(-\infty,a_{1}^{2}]\times\mathbb{R}$. In particular
$h(\nu_{a_{1}^{1}})=\nu_{a_{1}^{2}}$. In order to organize the remaining of
the proof, we divide it in 12 steps.

_Step 1:_ _We claim that $k_{1}=k_{2}$, i.e., $Z_{p_{1}}$ and $Z_{p_{2}}$ have
the same cardinality, and that $h(\nu_{a_{j}^{1}})=\nu_{a_{j}^{2}}$,
$j=1,\ldots,k$, with $k=k_{1}=k_{2}$. In particular, $h$ sends the region
$(\nu_{a_{j}^{1}},\nu_{a_{j+1}^{1}})$ onto
$(\nu_{a_{j}^{2}},\nu_{a_{j+1}^{2}})$, $j=0,\ldots,k$, where
$a_{0}^{i}=-\infty$ and $a_{k}^{i}=\infty$, $i=1,2$_.

If $k_{1}=1$, then the open region
$(a_{1}^{2},\infty)\times\mathbb{R}=h\left((a_{1}^{1},\infty)\times\mathbb{R}\right)$
has exactly one separatrix, and so $k_{2}=k_{1}=1$, proving Step 1 in this
case. So we assume that $k_{1},k_{2}>1$ and that
$h(\nu_{a_{l}^{1}})=\nu_{a_{l}^{2}}$ for $l=1,\ldots,j<k_{1}$. It is enough to
prove that $h(\nu_{a_{j+1}^{1}})=\nu_{a_{j+1}^{2}}$. Indeed, the closed region
$[\nu_{a_{j}^{1}},\nu_{a_{j+1}^{1}}]$ is mapped by $h$ onto the closed region
$[\nu_{a_{j}^{2}},h(\nu_{a_{j+1}^{1}})]$, contained in $\\{x\ |\ x\geq
a_{j}^{2}\\}\times\mathbb{R}$. According to the configurations (III) and (II)
studied in Section 3, besides $\nu_{a_{j}^{1}}$ and $\nu_{a_{j+1}^{1}}$, the
region $[\nu_{a_{j}^{1}},\nu_{a_{j+1}^{1}}]$ contains either one or two more
separatrices of $\mathscr{F}(p_{1})$ depending whether $\nu_{a_{j}^{1}}$ and
$\nu_{a_{j+1}^{i}}$ are or not in the same level set of $p_{1}$, respectively.
In the first case, being $\alpha$ the existing extra separatrix, it follows by
the configuration (III) that $\alpha$, $\nu_{a_{j}^{1}}$ and
$\nu_{a_{j+1}^{1}}$ are in the same level set of $p_{1}$. So by assumption (b)
it follows that $\nu_{a_{j}^{2}}$, $h(\alpha)$ and $h(\nu_{a_{j+1}^{1}})$
contained in $[\nu_{a_{j}^{2}},h(\nu_{a_{j+1}^{1}})]$ are in the same level
set of $p_{2}$. Since these three leaves are the only separatrices of
$\mathscr{F}(p_{2})$ contained in this set, we can conclude from a glance in
configurations (I), (II) and (III) that we must have
$h(\nu_{a_{j+1}^{1}})=\nu_{a_{j+1}^{2}}$. Analogously, assuming the second
case, we have from configuration (II) that one of the separatrices in
$(\nu_{a_{j}^{1}},\nu_{a_{j+1}^{1}})$, say $\alpha$, is such that
$p_{1}(\nu_{a_{j}^{1}})=p_{1}(\alpha)$, and the other, say $\beta$, satisfies
$p_{1}(\beta)=p_{1}(\nu_{a_{j+1}^{1}})$, with $p_{1}(\alpha)\neq
p_{1}(\beta)$. Hence by assumption (b), it follows that

(4) $p_{2}(\nu_{a_{j}^{2}})=p_{2}\left(h(\alpha)\right)\neq
p_{2}\left(h(\beta)\right)=p_{2}(h(\nu_{a_{j+1}^{1}})).$

Since these four separatrices of $\mathscr{F}(p_{2})$ are the only ones in the
region $[\nu_{a_{j}^{2}},h(\nu_{a_{j+1}^{1}})]$, and they satisfy (4), it
follows again by a glance in the configurations (I), (II) and (III) that we
must have $h(\nu_{a_{j+1}^{1}})=\nu_{a_{j+1}^{2}}$. So Step 1 is proved.

_Step 2:_ _Assumption(b) is also valid throughout
$\mathfrak{S}_{\mathscr{F}(p_{1})}$, i.e.,
$\sigma\left(p_{1}(\gamma)\right)=p_{2}\left(h(\gamma)\right)$ for any
separatrix $\gamma$ of $\mathscr{F}(p_{1})$_. Indeed, recall that two leaves
of $\mathscr{F}(p_{i})$ which are inseparable to each other are in the same
level set of $p_{i}$. Further, from Proposition 3.2, any separatrix of
$\mathscr{F}(p_{i})$ is inseparable to a separatrix of the form
$\nu_{a_{j}^{i}}$. Therefore, since $h$ sends pairs of inseparabel leaves of
$\mathscr{F}(p_{1})$ onto pairs of inseparable leaves of $\mathscr{F}(p_{2})$,
Step 2 follows by assumption (b).

_Step 3:_ We define $\ell:\mathbb{R}\to\mathbb{R}$ to be any homeomorphism
such that $\ell|_{r_{1}(Z_{p_{1}})}=\sigma$ or
$\ell|_{r_{1}(Z_{p_{1}})\cup\\{m\\}}=\overline{\sigma}$, in case
$r_{1}(Z_{p_{1}})$ is a singleton. Since $\sigma$ (or $\overline{\sigma}$) is
increasing, it follows that $\ell$ is orientation-preserving.

_Step 4:_ In order to finish the proof, it is enough to construct an
isomorphism $g:\mathscr{F}(p_{1})\to\mathscr{F}(p_{2})$ such that
$\ell\left(p_{1}(\gamma)\right)=p_{2}\left(g(\gamma)\right)$ for each
$\gamma\in\mathscr{F}(p_{1})$. Because with this isomorphism in hands it
follows by Theorem 2.1 that there exists an o-equivalence homeomorphism
$H:\mathbb{R}^{2}\to\mathbb{R}^{2}$ between ${\mathscr{F}(p_{1})}$ and
${\mathscr{F}(p_{2})}$ such that $H(\gamma)=g(\gamma)$ for each
$\gamma\in\mathscr{F}(p_{1})$. So for any $x\in\mathbb{R}^{2}$, by denoting
$\gamma_{x}$ the leaf of $\mathscr{F}(p_{1})$ containing $x$, it follows that

$\ell(p_{1}(x))=\ell(p_{1}(\gamma_{x}))=p_{2}(g(\gamma_{x}))=p_{2}(H(\gamma_{x}))=p_{2}(H(x)),$

hence $p_{1}$ and $p_{2}$ are o-topologically equivalent.

In the following steps, we will construct the isomorphism $g$.

_Step 5:_ We first define

$g(\gamma)=h(\gamma)$

for any $\gamma$ in the set of separatrices of $\mathscr{F}(p_{1})$, and also
for $\gamma=\xi$ when $r_{1}\left(Z_{p_{1}}\right)$ is a singleton. By Step 2,
it follows that $\ell(p_{1}(\gamma))=p_{2}(g(\gamma))$.

_Step 6:_ Here for each $j=0,\ldots,k$, we define a bijection $g_{j}$ from the
set of leaves $\mathscr{F}(p_{1})$ foliating the strip region
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ onto the set of leaves of
$\mathscr{F}(p_{2})$ foliating $(a_{j}^{2},a_{j+1}^{2})\times\mathbb{R}$ (see
Step 1) as follows:

We let $\alpha^{1}(t)$ be a transversal curve in
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ parametrized by the level sets of
$p_{1}$, i.e., such that $p_{1}(\alpha^{1}(t))=t$ (for instance, for any
$x_{0}\in(a_{j}^{1},a_{j+1}^{1})$, we can take
$\alpha^{1}(t)=\big{(}x_{0},N_{t}(x_{0})\big{)}$, where $N_{t}(x)$ is defined
in Lemma 3.1 for function $p_{1}$). We define $\alpha^{2}(t)$ having the same
meaning in $(a_{j}^{2},a_{j+1}^{2})\times\mathbb{R}$ for function $p_{2}$.

Each leaf $\gamma^{i}$ of $\mathscr{F}(p_{i})$ foliating
$(a_{j}^{j},a_{j+1}^{i})\times\mathbb{R}$ can be uniquely denoted by
$\gamma^{i}_{t}$ meaning the only leaf of $\mathscr{F}(p_{i})$ crossing the
transversal $\alpha^{i}$ in the point $\alpha^{i}(t)$. So we define

$g_{j}(\gamma^{1}_{t})=\gamma^{2}_{\ell(t)},$

for each $t\in\mathbb{R}$. This is a bijection between the leaves of
$\mathscr{F}(p_{1})$ foliating $(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ onto
the leaves of $\mathscr{F}(p_{2})$ foliating
$(a_{j}^{2},a_{j+1}^{2})\times\mathbb{R}$. By assumption (b),
$g_{j}(\gamma)=h(\gamma)$ for any separatrix $\gamma$ in
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$, as well as if $\gamma$ is $\xi$ (if
it is the case), hence $g_{j}$ agrees with $g$ defined in Step 5 in these
cases. By construction, $g_{j}$ satisfies $\ell\circ p_{1}=p_{2}\circ g_{j}$.

_Step 7:_ We then define $g:\mathscr{F}(p_{1})\to\mathscr{F}(p_{2})$ by
extending the bijection defined in Step 5 to the entire $\mathscr{F}(p_{1})$
as

$g(\gamma)=g_{j}(\gamma)$

if $\gamma$ is contained in the strip
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$, $j=0,\ldots,k$, according to Step
6. By construction it follows that $g$ is a well defined bijection and also
that it satisfies $\ell\circ p_{1}=p_{2}\circ g$.

It remains to prove that $g$ is an isomorphism. I.e., given
$\gamma_{1},\gamma_{2},\gamma_{3}\in\mathscr{F}(p_{1})$, we have to show that
the chordal relations of them are the same than the chordal relations of
$g(\gamma_{1}),g(\gamma_{2}),g(\gamma_{3})\in\mathscr{F}(p_{2})$.

_Step 8:_ If
$\gamma_{1}=\gamma_{t_{1}},\gamma_{2}=\gamma_{t_{2}},\gamma_{3}=\gamma_{t_{3}}$
are contained in the same strip region
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ with $t_{1}<t_{2}<t_{3}$, we have
$\gamma_{t_{1}}|\gamma_{t_{2}}|\gamma_{t_{3}}$. The definition of $g$ in Step
7 proves that $g(\gamma_{t_{1}})|g(\gamma_{t_{2}})|g(\gamma_{t_{3}})$, and so
$\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ have the same chordal relations
than $g(\gamma_{1})$, $g(\gamma_{2})$ and $g(\gamma_{3})$. Also, since $g=h$
in the set of separatrices, and $h$ is an isomorphism, the same happens if
$\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are separatrices. Hence _we can
assume that $\gamma_{1}$, $\gamma_{2}$ and $\gamma_{3}$ are not in the same
strip region and that one of them is not a separatrix_.

Before to proceed, we need some properties of $h$ in the next three steps.
With them in hands, we will be able to prove that $g$ preserves chordal
relations by using the fact that $h$ preserves chordal relations in Step 12.

_Step 9:_ Since each level set of $p_{i}$ has exactly one connected component
in $(a_{j}^{i},a_{j+1}^{i})\times\mathbb{R}$, it follows that
_$\gamma_{1}|\gamma_{2}|\gamma_{3}$ is equivalent to
$p_{i}(\gamma_{1})<p_{i}(\gamma_{2})<p_{i}(\gamma_{3})$ or to
$p_{i}(\gamma_{3})<p_{i}(\gamma_{2})<p_{i}(\gamma_{1})$ for leaves
$\gamma_{1},\gamma_{2},\gamma_{3}\in\mathscr{F}(p_{i})$ in the same strip
region $(a_{j}^{i},a_{j+1}^{i})\times\mathbb{R}$_.

_Step 10:_ _Given $\gamma_{1},\gamma_{2}\in\mathscr{F}(p_{1})$ in the same
strip region $(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ such that
$p_{1}(\gamma_{1})<p_{1}(\gamma_{2})$, then
$p_{2}(h(\gamma_{1}))<p_{2}(h(\gamma_{2}))$_. Indeed, we first assume that in
the strip region $(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ there are leaves
$\alpha,\beta$ both of them separatrices or one separatrix and the other being
$\xi$ (assumption (b) guarantees the existence of at least one strip region
like this). So we have, say, $p_{1}(\alpha)<p_{1}(\beta)$, and it follows that
$p_{2}(h(\alpha))=\ell\circ p_{1}(\alpha)<\ell\circ
p_{1}(\beta)=p_{2}(h(\beta))$, as $\ell$ is increasing. Therefore, since $h$
preserves chordal relations, it follows by Step 9 that for a given leaf
$\gamma$ in this strip region: $p_{1}(\gamma)<p_{1}(\alpha)$, or
$p_{1}(\alpha)<p_{1}(\gamma)<p_{1}(\beta)$ or $p_{1}(\beta)<p_{1}(\gamma)$
implies respectively that $p_{2}(h(\gamma))<p_{2}(h(\alpha))<p_{2}(h(\beta))$,
or $p_{2}(h(\alpha))<p_{2}(h(\gamma))<p_{2}(h(\beta))$ or
$p_{2}(h(\beta))<p_{2}(h(\gamma))$. So we conclude, using once more Step 9,
that if $p_{1}(\gamma_{1})<p_{1}(\gamma_{2})$, then
$p_{2}(h(\gamma_{1}))<p_{2}(h(\gamma_{2}))$ for any leaves
$\gamma_{1},\gamma_{2}$ in this strip region. I.e., Step 10 is true in a strip
region containing leaves $\alpha$ and $\beta$ as above.

Now, in order to prove Step 10 for any strip region, it is enough to conclude
that given a strip region containing $\alpha$ and $\beta$ as above, we are
able to construct $\overline{\alpha}$ and $\overline{\beta}$ in both the
_adjacent_ strip regions (when it is the case) such that
$p_{1}(\overline{\alpha})<p_{1}(\overline{\beta})$ and
$p_{2}(h(\overline{\alpha})<p_{2}(\overline{\beta})$. Because then we apply
the same reason than in the former case in order to prove that
$p_{1}(\gamma_{1})<p_{1}(\gamma_{2})$ implies
$p_{2}(h(\gamma_{1}))<p_{2}(h(\gamma_{2}))$ for $\gamma_{1},\gamma_{2}$ in
this strip. So we proceed to construct $\overline{\alpha}$ and
$\overline{\beta}$ in the region
$(a_{j_{0}}^{1},a_{j_{0}+1}^{1})\times\mathbb{R}$ by assuming the existence of
$\alpha$ and $\beta$ in the region
$(a_{j_{0}-1}^{1},a_{j_{0}}^{1})\times\mathbb{R}$. The other situation is
analogous. Centered at a point $b\in\nu_{a_{j_{0}}^{1}}$ there is an open ball
$B$ such that

$B^{-}=B\cap\left((a_{j_{0}-1}^{1},a_{j_{0}}^{1})\times\mathbb{R}\right),\ \ \
\ B^{+}=B\cap\left((a_{j_{0}}^{1},a_{j_{0}+1}^{1})\times\mathbb{R}\right)$

are contained in canonical regions of $\mathscr{F}(p_{1})$. For any $a\in
B^{-}$ and $c\in B^{+}$, it follows that the signal of $p_{1}(c)-p_{1}(a)$ is
constant. We assume without loss of generality that
$p_{1}(a)<p_{1}(b)<p_{1}(c)$ for every $a\in B^{-}$ and for every $c\in
B^{+}$. We have $p_{1}(\alpha)=p_{1}(b)$ or $p_{1}(\beta)=p_{1}(b)$. We assume
$p_{1}(\alpha)=p_{1}(b)$. So by taking $\gamma_{a}$ the leaf containing an
$a\in B^{-}$ it follows that $p_{1}(\gamma_{a})<p_{1}(\alpha)$ and so
$p_{2}(h(\gamma_{a}))<p_{2}(h(\alpha))=p_{2}(h(b))$. With this we conclude
that

$p_{2}(h(a))<p_{2}(h(b))<p_{2}(h(c))$

for any $a\in B^{-}$ and $c\in B^{+}$, because analogously the signal of
$p_{2}(h(c))-p_{2}(h(a))$ is constant. We fix $c_{0}\in B^{+}$ and take
$\overline{\beta}=\gamma_{c_{0}}$, the leaf of $\mathscr{F}(p_{1})$ containing
$c_{0}$, and $\overline{\alpha}$ to be the separatrix contained in
$(a_{j_{0}}^{1},a_{j_{0}+1}^{1})\times\mathbb{R}$ which is inseparable with
$\gamma_{a_{j_{0}}^{1}}$. By construction we have
$p_{1}(\overline{\alpha})=p_{1}(b)<p_{1}(\overline{\beta})$ and
$p_{2}(h(\overline{\alpha}))=p_{2}(h(b))<p_{2}(h(c_{0}))=p_{2}(h(\overline{\beta})$,
finishing the proof of Step 10.

_Step 11:_ _For any ordinary leaf $\gamma\in\mathscr{F}(p_{1})$, the leaves
$h(\gamma)$ and $g(\gamma)$ are in the same canonical region_. Indeed, if
$\alpha$ is a separatrix contained in the same strip region
$(a_{j}^{1},a_{j+1}^{1})\times\mathbb{R}$ than $\gamma$, we have
$p_{1}(\gamma)<p_{1}(\alpha)$ or $p_{1}(\gamma)>p_{1}(\alpha)$. From Step 10
it follows that $p_{2}(h(\gamma))<p_{2}(h(\alpha))$ or
$p_{2}(h(\gamma))>p_{2}(h(\alpha))$, respectively. Since
$p_{2}(g(\gamma))<p_{2}(g(\alpha))$ or $p_{2}(g(\gamma))>p_{2}(g(\alpha))$,
respectively, we are done.

_Step 12:_ _The bijection $g$ is an isomorphism_. According to Step 8, it
remains to analyze two possibilities for $\gamma_{1},\gamma_{2},\gamma_{3}$:
(i) $\gamma_{1}$ is an ordinary leaf in a canonical region containing neither
$\gamma_{2}$ nor $\gamma_{3}$ and $\gamma_{2}$ and $\gamma_{3}$ are not
contained in the same canonical region, and (ii) $\gamma_{1}$ and $\gamma_{2}$
are ordinary leaves in the same canonical region not containing $\gamma_{3}$.
For both cases, we will show that the chordal relations of the triples
$g(\gamma_{1}),g(\gamma_{2}),g(\gamma_{3})$ and
$h(\gamma_{1}),h(\gamma_{2}),h(\gamma_{3})$ are the same. This will conclude
that $g$ is an isomorphism, because $h$ is an isomorphism.

In case (i), it follows from Step 11 that $g(\gamma_{1})$ and $h(\gamma_{1})$
are in the same canonical region. Analogously, we have that $g(\gamma_{i})$
and $h(\gamma_{i})$ are in the same canonical region if $\gamma_{i}$ is an
ordinary leaf, for $i=2,3$. Since $\gamma_{2}$ and $\gamma_{3}$ are not both
ordinary leafs contained in the _same_ canonical region, it then follows by
applying (possible more than once) Lemma 5.1 together with the fact that $g=h$
in the separatrices that $g(\gamma_{1})$, $g(\gamma_{2})$, $g(\gamma_{3})$
have the same chordal relations than $h(\gamma_{1})$, $h(\gamma_{2})$,
$h(\gamma_{3})$.

In case (ii), by (a) of Lemma 5.1, we can assume
$\gamma_{1}|\gamma_{2}|\gamma_{3}$. It is enough to prove that
$g(\gamma_{1})|g(\gamma_{2})|h(\gamma_{3})$, because
$h(\gamma_{3})=g(\gamma_{3})$ in case $\gamma_{3}$ is a separatrix, or
$h(\gamma_{3})$ and $g(\gamma_{3})$ are in the same canonical region by Step
11 otherwise, and then (b) of Lemma 5.1 applies. We suppose on the contrary
that this is not true. By (a) of Lemma 5.1 again, we have
$g(\gamma_{2})|g(\gamma_{1})|h(\gamma_{3})$.

Up to multiplying $p_{1}$ and $p_{2}$ by $-1$ we can assume that
$p_{1}(\gamma_{1})<p_{1}(\gamma_{2})$, hence
$p_{2}(g(\gamma_{1}))<p_{2}(g(\gamma_{2}))$ and, by Step 10,
$p_{2}(h(\gamma_{1}))<p_{2}(h(\gamma_{2}))$. By analyzing the possible
positions of $g(\gamma_{i})$ and $h(\gamma_{i})$, i.e., the signals of
$p_{2}(g(\gamma_{i}))-p_{2}(h(\gamma_{j}))$, $i,j=1,2$, we see that in order
to not contradict $h(\gamma_{1})|h(\gamma_{2})|h(\gamma_{3})$ (because $h$ is
an isomorphism) and $g(\gamma_{2})|g(\gamma_{1})|h(\gamma_{3})$ we must have
$p_{2}(h(\gamma_{2}))<p_{2}(g(\gamma_{1}))$. But then we conclude that
$h(\gamma_{3})$ is contained in $(h(\gamma_{2}),g(\gamma_{1}))$, which is an
open connected set contained in the same canonical region containing
$g(\gamma_{i}),h(\gamma_{i})$, $i=1,2$, a contradiction with our assumption
(ii). This finishes the proof. ∎

######  Corollary 5.3.

Let $p_{i}(x,y)=r_{i}(x)+s_{i}(x)y$, $i=1,2$, be two finite linear-like
submersions. Then $p_{1}$ is topologically equivalent to $p_{2}$ if and only
if $Z_{p_{1}}=Z_{p_{2}}=\emptyset$ or each of the following conditions hold:

  1. (a)

$\mathscr{F}(p_{1})$ and $\mathscr{F}(p_{2})$ are topologically equivalent.

  2. (b)

There exists a monotone bijection $\sigma:r_{1}(Z_{p_{1}})\to
r_{2}(Z_{p_{2}})$ such that

$\sigma(p_{1}(\nu_{a}))=p_{2}(h(\nu_{a})),\ \ \ \ a\in Z_{p_{1}},$

where $h$ is an equivalence homeomorphism between $\mathscr{F}(p_{1})$ and
$\mathscr{F}(p_{2})$.

###### Proof.

(Necessity) If there are homeomorphisms $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$
and $\ell:\mathbb{R}\to\mathbb{R}$ such that $\ell\circ p_{1}=p_{2}\circ h$,
it follows that $h$ is an equivalence homeomorphism between
$\mathscr{F}(p_{1})$ and $\mathscr{F}(p_{2})$, proving (a). If
$Z_{p_{1}}=Z_{p_{2}}=\emptyset$ there is nothing to do. On the other hand,
taking $\sigma=\ell|_{r_{1}(Z_{p_{1}})}$, and considering Lemma 2.6 and
Theorem 3.5, it follows that $\sigma$ satisfies (b).

(Sufficiency) We take $p_{3}(x,y)=p_{2}(x,\tau y)$ with $\tau=1$ or $-1$ if
$h$ preserves or reverses orientation, respectively. In any case $p_{3}$ is
topologically equivalent to $p_{2}$, with $Z_{p_{3}}=Z_{p_{2}}$ and
$r_{3}(Z_{p_{3}})=r_{2}(Z_{p_{2}})$.

If $Z_{p_{1}}$ is empty, it follows by Theorem 5.2 that $p_{1}$ and $p_{3}$
are o-topologically equivalent. If $r_{1}(Z_{p_{1}})$ has at least two
elements, we take $\kappa=\pm 1$ such that
$\kappa\sigma:r_{1}(Z_{p_{1}})\to\kappa r_{3}(Z_{p_{3}})$ is increasing. Then
again by Theorem 5.2 it follows that $p_{1}$ and $\kappa p_{3}$, which is
topologically equivalent to $p_{3}$, are o-topologically equivalent.

So we assume that $r_{1}(Z_{p_{1}})=\\{c\\}$, where
$c=r_{1}(a)=p_{1}(\nu_{a})$ for any $a\in Z_{p_{1}}$. We take any ordinary
leaf $\xi\in\mathscr{F}(p_{1})$ such that $m=p_{1}(\xi)>c$ and let $\kappa=\pm
1$ be such that $\kappa p_{3}(h(\xi))>\kappa p_{3}(h(\nu_{a}))$. Then
$\widetilde{\sigma}:\\{c,m\\}\to\\{\kappa\sigma(c),\kappa p_{3}(h(\xi))\\}$,
with $\widetilde{\sigma}(c)=\kappa\sigma(c)$ and $\widetilde{\sigma}(m)=\kappa
p_{3}(h(\xi))$, is increasing, and Theorem 5.2 applies once more to conclude
that $p_{1}$ and $\kappa p_{3}$ are o-topologically equivalent. ∎

Theorem 1.3 and Corollary 1.4 are direct consequences of Theorem 5.2 and
Corollary 5.3, respectively, after applying Markus Theorem 2.3.

###### Proof of Theorem 1.1.

In case $B(p_{1})=B(p_{2})=\emptyset$, the result follows by Theorem 5.2. We
assume $B(p_{1})\neq\emptyset$.

For the necessity, if $\ell\circ p=q\circ h$, with
$\ell:\mathbb{R}\to\mathbb{R}$ and $h:\mathbb{R}^{2}\to\mathbb{R}^{2}$
orientation preserving homeomorphisms, then $h$ is an o-equivalence
homeomorphism between $\mathscr{F}(p)$ and $\mathscr{F}(q)$, proving (a).
Statement (b) follows trivially by defining $\sigma=\ell|_{B(p)}$ (or
$\overline{\sigma}=\ell|_{B(p)\cup\\{m\\}}$ for any $m\notin B(p_{1})$ in case
$B(p_{1})$ is a singleton) and considering Lemma 2.6.

Now the sufficiency will follow from Theorem 5.2 by showing that assumption
(b) of Theorem 1.1 implies assumption (b) of Theorem 5.2 with $p_{1}=p$ and
$p_{2}=q$. Indeed, for each $a\in Z_{p_{1}}$, the leaf $h(\nu_{a})$ is a
separatrix of $\mathscr{F}(p_{2})$ and so by Proposition 3.2 and Theorem 3.5
it follows that $q(h(\nu_{a}))=\\{c\\}$ is contained in $B(q)$ (actually, as
in the proof of Theorem 5.2, we have that $h(\nu_{a}))=\nu_{b}$ for certain
$b\in Z_{p_{2}}$). Then from assumption (b) of Theorem 1.1 it follows that
$h^{-1}(q^{-1}(c))=p^{-1}(\sigma^{-1}(c))$, hence $\nu_{a}\subset
p^{-1}(\sigma^{-1}(c))$, and so $\sigma(p(\nu_{a}))=\\{c\\}$. This finishes
the proof in case $B(p_{1})$ is not a singleton. If $B(p_{1})$ has only one
element, any $\xi\subset p_{1}^{-1}(m)$, with $m$ given in (b) of Theorem 1.1
will apply to prove (b) of Theorem 5.2. ∎

###### Proof of Corollary 1.2.

It follows similarly as the proof of Corollary 5.3 above. ∎

We think that results like theorems 1.1 and 1.3 as well as their corollaries
are true in a wider class of submersion functions, namely the ones where we do
not have the vanishing at infinity phenomenon, that already does not appear in
our class of functions, according to Corollary 3.6. Anyway, our proofs need
special regions of the foliation $\mathscr{F}(p)$ where $p$ assumes all the
values in $\mathbb{R}$, as the reasons in this section make clear. New
techniques must be developed in order to control this in the general case.

##  6\. Acknowledgements

We thank Professor Luis Renato Gonçalves Dias for reading and commenting on a
previous version of this article. The first named author is partially
supported by the grants 2019/07316-0 and 2020/14498-4, São Paulo Research
Foundation (FAPESP). The second named author is partially supported by
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) -
Finance Code 001.

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