∎

11institutetext: T. Medvedev 22institutetext: Laboratory of Algorithms and
Technologies for Network Analysis; HSE University
136 Rodionova Street, Niznhy Novgorod, Russia
22email: mtv2001@mail 33institutetext: O. Pochinka 44institutetext:
International Laboratory of Dynamical Systems and Applications; HSE
University, 25/12 Bolshaya Pecherckaya Street, Niznhy Novgorod, Russia

# A quasi-energy function for Pixton diffeomorphisms defined by generalized
Mazur knots

Timur Medvedev  Olga Pochinka

(Received: date / Accepted: date)

###### Abstract

In this paper we give a lower estimate for the number of critical points of
the Lyapunov function for Pixton diffeomorphisms (i.e. Morse-Smale
diffeomorphisms in dimension 3 whose chain recurrent set consists of four
points: one source, one saddle and two sinks). Ch. Bonatti and V. Grines
proved that the class of topological equivalence of such diffeomorphism $f$ is
completely defined by the equivalency class of the Hopf knot $L_{f}$ that is
the knot in the generating class of the fundamental group of the manifold
$\mathbb{S}^{2}\times\mathbb{S}^{1}$. They also proved that there are
infinitely many such classes and that any Hopf knot can be realized by a
Pixton diffeomorphism. D. Pixton proved that diffeomorphisms defined by the
standard Hopf knot $L_{0}=\\{s\\}\times\mathbb{S}^{1}$ have an energy function
(Lyapunov function) whose set of critical points coincide with the chain
recurrent set whereas the set of critical points of any Lyapunov function for
Pixton diffeomorphism with nontrivial (i.e. non equivalent to the standard)
Hopf knot is strictly larger than the chain recurrent set of the
diffeomorphism. The Lyapunov function for Pixton diffeomorphism with minimal
number of critical points is called the quasi-energy function. In this paper
we construct a quasi-energy function for Pixton diffeomorphisms defined by a
generalized Mazur knot.

###### Keywords:

Hopf knot Mazur knot Pixton diffeomorphism quasi-energy function

###### MSC:

37C15 37D15

††journal:  ††thanks: The research was done with the support of Russian
National Foundation (project 21-11-00010) except construction of the quasi-
energy function which was supported by International Laboratory of Dynamical
Systems and Applications of National Research University Higher School of
Economics, grant of Government of Russian Federation 075-15-2022-1101.

##  1 Introduction and the main results

Let $M^{n}$ be a smooth closed $n$-manifold with a metric $d$ and let
$f:M^{n}\to M^{n}$ be a diffeomorphism. For two given points $x,y\in M^{n}$ a
sequence of points $x=x_{0},\dots,x_{m}=y$ is called an $\varepsilon$-chain of
length $m\in\mathbb{N}$ connecting $x$ to $y$ if
$d(f(x_{i-1}),x_{i})<\varepsilon$ for $1\leqslant i\leqslant m$ (Fig. 1).

Figure 1: An $\varepsilon$-chain of length $m\in\mathbb{N}$

A point $x\in M^{n}$ is called chain recurrent for the diffeomorphism $f$ if
for every $\varepsilon>0$ there is an $\varepsilon$-chain of length $m$
connecting $x$ to itself for some $m$ ($m$ depends on $\varepsilon>0$). The
chain recurrent set, denoted by $\mathcal{R}_{f}$, is the set of all chain
recurrent points of $f$. Define the equivalence on $\mathcal{R}_{f}$ by the
rule: $x\sim y$ if for every $\varepsilon>0$ there is are $\varepsilon$-chains
connecting $x$ to $y$ and $y$ to $x$. This equivalence relation defines
equivalence classes called chain components.

If the chain recurrent set of a diffeomorphism $f$ is finite then it consists
of periodic points. A periodic point $p\in\mathcal{R}_{f}$ of period $m_{p}$
is said to be hyperbolic if absolute values of all the eigenvalues of the
Jacobian matrix $\left(\frac{\partial f^{m_{p}}}{\partial x}\right)|_{p}$ are
not equal to 1. If absolute values of all these eigenvalues are greater (less)
than 1 then $p$ is called a sink (a source). Sinks and sources are called
knots. If a periodic point is not a knot then it is called a saddle.

Let $p$ be a hyperbolic periodic point of a diffeomorphism $f$ whose chain
recurrent set is finite. The Morse index of $p$, denoted by $\lambda_{p}$, is
the number of eigenvalues of Jacobian matrix whose absolute values are greater
than 1. The stable manifold $W^{s}_{p}=\\{x\in
M^{n}:\lim\limits_{k\to+\infty}d(f^{km_{p}}(x),p)=0\\}$ and the unstable
manifold $W^{u}_{p}=\\{x\in
M^{n}:\lim\limits_{k\to+\infty}d(f^{-km_{p}}(x),p)=0\\}$ of $p$ are smooth
manifolds diffeomorphic to $\mathbb{R}^{\lambda_{p}}$ and
$\mathbb{R}^{n-\lambda_{p}}$, respectively. Stable and unstable manifolds are
called invariant manifolds. A connected component of the set
$W^{u}_{p}\setminus p$ ($W^{s}_{p}\setminus p$) is called a unstable (stable)
separatrice of $p$.

A diffeomorphism $f:M^{n}\to M^{n}$ is called a Morse-Smale diffeomorphism if

  1. 1.

its chain recurrent set $\mathcal{R}_{f}$ consists of finite number of
hyperbolic points;

  2. 2.

for any two points $p$, $q\in\mathcal{R}_{f}$ the manifolds $W^{s}_{p}$,
$W^{u}_{q}$ intersect transversally.

C Conley in Conley1978  gave the following definition: a Lyapunov function for
a Morse-Smale diffeomorphism $f:M^{n}\to M^{n}$ is a continuous function
$\varphi:M^{n}\to\mathbb{R}$ satisfying

  * •

$\varphi(f(x))<\varphi(x)$ if $x\notin R_{f}$;

  * •

$\varphi(f(x))=\varphi(x)$ if $x\in R_{f}$.

Notice that every Morse-Smale diffeomorphism $f$ has a Morse-Lyapunov function
111This function can be constructed, for example, by suspension. Consider the
topological flow $\hat{f}^{t}$ on the manifold $M^{n}\times\mathbb{R}$ defined
by $\hat{f}^{t}(x)=x+t$. Define the diffeomorphism $g:M^{n}\times\mathbb{R}\to
M^{n}\times\mathbb{R}$ by $g(x,\tau)=(f(x),\tau-1)$ and let
$G=\\{g^{k}~{},k\in\mathbb{Z}\\}$ and $W=(M^{n}\times\mathbb{R})/G$. Denote by
$p_{{}_{W}}:M^{n}\times\mathbb{R}\to W$ the natural projection and denote by
$f^{t}$ the flow on $W$ defined by
$f^{t}(x)=p_{{}_{W}}(\hat{f}^{t}(p^{-1}_{{}_{W}}(x)))$. The flow $f^{t}$ is
called the suspension over $f$. By construction the chain recurrent set of
$f^{t}$ consists of the finite number of periodic orbits
$\delta_{i}=p_{{}_{W}}(\mathcal{O}_{i}\times\mathbb{R}),\,i\in\\{1,\dots,k_{f}\\}$
and this means that the suspension $f^{t}$ is a Morse-Smale flow. A Lyapunov
function for these flows is constructed in Meyer1968 . Then the restriction of
this function on $M$ is the desired Lyapunov function for $f$., i.e. a
Lyapunov function $\varphi:M^{n}\to\mathbb{R}$ which is a Morse function such
that each periodic point $p\in\mathcal{R}_{f}$ is its non-degenerate critical
point of index $\lambda_{p}$ with Morse coordinates $(V_{p},\phi_{p}:y\in
V_{p}\mapsto(x_{1}(y),\dots,x_{n}(y))\in\mathbb{R}^{n}$ and

$\phi_{p}^{-1}(Ox_{1}\dots x_{\lambda_{p}})\subset
W^{u}_{p},\,\,\phi_{p}^{-1}(Ox_{\lambda_{p}+1}\dots x_{n})\subset
W^{s}_{p}.\quad\quad\quad(*)$

If the function $\varphi$ has no critical points outside $\mathcal{R}_{f}$
then following Pixton1977  we call it the energy function for the Morse-Smale
diffeomorphism $f$.

The proof of existence of an energy Morse function for a Morse-Smale
diffeomorphism of the circle is an easy exercise. D. Pixton Pixton1977  in
1977 proved that every Morse-Smale diffeomorphism of a surface has an energy
function. There he also constructed an example of a Morse-Smale diffeomorphism
on the 3-sphere which admits no energy function. The obstacle to existence of
an energy function in his example was the wild embedding of the saddle
separatrices in the ambient manifold (i.e. the closure of the separatrice is
not a submanifold of the ambient space). From MePo2018  it follows that there
are Morse-Smale diffeomorphisms with no energy function on manifolds of any
dimension $n>2$. Therefore, following GrLauPo2009  for a Morse-Smale
diffeomorphism $f$ we call a Morse-Lyapunov function with the minimal number
of critical points (denote it by $\rho_{{}_{f}}$) a quasi-energy function.
Notice that $\rho_{{}_{f}}$ is a topological invariant, i.e. if two
diffeomorphisms $f,f^{\prime}:M^{n}\to M^{n}$ are topologically conjugate
(that is there is a diffeomorphism $h:M^{n}\to M^{n}$ such that $h\circ
f=f^{\prime}\circ h$) then $\rho_{{}_{f}}=\rho_{{}_{f^{\prime}}}$.

In this paper we give a lower estimate of $\rho_{{}_{f}}$ for Pixton
diffeomorphisms. The class of Pixton diffeomorphisms $\mathcal{P}$ is defined
in the following way. Every diffeomorphism $f\in\mathcal{P}$ is a Morse-Smale
3-diffeomorphism whose chain recurrent set consists of four points: one
source, one saddle and two sinks (for details see section 2). Notice that
Pixton’s example is a diffeomorphism of this class. According to BoGr  the
class of topological conjugacy of a diffeomorphism $f\in\mathcal{P}$ is
completely defined by the equivalence class of the Hopf knot $L_{f}$, i.e. the
knot in the generating class of the fundamental group of the manifold
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ (see Proposition 1). Moreover, any Hopf
knot can be realized as a Pixton diffeomorphism.

Recall that a knot in $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is a smooth
embedding $\gamma:\mathbb{S}^{1}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ or the
image of this embedding $L=\gamma(\mathbb{S}^{1})$. Two knots $L,L^{\prime}$
are said to be equivalent if there is a homeomorphism
$h:\mathbb{S}^{2}\times\mathbb{S}^{1}\to\mathbb{S}^{2}\times\mathbb{S}^{1}$
such that $h(L)=L^{\prime}$. Two knots $\gamma,\gamma^{\prime}$ are smoothly
homotopic if there exists a smooth map
$\Gamma:\mathbb{S}^{1}\times[0,1]\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ such
that $\Gamma(s,0)=\gamma(s)$ and $\Gamma(s,1)=\gamma^{\prime}(s)$ for every
$s\in\mathbb{S}^{1}$. If $\Gamma|_{\mathbb{S}^{1}\times\\{t\\}}$ is an
embedding for every $t\in[0,1]$ then the knots are said to be isotopic.

Any Hopf knot $L\subset\mathbb{S}^{2}\times\mathbb{S}^{1}$ is smoothly
homotopic to the standard Hopf knot $L_{0}=\\{s\\}\times\mathbb{S}^{1}$ (see,
for example, K-L ) but generally it is neither isotopic nor equivalent to it.
B. Mazur Mazur  constructed the Hopf knot $L_{M}$ which we call the Mazur knot
and which is non-equivalent and non-isotopic to $L_{0}$ (see Fig. 2).

Figure 2: Two non-isotopic and non equivalent Hopf knots $L_{0}$ and $L_{M}$:
a) the standard Hopf knot $L_{0}$; b) the Mazur knot $L_{M}$

It follows from the results of AMP  that there exists a countable family of
pairwise non-equivalent Hopf knots $L_{n},\,n\in\mathbb{N}$ which are
generalized Mazur knots (Fig. 3).

Figure 3: A generalized Mazur knot $L_{n}$

According to GrLauPo2008  a Pixton diffeomorphism $f$ admits an energy Morse
function if and only if the knot $L_{f}$ is trivial (i.e. equivalent to the
standard one). If the knot $L_{f}$ is not trivial then the number
$\rho_{{}_{f}}$ of the critical points of a quasi-energy Morse function of $f$
is evidently even and

$\rho_{{}_{f}}\geqslant 6.$

The main result of this paper is the proof of Theorem 1.1.

###### Theorem 1.1

Let $f$ be a Pixton diffeomorphism ($f\in\mathcal{P}$) and let
$L_{n},\,n\in\mathbb{N}$ be its knot. Then the number $\rho_{{}_{f}}$ of
critical points of a quasi-energy function of $f$ is calculated by222For $n=1$
Theorem 1.1 is proved in GrLauPo2009 .

$\rho_{{}_{f}}=4+2n.$

##  2 Construction of Pixton diffeomorphisms

In dynamics a wild Artin-Fox arc was for the first time introduced by D.
Pixton in Pixton1977  where he constructed a Morse-Smale diffeomorphism on the
3-sphere with the unique saddle whose invariant manifolds form an Artin-Fox
arc. We give the modern construction of these diffeomorphisms following Ch.
Bonatti and V. Grines BoGr  where Pixton diffeomorphisms were also classified
(see also GrMePo2016 , MePo2018 ).

For ${\bf x}=(x_{1},\dots,x_{n})\in\mathbb{R}^{n}$ denote $||{\bf
x}||=\sqrt{x_{1}^{2}+\dots+x_{n}^{2}}$. Let
$h:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be the diffeomorphism defined by
$h(x_{1},x_{2},x_{3})=\left(\frac{x_{1}}{2},\frac{x_{2}}{2},\frac{x_{3}}{2}\right)$.
Define the map $p:\mathbb{R}^{3}\setminus
O\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ by

$p(x_{1},x_{2},x_{3})=\left(\frac{x_{1}}{||{\bf x}||},\frac{x_{2}}{||{\bf
x}||},\log_{2}(||{\bf x}||)\pmod{1}\right).$

Let $L\subset(\mathbb{S}^{2}\times\mathbb{S}^{1})$ be a Hopf knot and let
$U(L)$ be its tubular neighborhood. Then the set $\bar{L}=p^{-1}(L)$ is the
$h$-invariant arc in $\mathbb{R}^{3}$ and $U(\bar{L})=p^{-1}(U(L))$ is its
$h$-invariant neighborhood diffeomorphic to
$\mathbb{D}^{2}\times\mathbb{R}^{1}$ (Fig. 4).

Figure 4: Suspension of a Hopf knot

Let
$C=\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}~{}:~{}x_{2}^{2}+x_{3}^{2}\leqslant
4\\}$ and let $g^{t}:C\to C$ be the flow defined by

$g^{t}(x_{1},x_{2},x_{3})=(x_{1}+t,x_{2},x_{3}).$

Then there is a diffeomorphism ${\zeta}:{U(L)}\to C$ that conjugates
$h|_{{U(L)}}$ and $g=g^{1}|_{C}$. Define the flow $\phi^{t}$ on $C$ by:

$\begin{cases}\dot{x}_{1}=\begin{cases}1-\frac{1}{9}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4)^{2},\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant 4\cr 1,\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}>4\end{cases}\cr\dot{x}_{2}=\begin{cases}\frac{x_{2}}{2}\big{(}\sin\big{(}\frac{\pi}{2}\big{(}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-3\big{)}\big{)}-1\big{)},\quad
2<x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant 4\cr-x_{2},\quad\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant 2\cr 0,\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}>4\end{cases}\cr\dot{x}_{3}=\begin{cases}\frac{x_{3}}{2}\big{(}\sin\big{(}\frac{\pi}{2}\big{(}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-3\big{)}\big{)}-1\big{)},\quad
2<x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant 4\cr-x_{3},\quad\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant 2\cr 0,\quad
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}>4.\end{cases}\end{cases}$

Figure 5: Trajectories of the flow $\phi^{t}$

By construction the diffeomorphism $\phi=\phi^{1}$ has two fixed points: the
saddle $P(1,0,0)$ and the sink $Q(-1,0,0)$ (Fig. 5), both being hyperbolic.
One unstable separatrice of the saddle $P$ coincides with the open interval
$\left\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:\,|x_{1}|<1,\,x_{2}=x_{3}=0\right\\}$
in the basin of the sink $Q$ while the other coincides with the ray
$\left\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:\,x_{1}>1,\,x_{2}=x_{3}=0\right\\}$.
Notice that $\phi$ coincides with the diffeomorphism $g=g^{1}$ outside the
ball $\\{(x_{1},x_{2},x_{3})\in C:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leqslant
4\\}$.

Define the diffeomorphism $\bar{f}_{L}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ so
that $\bar{f}_{L}$ coincides with $h$ outside ${U({L})}$ and it coincides with
${\zeta}^{-1}\phi{\zeta}$ on ${U({L})}$. Then $\bar{f}_{L}$ has in ${U({L})}$
two fixed points: the sink ${\zeta}^{-1}(Q)$ and the saddle ${\zeta}^{-1}(P)$,
both being hyperbolic. The unstable separatrice of the saddle
${\zeta}^{-1}(P)$ lies in ${L}$ (Fig. 6).

Figure 6: The phase portrait of the diffeomorphism $\bar{f}_{L}$

Now project the dynamics onto the $3$-sphere. Denote by $N(0,0,0,1)$ the North
Pole of the sphere $\mathbb{S}^{3}=\\{{\bf
x}=(x_{1},x_{2},x_{3},x_{4}):\,||{\bf x}||=1\\}$. For every point ${\bf
x}\in(\mathbb{S}^{3}\setminus\\{N\\})$ there is the unique line passing
through $N$ and ${\bf x}$ in $\mathbb{R}^{4}$. This line intersects
$\mathbb{R}^{3}$ in exactly one point $\vartheta_{+}({\bf x})$ (Fig. 7). The
point $\vartheta_{+}({\bf x})$ is the stereographic projection of the point
${\bf x}$. One can easily check that

$\vartheta_{+}(x_{1},x_{2},x_{3},x_{4})=\left(\frac{x_{1}}{1-x_{4}},\frac{x_{2}}{1-x_{4}},\frac{x_{3}}{1-x_{4}}\right).$

Thus, the stereographic projection
$\vartheta_{+}:\mathbb{S}^{3}\setminus\\{N\\}\to\mathbb{R}^{3}$ is a
diffeomorphism.

Figure 7: The stereographic projection.

By construction $\bar{f}_{L}$ coincides with $h$ in some neighborhood of the
point $O$ and in some neighborhood of the infinity. Therefore, it induces on
$\mathbb{S}^{3}$ the Morse-Smale

$f_{{L}}({\bf
x})=\begin{cases}\vartheta_{+}^{-1}(\bar{f}_{L}(\vartheta_{+}({\bf
x}))),~{}{\bf x}\neq N;\cr N,~{}{\bf x}=N\end{cases}.$

It follows directly from the construction that the non-wandering set of
$f_{{L}}$ consists of exactly four fixed hyperbolic points: two sinks
$\omega=\vartheta_{+}^{-1}({\zeta}^{-1}(Q))$, $S$, one saddle
$\sigma=\vartheta_{+}^{-1}({\zeta}^{-1}(P))$ and one source $N$. We say the
constructed diffeomorphism to be model and it is of Pixton class.

######  Proposition 1 (BoGr )

  * •

Any diffeomorphism $f\in\mathcal{P}$ is topologically conjugate to some model
diffeomorphism $f_{L}$.

  * •

Two model diffeomorphisms $f_{L},f_{L^{\prime}}$ are topologically conjugate
if and only if their knots $L,L^{\prime}$ are equivalent.

##  3 Genus of Hopf knot

In this section we introduce the notion of genus for a Hopf knot and use it to
estimate the number of critical points of the quasi-energy function of the
Pixton diffeomorphism defined by this knot.

Let $L$ be a Hopf knot and let $\bar{L}=p^{-1}(L)$ be its cover in
$\mathbb{R}^{3}\setminus O$. We say a closed orientable surface
$\Sigma\subset\mathbb{S}^{2}\times\mathbb{S}^{1}$ to be a secant surface of
the knot $L$ if it intersects $L$ in a unique point and there is an
$h$-compressible 3-manifold $Q_{\Sigma}\subset\mathbb{R}^{3}$ (that is
$h(Q_{\Sigma})\subset int\,Q_{\Sigma}$) with the boundary $\bar{\Sigma}$ such
that $\Sigma=p(\bar{\Sigma})$ and the intersection $\bar{L}\cap\bar{\Sigma}$
is the unique point $\bar{y}$. The minimally possible genus $g_{{}_{L}}$ of
the secant surface is called the genus of the knot $L$. The secant surface of
$L$ of genus $g_{{}_{L}}$ is said to be minimal.

###### Lemma 1

If $\Sigma$ is a minimal secant surface of the knot $L$ then the surface
$\bar{\Sigma}\setminus\bar{y}$ is non-compressible in
$\mathbb{R}^{3}\setminus(O\cup\bar{L})$, i.e. any simple closed curve
$c\subset int~{}(\bar{\Sigma}\setminus\bar{y})$ is contractible on
$\bar{\Sigma}\setminus\bar{y}$ if it bounds a smoothly embedded 2-disk
$D\subset int~{}(\mathbb{R}^{3}\setminus(O\cup\bar{L}))$ such that
$D\cap(\bar{\Sigma}\setminus\bar{y})=\partial D=c$.

###### Proof

Let $\Sigma$ be a minimal secant surface of $L$ and let $\bar{y}$ be the
unique point of the intersection $\bar{L}\cap\bar{\Sigma}$. Assume the
opposite: the surface $\bar{\Sigma}\setminus\bar{y}$ is compressible in
$\mathbb{R}^{3}\setminus(O\cup\bar{L})$. Then there is a non-contractible
simple closed curve $c\subset int~{}(\bar{\Sigma}\setminus\bar{y})$ and there
is the smoothly embedded 2-disk $D\subset
int~{}(\mathbb{R}^{3}\setminus(O\cup\bar{L}))$ such that
$D\cap(\bar{\Sigma}\setminus\bar{y})=\partial D=c$ (see, for example, Neu ).
Then we have two possibilities:

$(int\,D)\cap\left(\bigcup\limits_{k\in\mathbb{Z}}h^{k}(\bar{\Sigma})\right)=\emptyset,$
(1)
$(int\,D)\cap\left(\bigcup\limits_{k\in\mathbb{Z}}h^{k}(\bar{\Sigma})\right)\neq\emptyset.$
(2)

In case (1) two subcases are possible: (1a) $D\subset Q_{\Sigma}$, (1b)
$D\subset(\mathbb{R}^{3}\setminus int\,Q_{\Sigma})$. For case 1a) let
$N(D)\subset Q_{\Sigma}$ be a tubular neighborhood of the disk $D$. Then
exactly one connected component of the set $Q_{\Sigma}\setminus int\,N(D)$
intersects $\bar{L}$. According to (1) this component is $h$-compressible and
its boundary intersects $\bar{L}$ at a unique point. The projection of this
boundary into $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is, therefore, the secant
surface of $L$ of genus less than $g_{L}$. This contradicts the fact that the
surface $\Sigma$ is minimal. In case 1b) let
$N(D)\subset(\mathbb{R}^{3}\setminus int\,Q_{\Sigma})$ be a tubular
neighborhood of $D$. Then due to (1) the set $Q_{\Sigma}\cup N(D)$ is
$h$-compressible and its boundary intersects $\bar{L}$ at a unique point. The
projection of this boundary into $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is,
therefore, the secant surface of $L$ of genus less than $g_{L}$ and we have
the same contradiction.

In case (2) without loss of generality assume the intersection
$int\,D\cap(\bigcup\limits_{k\in\mathbb{Z}}h^{k}(\bar{\Sigma}))$ to be
transversal and denote it by $\Gamma$. Let $\gamma$ be a curve from $\Gamma$.
We say the curve $\gamma$ to be innermost if it is the boundary of the disk
$D_{\gamma}\subset D$ such that $int\,D_{\gamma}$ contains no curves of
$\Gamma$. Consider this innermost curve $\gamma\subset f^{k}(\Sigma)$. There
are two subcases: a) $\gamma$ is essential on $f^{k}(\Sigma)$ and b) $\gamma$
is contractible on $f^{k}(\Sigma)$. In case a) the arguments of the case (1)
apply for the body $f^{k}(Q_{\Sigma})$ and the disk $D_{\gamma}$ and we get
the contradiction to the minimality of the surface $\Sigma$. In case b) denote
by $d_{\gamma}\subset f^{k}(\Sigma)$ the 2-disk bounded by $\gamma$ and denote
by $B_{\gamma}\subset(\mathbb{R}^{3}\setminus O)$ the 3-ball bounded by the
2-sphere $D_{\gamma}\cup d_{\gamma}$. Consider: b1) $B_{\gamma}\subset
f^{k}(Q_{\Sigma})$ and b2) $B_{\gamma}\subset(\mathbb{R}^{3}\setminus
int\,f^{k}(Q_{\Sigma}))$. For b1) let $N(B_{\gamma})\subset f^{k}(Q_{\Sigma})$
be a tubular neighborhood of $B_{\gamma}$. Then the set $Q_{\Sigma}\setminus
int\,N(B_{\gamma})$ is $h$-compressible because the curve $\gamma$ lies in its
interior and the boundary of $Q_{\Sigma}\setminus int\,N(B_{\gamma})$
intersects $\bar{L}$ at a unique point. The projection of this boundary into
$\mathbb{S}^{2}\times\mathbb{S}^{1}$ is, therefore, the secant surface of the
knot $L$ of genus $g_{L}$ for which the number of connected components of the
set $\Gamma$ is less. We get the same result for b2) for the set
$Q_{\Sigma}\cup N(B_{\gamma})$. Thus, iterating the process we come either to
the case a) or to the case (1) and get a contradiction.

###### Lemma 2

For any diffeomorphism $f\in\mathcal{P}$ the following estimation holds

$\rho_{{}_{f}}\geqslant 4+2g_{{}_{L_{f}}}.$ (3)

###### Proof

Since Proposition 1 is true and since the number $\rho_{f}$ of the critical
points of a quasi-energy function of $f\in\mathcal{P}$ is invariant, from now
on we consider model Pixton diffeomorphisms$f_{L}$ with the Hopf knot $L$.
Denote by $\ell$ the non-stable separatrice of the saddle $\sigma$ lying in
the basin of the sink $S$. Let

$p_{S}:W^{s}_{S}\setminus S\to\mathbb{S}^{2}\times\mathbb{S}^{1}$

be the natural projection sending a point $w\in(W^{s}_{S}\setminus S)$ to the
point $p(f^{k_{w}}(w)),\,f^{k_{w}}(w)\in V_{S}$. Since the diffeomorphism
$f_{L}$ coincides with the homothety $h$ in some neighborhood $V_{S}$ of $S$,
the natural projection $p_{S}$ is well defined and $p_{S}(\ell)=L$ by
construction.

Consider an arbitrary Morse-Lyapunov function
$\varphi:\mathbb{S}^{3}\to\mathbb{R}$ of the diffeomorphism $f_{L}$. To be
definite let $\varphi(S)=0$, $\varphi(\sigma)=1$ and $\varphi(N)=3$. From the
definition of the Morse-Lyapunov function it follows that $\varphi|_{\ell}$
monotonically decreases in some neighborhood of the saddle $\sigma$.
Therefore, there is $\varepsilon_{1}\in(0,1)$ such that the interval
$(1-\varepsilon_{1},1)$ contains no critical values of $\varphi$ and the
connected component $\bar{\Sigma}_{1}$ of the level set
$\varphi^{-1}(1-\varepsilon_{1})$ intersects the separatrice $\ell$ at the
unique point. Denote this point by $w_{1}$.

Let $\bar{Q}_{1}$ be the connected component of the set
$\varphi^{-1}([0,1-\varepsilon_{1}])$ which contains the segment $[w_{1},S]$
of the closure of the separatrice $\ell$. Since $\varphi$ decreases along the
trajectories of $f$, the values of $\varphi$ on $W^{s}_{\sigma}$ are greater
than 1. Therefore, the manifold $\bar{Q}_{1}$ lies in the manifold $W^{s}_{S}$
diffeomorphic to $\mathbb{R}^{3}$. Let the function $\varphi|_{\bar{Q}_{1}}$
have $k_{q},\,q\in\\{0,\dots,3\\}$ critical points of index $q$. Due to
(Fomad, , Theorem 6.1) on the manifold $\bar{Q}_{1}$ there exists a self-
indexing Morse function $\psi$ (the value of the function in a critical point
equals the index of this point) which has $k_{q}$ critical points of index $q$
and which is constant on $\partial{\bar{Q}_{1}}$. Thus, the manifold
$\bar{Q}_{1}$ is the surface $\tilde{Q}_{1}$ of genus $g_{1}=1+k_{1}-k_{0}$
with attached handles of indexes 2 and 3. Then the genus of any surface of the
set $\partial\bar{Q}_{1}$ cannot be greater than $g_{1}$.

On the other hand, the number of critical points of $\varphi|_{\bar{Q}_{1}}$
is not less than $k_{0}+k_{1}$. If $k_{0}\geqslant 1$ and
$g_{1}=1+k_{1}-k_{0}$ then one gets $k_{0}+k_{1}=g_{1}+2k_{0}-1\geqslant
g_{1}+1$. Thus, $\varphi|_{\bar{Q}_{1}}$ has at least $g_{1}+1$ critical
points.

Denote by $\bar{\Sigma}_{1}$ the connected component of $\partial\bar{Q}_{1}$
which intersects the separatrice $\ell$. Then the surface $\bar{\Sigma}_{1}$
divides the manifold $W^{s}_{S}\cong\mathbb{R}^{3}$ into two parts, one of
which $Q_{1}$ being an $h$-compressible body. This means that
$\Sigma_{1}=p_{S}(\bar{\Sigma}_{1})$ is the secant surface of $L$ and,
therefore,

$g_{1}\geqslant g_{{}_{L}}.$

Analogously, there is $\varepsilon_{2}\in(0,1)$ for which the interval
$(1,1+\varepsilon_{2})$ contains no critical points of $\varphi$ and the
connected component $\bar{Q}_{2}$ of the level set
$\varphi^{-1}([0,1+\varepsilon_{2})]$ contains $cl(W^{u}_{\sigma})$ in its
interior while the intersection $\bar{Q}_{2}$ with $W^{s}_{\sigma}$ is the
unique 2-disk. Due to construction the function $\varphi|_{\bar{Q}_{2}}$ has
at least $g_{1}+3$ critical points and genus of the connected components of
$\partial\bar{Q}_{2}$ is less or equals $g_{1}$. Denote by $\bar{\Sigma}_{2}$
the connected component of $\partial\bar{Q}_{2}$ which intersects
$W^{s}_{\sigma}$ and denote by $g_{2}$ its genus. The surface
$\bar{\Sigma}_{2}$ divides the manifold $W^{u}_{N}\cong\mathbb{R}^{3}$ into
two parts, one of which $Q_{2}$ being an $h^{-1}$-compressible body. Arguing
as above one comes to conclusion that the number of critical points of
$\varphi|_{Q_{2}}$ is at least $g_{2}+1$. Therefore, the total number of
critical points of $\varphi$ is greater or equal to

$g_{1}+3+g_{2}+1\geqslant 4+2g_{1}\geqslant 4+2g_{{}_{L_{f}}}.$

##  4 The generalized Mazur knot $L_{n}$

In this section we show that the genus $g_{{}_{L_{n}}}$ of a generalized Mazur
knot equals ${n}$. At first we give a detailed description of construction of
$L_{n}$.

###  4.1 Construction of the generalized Mazur knot $L_{n}$

Recall that $h:\mathbb{R}^{3}\to\mathbb{R}^{3}$ is the homothety defined by

$h(x_{1},x_{2},x_{3})=\left(\frac{x_{1}}{2},\frac{x_{2}}{2},\frac{x_{3}}{2}\right)$

and $p:\mathbb{R}^{3}\setminus O\to\mathbb{S}^{2}\times\mathbb{S}^{1}$ is the
natural projection defined by

$p(x_{1},x_{2},x_{3})=\left(\frac{x_{1}}{||{\bf x}||},\frac{x_{2}}{||{\bf
x}||},\log_{2}(||{\bf x}||)\pmod{1}\right).$

Consider the annulus

$K=\left\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:\frac{1}{4}\leq
x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\leq 1\right\\}$

bounded by the spheres

$\mathbb{S}^{2}=\left\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\right\\},\,h(\mathbb{S}^{2}).$

Pick on the circle

$\mathbb{S}^{1}=\left\\{(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}:x_{1}^{2}+x_{2}^{2}=1,\,x_{3}=0\right\\}$

pairwise distinct points $\alpha_{1},\dots,\alpha_{2n+1}$ numbered in counter-
clockwise order (Fig. 8). Let $a_{i},\,i\in\\{1,\dots,2n\\}$ be the arc of the
circle $\mathbb{S}^{1}$ bounded by $\alpha_{i}$, $\alpha_{i+1}$ whose interior
contains no points of $\\{\alpha_{1},\dots,\alpha_{2n+1}\\}$. Let
$B,\,A_{i}\subset int\,K,\,i\in\\{1,\dots,2n\\}$ be pairwise disjoint smooth
arcs such that:

  1. 1.

the boundary points of $B$ are $\alpha_{2n+1}$, $h(\alpha_{1})$; the boundary
points of $A_{2j-1}$ are $\alpha_{2j-1}$, $\alpha_{2j}$ and the boundary
points of $A_{2j}$ are $h(\alpha_{2j})$, $h(\alpha_{2j+1})$ for
$j\in\\{1,\dots,n\\}$;

  2. 2.

the closed curves $c_{2j-1}=cl(a_{2j-1}\cup
A_{2j-1}),\,c_{2j}=cl(h(a_{2j})\cup A_{2j})$ bound the 2-disks
$d_{2j-1},\,d_{2j}$, the transversal intersection of these disks being the arc
$l_{j}$ with the boundary points $b_{2j-1}=d_{2j-1}\cap A_{2j}$ and
$b_{2j}=d_{2j}\cap A_{2j-1}$;

  3. 3.

the arc $cl(h(A_{1})\cup A_{2}\cup\dots\cup h(A_{2n-1})\cup A_{2n}\cup B)$ is
smooth.

Figure 8: Construction of the knot $L_{n}$

Let

$\bar{L}_{n}=\bigcup\limits_{k\in\mathbb{Z}}h^{k}(B\cup A_{1}\cup\dots\cup
A_{2n}),\,L_{n}=p(\bar{L}_{n}).$

###  4.2 The genus of the knot $L_{n}$

###### Lemma 3

The genus $g_{{}_{L_{n}}}$ of the knot $L_{n}$ equals $n$.

###### Proof

Since there is a secant surface of $L_{n}$ of genus $n$, we have
$g_{{}_{L_{n}}}\leqslant n$ (Fig. 9). Now we show that
$g_{{}_{L_{n}}}\geqslant n$.

Figure 9: A secant surface of $L_{n}$ of genus $n$

To that end we prove that for $L_{n}$ there exists a minimal secant surface
$\Sigma$ such that $\bar{\Sigma}\subset K$ and
$\bar{L}_{n}\cap\bar{\Sigma}=h(\alpha_{1})$.

Indeed, let $\Sigma_{0}$ be some minimal secant surface of $L_{n}$. Then there
exists the connected component $\bar{\Sigma}_{0}$ of $p^{-1}(\Sigma_{0})$ such
that it intersects the curve $\bar{L}_{n}$ at the point $\bar{y}_{0}$ situated
on $\bar{L}_{n}$ between $\alpha_{1},h(\alpha_{1})$ and that bounds the
$h$-compressible body $Q_{\Sigma_{0}}$. Without loss of generality let
$\bar{y}_{0}=h(\alpha_{1})$ (otherwise the desired surface is constructed by
removing the tubular neighborhood of the arc
$[\bar{y}_{0},h(\alpha_{1})]\subset\bar{L}_{n}$ from $Q_{\Sigma_{0}}$).

Denote by $k_{+},k_{-}\geqslant 0$ the maximal integers for
which$f^{k}(\bar{\Sigma}_{0})\cap\bar{\Sigma}_{0}\neq\emptyset,\,f^{-k}(\bar{\Sigma}_{0})\cap\bar{\Sigma}_{0}\neq\emptyset,\,k\geqslant
0$, respectively. If $k_{+}=k_{-}=0$ then $\bar{\Sigma}_{0}$ is the desired
surface. Otherwise we show the way to decrease by 1 the number $k_{+}>0$ (for
$k_{-}$ the arguments are the same) using isotopy of the secant surface.

Notice that $\bar{\Sigma}_{0}\cap
f^{k_{+}}(c_{2j-1})=\emptyset,\,j\in\\{1,\dots,n\\}$. Without loss of
generality let the intersection
$\Gamma=\bigcup\limits_{j=1}^{n}f^{k_{+}}(d_{2j-1})\cap\bar{\Sigma}_{0}$ be
transversal. Let $\gamma$ be a curve from $\Gamma$. Then $\gamma$ bounds the
unique disk $D_{\gamma}\subset f^{k_{+}}(d_{2j-1})$. There are two
possibilities: 1) $b_{2j-1}\notin D_{\gamma}$, 2) $b_{2j-1}\in D_{\gamma}$. In
case 1) we say the curve $\gamma$ to be innermost if it bounds the disk
$D_{\gamma}\subset f^{k_{+}}(d_{2j-1})$ such that $int\,D_{\gamma}$ contains
no curves of $\Gamma$. Consider this innermost curve $\gamma$. Due to Lemma 1
the surface $\bar{\Sigma}_{0}\setminus\bar{y}_{0}$ is non-compressible in
$\mathbb{R}^{3}\setminus(O\cup\bar{L}_{n})$ and, therefore, there exists the
disk $d_{\gamma}\subset(\bar{\Sigma}_{0}\setminus\bar{y}_{0})$ bounded by
$\gamma$. Denote by
$B_{\gamma}\subset(\mathbb{R}^{3}\setminus(O\cup\bar{L}_{n}))$ the 3-ball
bounded by the 2-sphere $D_{\gamma}\cup d_{\gamma}$. Consider two subcases:
1a) $B_{\gamma}\subset Q_{\Sigma_{0}}$ and 1b)
$B_{\gamma}\subset(\mathbb{R}^{3}\setminus int\,Q_{\Sigma_{0}})$.

In case 1a) let $N(B_{\gamma})\subset Q_{\Sigma_{0}}$ be a tubular
neighborhood of the ball $B_{\gamma}$. Then the set $Q_{\Sigma}\setminus
int\,N(B_{\gamma})$ is $h$-compressible because the curve $\gamma$ lies in its
interior and its boundary intersects $\bar{L}_{n}$ at a unique point. The
projection of this boundary to $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is,
therefore, a secant surface of $L_{n}$ of the same genus as $\Sigma_{0}$. For
it the number of the connected components of the set $\Gamma$ is less. One
gets the same result in case 1b) for the set $Q_{\Sigma_{0}}\cup
N(B_{\gamma})$.

If we continue this process then we get the secant surface of $L_{n}$ of the
same genus as $\Sigma_{0}$ and for which the set $\Gamma$ contains no curves
of type 1). Denote the resulting surface again by $\Sigma_{0}$. Now the set
$\Gamma$ consists only of the curves $\gamma$ bounding the disk
$D_{\gamma}\subset b_{2j-1}$ which contains the point $b_{2j-1}$. Since
$(b_{2j-1}\sqcup c_{2j-1})\subset(\mathbb{R}^{3}\setminus Q_{\Sigma_{0}})$,
the number of these curves on the disk $d_{2j-1}$ is even. Since the surface
$\bar{\Sigma}_{0}\setminus\bar{y}_{0}$ is non-compressible in
$\mathbb{R}^{3}\setminus(O\cup\bar{L}_{n})$, all these curves are pairwise
homotopic on $\bar{\Sigma}_{0}\setminus\bar{y}_{0}$ and, therefore, they lie
in the annulus $\kappa\subset(\bar{\Sigma}_{0}\setminus\bar{y}_{0})$ bounded
by the pair of these curves $\gamma_{1},\gamma_{2}$. Denote by
$\tilde{\kappa}\subset d_{2j-1}$ the annulus bounded by the same curves on the
disk $d_{2j-1}$. Let
$\tilde{\Sigma}_{0}=\bar{\Sigma}_{0}\setminus\kappa\cup\tilde{\kappa}$. Due to
construction the surface $\tilde{\Sigma}_{0}$ is of the same genus as the
surface $\bar{\Sigma}_{0}$ and it bounds an $h$-compressible body. Having
removed a tubular neighborhood of the annulus $\tilde{\kappa}$ from this body
we get a $h$-compressible body whose boundary does not intersect the disk
$d_{2j-1}$ and whose projection to $\mathbb{S}^{2}\times\mathbb{S}^{1}$ is the
secant surface of the knot $L_{n}$ of the same genus as $\Sigma_{0}$.

If we continue this process then we get a secant surface of $L_{n}$ of the
same genus as $\Sigma_{0}$ and for which the set $\Gamma$ is not empty. Denote
this surface again by $\Sigma_{0}$. Without loss of generality let the
intersections of the surface $\bar{\Sigma}_{0}$ with the spheres
$f^{k}(\mathbb{S}^{2})$ be transversal. Denote by $\mathcal{F}$ the set of the
connected components of the intersection $f^{k_{+}}(K)\cap\bar{\Sigma}_{0}$.
Now we show the way to reduce by 1 the number of the components in
$\mathcal{F}$ using isotopy of the secant surface.

Denote by $Q$ the set obtained by removal from the annulus $f^{k_{+}}(K)$ of
the tubular neighborhoods of the disks $d_{2j-1}$ as well as the tubular
neighborhoods of the curves $A_{2j},\,j\in\\{1,\dots,n\\}$. Then $Q$ is
homeomorphic to the direct product of the 2-sphere with $2n+1$ deleted points
and the segment. Since
$Q\cap\bar{\Sigma}_{0}=f^{k_{+}}(K)\cap\bar{\Sigma}_{0}$ and since
$\bar{\Sigma}_{0}\setminus\bar{y}_{0}$ is non-compressible in
$\mathbb{R}^{3}\setminus(O\cup\bar{L}_{n})$, each connected component of
$F\in\mathcal{F}$ is non-compressible in $Q$. Due to (Wald, , Corollary 3.2)
there exists a surface $\tilde{F}\subset f^{k_{+}-1}(\mathbb{S}^{2})$
diffeomorphic to $F$ for which $\partial F=\partial{\tilde{F}}$ and the
surface $F\cup\tilde{F}$ bounds in $Q$ the body $\Delta$ diffeomorphic to the
direct product $F\times[0,1]$. Then we replace the part $F$ of
$\bar{\Sigma}_{0}$ with $\tilde{F}$. If we continue the process we get the
desired secant surface $\Sigma\subset K$.

Figure 10: Generators of the group $\pi_{1}(K\setminus\bar{L}_{n})$

Notice (see, for instance, (Dav, , Exercise 2.8.1)) that the fundamental group
$\pi_{1}(K\setminus\bar{L}_{n})$ has $2n$ generators
$\gamma_{1},\dots,\gamma_{2n}$, each of which $\gamma_{i}$,
$i\in\\{1,\dots,2n\\}$ being the generator of the punctured disk
$d_{i}\setminus b_{i}$ (Fig. 10). Since $b_{2j-1}\in int\,Q_{\Sigma}$ and
$c_{2j-1}\cap Q_{\Sigma}=\emptyset$, there exists the connected component of
$\tilde{d}_{2j-1}$ of the intersection $d_{2j-1}\cap Q_{\Sigma}$ which
contains the point $b_{2j-1}$. This component is the 2-disk bounded by the
curve $\tilde{\gamma}_{2j-1}\subset(\bar{\Sigma}\setminus h(\alpha_{1}))$ with
holes and the curves $\gamma_{2j-1},\,\tilde{\gamma}_{2j-1}$ are homotopic on
the punctured disk $d_{2j-1}\setminus b_{2j-1}$. In the same way one finds the
curves $\tilde{\gamma}_{2j}\subset(\bar{\Sigma}\setminus h(\alpha_{1}))$
homotopic to the curves $\gamma_{2j}$ on the punctured disk $d_{2j}\setminus
b_{2j}$ (Fig. 10). Due to Lemma 1 the surface $\bar{\Sigma}\setminus
h(\alpha_{1})$ is non-compressible in $K\setminus\bar{L}_{n}$. Then the curves
$\tilde{\gamma}_{1},\dots,\tilde{\gamma}_{2n}$ are pairwise non-homotopic to
the generators on the surface $\bar{\Sigma}\setminus h(\alpha_{1})$.
Therefore, the genus of the surface $\bar{\Sigma}$ cannot be less than $n$.

##  5 Construction of a quasi-energy function for a Pixton diffeomorphism with
the Hopf knot $L_{n}$

Let $f$ be a Pixton diffeomorphism constructed for a generalized Mazur knot
$L_{n}$. Then its non-wandering set $\Omega_{f}$ consists of four points: two
sinks $\omega,\,S$, a source $N$ and a saddle $\sigma$. Then
$W^{u}_{\sigma}\setminus\sigma$ consists of two separatrices
$\ell_{\omega},\,\ell_{S}$ respective closures of which contain the sinks
$\omega,\,S$, the separatrice $\ell_{\omega}$ being tame while $\ell_{S}$
being wild. Let $\bar{\Sigma}$ be the surface of genus $n$ bounding the
handle-body $Q_{\Sigma}$ of the same genus. Now we construct for $f$ a Morse-
Lyapunov function with $6+2n$ critical points.

Our construction of a quasi-energy function is analogous to the construction
of an energy function in GrLauPo2009 .

  1. 1.

Choose an energy function $\varphi_{p}:U_{p}\to\mathbb{R}$ in the neighborhood
of each fixed point $p$ of $f$ so that $\varphi_{p}(p)=\dim\,W^{u}_{p}$. Let
$B_{\omega},\,B_{S}$ be the 3-balls which are the level sets of respective
functions $\varphi_{\omega},\,\varphi_{S}$ and such that $B_{S}\subset
int\,Q_{\Sigma}$. Choose a tubular neighborhood $T_{\sigma}$ of the arc
$W^{u}_{\sigma}\setminus(B_{\omega}\cup Q_{\Sigma})$ so that the handle-body
$B_{\omega}\cup Q_{\Sigma}\cup T_{\sigma}$ of genus $n$ is $f$-compressible
and its intersection with $W^{s}_{\sigma}$ is the 2-disk. Denote by $P^{+}$
the smoothing of this body by addition of a small exterior collar.

  2. 2.

Due to (GrLauPo2009, , Section 4.3) there exists an energy function
$\varphi:P^{+}\setminus int\,Q_{\Sigma}$ whose value on $\partial P^{+}$ is
$4/3$, whose value on $\bar{\Sigma}$ is $2/3$ and which has exactly two
critical points $\omega,\,\sigma$ of respective Morse indexes $0,1$. The disks
$d_{1},\dots,d_{2n-1}$ cut the handle-body $Q_{\Sigma}$ making the 3-ball.
Denote by $B_{\Sigma}$ the smoothing of this ball by removal of the interior
collar. The results of the classic Morse theory (see, for example, Milnor2016
) allow to extend the function $\varphi$ to the set $Q_{\Sigma}\setminus
int\,B_{\Sigma}$ in such way that it has $n$ critical points of Morse index 1,
one point lying on each disk $d_{1},\dots,d_{2n-1}$, while the value of
$\varphi$ on $\partial B_{\Sigma}$ is $1/3$. Due to (GrLauPo2009, , Lemma 4.2)
the function $\varphi$ can be extended to the ball $B_{\Sigma}$ by an energy
function with the unique critical point $S$ of Morse index 0. Since
$f(Q_{\Sigma})\subset int\,B_{\Sigma}$, the constructed function decreases
along the trajectories of the diffeomorphism $f$.

  3. 3.

It follows from the definition of the knot $L_{n}$ that
$P^{-}=\mathbb{S}^{3}\setminus int\,P^{+}$ is the handle-body of genus $n$.
Moreover, the disks $d_{2},\dots,d_{2n}$ cut $P^{-}$ making the 3-ball. Denote
by $B^{-}$ smoothing of this ball by removal of the interior collar. The
results of the classic Morse theory (see, for example, Milnor2016 ) allow
extension of the function $\varphi$ to the set $P^{-}\setminus int\,B^{-}$ in
such way that it has $n$ critical points of Morse index 2, one point lying on
each disk $d_{2},\dots,d_{2n}$, while the value of $\varphi$ on $\partial
B^{-}$ is $5/3$. According to (GrLauPo2009, , Lemma 4.2) the function
$\varphi$ can be extended to the ball $B^{-}$ by an energy function with
unique critical point $N$ of Morse index 3. Since $f(B^{-})\subset
int\,P^{-}$, the constructed function decreases along the trajectories of the
diffeomorphism $f$ and, therefore, it is the desired quasi-energy function.

## Conflict of interest

The authors declare that they have no conflict of interest.

## References

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# Enhancing Saliency Prediction in Monitoring Tasks: The Role of Visual
Highlights

Zekun Wu<EMAIL_ADDRESS>0000-0002-5233-2352 Saarland
UniversityCampus, 66123 SaarbrückenSaarbrückenSaarlandGermany66123 and  Anna
Maria Feit<EMAIL_ADDRESS>Saarland University, Saarland Informatics
CampusE1 7, 66123 SaarbrückenSaarbrückenGermany

(2024)

###### Abstract.

This study examines the role of visual highlights in guiding user attention in
drone monitoring tasks, employing a simulated interface for observation. The
experiment results show that such highlights can significantly expedite the
visual attention on the corresponding area. Based on this observation, we
leverage both the temporal and spatial information in the highlight to develop
a new saliency model: the highlight-informed saliency model (HISM), to infer
the visual attention change in the highlight condition. Our findings show the
effectiveness of visual highlights in enhancing user attention and demonstrate
the potential of incorporating these cues into saliency prediction models.

Human-centered Computing, Human computer interaction (HCI), Visualization, Eye
Tracking

††copyright: acmcopyright††journalyear: 2024††conference: ETRA ’24; June
04–07, 2024; Glasgow, UK††ccs: Human-centered computing

Figure 1. Visual highlights can effectively direct the user’s attention to
relevant areas. On average, it takes around 1s until the highlight is noticed
and some user’s attention has already moved away after 4s, as shown by the
sequence of interface screenshots and saliency maps from the multi-drone
monitoring task we studied in this work. The images show a critical situation
(CS) occurring for the second drone and the corresponding saliency maps at
four-time points: 1s before CS, CS happening, 1s after CS, and 4s after CS.
Note that with the highlight’s help, an immediate focus of attention would be
put on the CS icon as visualized by the heatmap in the saliency map at T3 (1s
after CS). Image-based saliency models, as typically employed to predict
attention on user interfaces, cannot capture the temporal change in visual
attention due to visual highlighting.

Interface Frame and Saliency Map in different Timestamp

##  1\. Motivation and Study Design

In this study, we aim to explore how visual highlighting affects a user’s
visual attention during a monitoring task, and, more importantly, how we can
_predict_ the temporal changes in visual attention due to highlighting, using
saliency models. As indicated in Figure 1, we developed an interface
simulating the task of remotely monitoring four delivery drones. In the task,
the participants had to identify critical situations (CS) that could
potentially happen at any moment of the flight, while keeping an eye on all
the drones’ states to maintain the situation awareness (SA). When the CS
happened, the corresponding icon would be highlighted in yellow with a 50%
probability. Participants were instructed to press the space bar when they
detected a CS and at the same time monitor all parameters of all drones. Also,
to test the participants’ SA, we would stop the task randomly to ask multiple
choice questions targeting the specific drone state (for example, what is the
altitude for each drone now?), referring to the Situation Awareness Global
Assessment Technique (SAGAT) (Endsley, 2000). We have recruited 30
participants and kept effective data from 28 of them. The study procedure and
task were approved by the university’s ethics committee.

##  2\. Visual Highlights’ Influence on Gaze Behavior

We started with gaze data analysis for different highlight conditions.
Specifically, we chose the icon corresponding to CS and the parameter below it
as an area of interest (AOI) and used a set of gaze metrics as fixation count,
fixation duration, time to first fixation, and revisits based on prior work
(Feit et al., 2017) to give insights into the highlights’ influence on the
user’s visual attention. Our findings, illustrated in Figure 2, reveal no
significant differences in fixation count and fixation duration between the
two conditions. However, the time to first fixation was shorter for
highlighted situations than non-highlighted ones, suggesting that visual
highlighting effectively directs attention more rapidly in CS. The number of
revisits to the AOI showed no significant difference between the two
conditions, indicating that once a CS is detected, visual highlighting does
not further influence visual attention in terms of fixation behavior. These
findings are consistent with the previous studies (Karatas et al., 2020;
Schriver et al., 2017) and confirm that visual cues can facilitate faster
visual attention in the context of a monitoring task where the user has to
retain high SA of all drones while also detecting CS.

Figure 2. Hit Rates and Response Time in Different Highlight Conditions

##  3\. Highlight Informed Saliency Prediction

To predict how the visual highlight affects users’ visual attention over time,
we developed a new deep-learning highlight-informed saliency model (HISM), as
indicated in Figure 3. The HISM is targeted at predicting how salient the AOI
is relative to other visual elements in different timestamps with different
highlight conditions. Specifically, in the drone monitoring interface, we
chose the icons and the parameters below them as the potential visual
elements. We would normalize the saliency in each timestamp over all of the
elements to calculate the element-level saliency in the temporal sense (Gupta
et al., 2018; Aydemir et al., 2023).

The HISM takes the input information from the visual highlight in two
branches: the spatial information as the global stacked image that contains
both the global picture of the interface screen and the exact location of the
targeted AOI and the temporal information as a sequence of the local image of
the targeted AOI in different timestamps. As indicated in the Figure 3, the
HISM mainly consists of three components: the ResNet(He et al., 2016)
extracting the spatial features from the global stacked image, the LSTM(Yu et
al., 2019) extracting the temporal features from the highlight vector, which
is translated from the local image sequence with pre-trined binary classifier,
and the Saliency Model with three fully connected layers to predict the final
saliency. To examine the HISM performance, we chose two current state-of-art
saliency model: SimpleNet(Reddy et al., 2020) and TASED-Net (Min and Corso,
2019), as baselines. As both of them are predicting saliency on the pixel
level, we normalized their saliency prediction results over the AOI being
highlighted. As indicated in Figure 4, by considering both the temporal and
spatial information of the highlight, the HISM outperforms both baselines as
it exhibits enhanced predictive accuracy with a mean squared error (MSE) of
0.006, which is markedly lower than that of the TASED-Net and SimpleNet
models, which achieved performance with MSEs of 0.01 and 0.009, respectively.

Figure 3. Highlight Informed Saliency Model Framework, which includes three
key components: the ResNet50 to extract the spatial information, the two-layer
LSTM block to generate the temporal features, and the MLP takes both types of
features to predict the saliency of the AO Figure 4. Normalized saliency of
the critical AOI over time computed over 0.5s windows. The corresponding icon
was highlighted at second 5. Our HISM model captures the time to first
fixation of about 1s and the following decrease in attention. State-of-the-art
image- and video-based saliency models (SimpleNet and TASED-Net) could not
capture the temporal changes in users’ attention

##  4\. Conclusions and Future Work

In short, we developed a saliency model targeting predicting the visual
attention change on the visual element with different highlights in the
monitoring task. In the future, we will further examine the model performance
on the free-viewing gaze data from more general interfaces such as web pages,
etc.

###### Acknowledgements.

This work is funded by DFG grant 389792660 as part of TRR 248 – CPEC, see
https://perspicuous-computing.science

## References

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11institutetext: University of Technology and Education, Vietnam
22institutetext: FPT University Ha Noi, Vietnam 33institutetext: National
Cheng Kung University, Taiwan 44institutetext: Ho Chi Minh City Medicine and
Pharmacy University, Vietnam 55institutetext: Taipei Medical University,
Taiwan 66institutetext: Vietnamese-German University, Vietnam
66email<EMAIL_ADDRESS>
# Beyond Traditional Approaches: Multi-Task Network for Breast Ultrasound
Diagnosis

Dat T. Chung 11 Minh-Anh Dang 33 Mai-Anh Vu 22 Minh T. Nguyen 44 Thanh-Huy
Nguyen 55 Vinh Q. Dinh 66

###### Abstract

Breast Ultrasound plays a vital role in cancer diagnosis as a non-invasive
approach with cost-effective. In recent years, with the development of deep
learning, many CNN-based approaches have been widely researched in both tumor
localization and cancer classification tasks. Even though previous single
models achieved great performance in both tasks, these methods have some
limitations in inference time, GPU requirement, and separate fine-tuning for
each model. In this study, we aim to redesign and build end-to-end multi-task
architecture to conduct both segmentation and classification. With our
proposed approach, we achieved outstanding performance and time efficiency,
with 79.8% and 86.4% in DeepLabV3+ architecture in the segmentation task.

###### Keywords:

Breast ultrasound Cancer diagnosis Multi-task.

##  1 Introduction

Breast cancer is the leading cause of cancer deaths in women. It currently has
the highest incidence rate of cancer among women in the United States; 31% of
all newly diagnosed cases of cancer in 2022 were found to be related to it.
Because of its high incidence rate, early detection of breast cancer is
crucial to lowering death rates and increasing available treatment options.
Because breast ultrasound imaging is noninvasive, nonradioactive, and cost-
effective, it is a useful screening tool. Compared to previous studies using
x-rays to diagnose cancer mammograms [1, 2, 3], ultrasound diagnosis achieves
more promising overall.

Deep-learning techniques have overtaken traditional ones as the preferred
methods for segmenting breast ultrasound images as a result of advancements in
computer technology. One of the most well-liked fully convolutional network
models among them is the U-net model, which finds extensive application in the
field of medical image processing. The U-net model is an end-to-end, fully
convolutional network with skip layers between the synthesis and analysis
paths that run pixel-by-pixel. It gained popularity due to the ability to
reserve a large number of significant features using a small training dataset.
Consequently, U-Net is widely used as a baseline for breast ultrasound
diagnosis

In the real world, however, breast ultrasound diagnosis screening necessitates
performing two tasks at the same time: segmentation and classification. In
this work, we used the mechanism of multi-task learning by redesigning the
framework with two distinct heads. Our proposed framework effectively uses
auxiliary information from the cancer classification task to improve tumor
segmentation task performance. Furthermore, we ran extensive experiments on
our proposed framework under various architectures and backbones. The task
weights were also carefully tested to determine how much each branch
contributed to the overall loss during the training stage. Our multi-task
approach achieved its robustness performance, reaching a peak at 86.4% in the
segmentation task.

##  2 Related works

####  2.0.1 Medical Image Segmentation

Artificial intelligence-based medical image segmentation has become a matter
of concern in the Computer-Aided Diagnosis field. U-Net [4], which has been
widely applied in segmentation, is an autoencoder-based architecture with skip
connections to incorporate feature maps from the encoder and decoder.
Succeeding U-Net, several architectures have been proposed for segmentation,
including U-Net++ [5], DeepLabV3 [6], and Feature Pyramid Networks (FPN) [7]
combined with various backbones as ResNet [8], ResNeXt [9], EfficientNet [10].

####  2.0.2 Multi-task learning

Multi-task learning (MTL) is an approach that aims to perform multiple related
tasks simultaneously. Recently, this technique has emerged as an efficient
method to boost a particular task’s performance by learning valuable
information from other related ones. Song et al. [11] proposed a multi-task
framework that performs skin lesion detection, classification, and
segmentation tasks simultaneously and achieved higher scores than state-of-
the-art methods, especially for segmentation tasks. Amyar et al. [12] used a
multi-task model to jointly identify COVID-19 patients and segment COVID-19
lesions from CT images, obtaining better results in both segmentation and
classification. Influenced by these promising results, we implemented a multi-
task model using classification tasks to boost the main segmentation task in
our work.

##  3 Methodology

Figure 1: Visualization of our proposed architecture with multi-task learning
approach. The shared encoder (green area) takes weighted sum loss from the
segmentation branch (blue area) and classification branch (yellow area).

Four encoders and four decoders were chosen for their architectural diversity.
The multi-task framework will consist of a shared weight backbone model with
two heads handling classification and segmentation tasks. Breast ultrasound
image is the input, and the outputs are the boundary of breast tumors and the
cancerous attribute of breast tumors for segmentation and classification
tasks, respectively.

###  3.1 Multi-task framework

We have implemented a multi-task framework that can be end-to-end trained for
both segmentation and classification tasks shown in Fig 1. Breast ultrasound
images of tumors frequently show a distinct echo pattern from the surrounding
tissue. To effectively capture these visual features, we use ResNet-50 [8],
ResNeXt-50 [9], Wide ResNet-50 [13], and EfficientNet-B4 [10], as a shared
backbone. Two separate task-specific heads are then utilized to predict breast
tumor boundary and attributes.

####  3.1.1 Classification head

In particular, in the task-specific heads of classification, the extracted
feature of the encoder fed into the classifier, including bias fully connected
and dropout layer as illustrated in classification head of Fig 1 for
prediction of either benign, malignant, or normal breast tumor.

####  3.1.2 Segmentation head

To identify the breast tumor boundaries throughout the entire image, we employ
the segmentation task as a simultaneous task. The extracted features from the
shared encoder will be provided through some of the most well-known decoders
in the medical imaging field, including Unet [4], Unet++ [5], and DeepLabV3
[6]. Each feature will sequentially be fed into upsampling and convolution
layers to recover its original size and generate the breast tumor’s boundary
prediction result.

###  3.2 Loss composition

####  3.2.1 Classification

A very prevalent issue that arises in medical imaging is an imbalance in the
dataset. The prediction performance of the models is skewed toward the
majority of samples since the number of negative samples is typically higher
than the number of positive ones. Influenced by balanced cross-entropy loss
and has an extra class weight parameter that can change the derivative of the
loss function, Focal loss is used for the classification task. This aids in
the model’s attention to classes with fewer examples or those that are
challenging.

$L_{classification}=FocalLoss(p_{t})=-\alpha_{t}(1-p_{t})^{\gamma}log(p_{t})$
(1)

where $p_{t}$ is the predicted probability of a ground truth class,
$\alpha_{t}$ is the balance factor for the true class, and $\gamma$ is the
hyperparameter that controls the degree of down-weighting for well-classified
samples.

####  3.2.2 Segmentation

We employ Dice Loss, which is determined by averaging the dice coefficients of
the background and foreground (breast tumor) classes.

$L_{segmentation}=DiceLoss(y,\hat{y})=\frac{y+\hat{y}+2y\hat{y}}{y+\hat{y}+\epsilon}$
(2)

where $y$ is the ground truth mask and $\epsilon$ is very small to prevent
loss function from division by zero.

Finally, we argue that joint training can enhance the feature representation
for each task. Thus, we define a multi-task loss as the weighted sum of two
tasks:

$L_{total}=\lambda
L_{segmentation}(y,\hat{y})+(1-\lambda)L_{classification}(p_{t})$ (3)

where $\lambda$ denotes the loss weight for each loss element.

##  4 Experiments Results

####  4.0.1 Experimental Setting.

We evaluated the performance of the multi-task framework using the public
dataset - BUSI [14]. It consists of 780 images with an average image size of
500x500 pixels. It is categorized into three classes which are benign,
malignant, and normal. In our experiments, we used an 8:2 ratio for training
and test sets and resized all images to 448 × 448 pixels as the input for the
model. RTX 3090Ti 24GB is used to train all models for 50 epochs with a batch
size of 8. Adam is used for optimization, with the initial learning rate set
to $10^{-4}$ and weight decay set to $10^{-5}$. We use accuracy and F1-score
metrics, which show the general effectiveness and predictive capacity of the
model under imbalanced data. F1-score, Intersection over Union (IoU), and Dice
Coefficient (Dice) are adopted in the evaluation of the segmentation task.

To determine the optimal value for hyperparameter $\lambda$ for the loss
function, we perform a grid search across values ranging from 0.1 to 0.9, as
depicted in 2. The best performance for both tasks is achieved at value
$\lambda$ = 0.7.

Table 1: Overall performance comparison on BUSI dataset.

Architecture | Backbone | Classification | Segmentation | Overall
---|---|---|---|---
Accuracy | F1-Score | IoU | Dice | F1-Score
Unet | Resnet50 | 0.682 | 0.648 | 0.682 | 0.756 | 0.756 | 0.690
Resnext50 | 0.873 | 0.663 | 0.692 | 0.758 | 0.774 | 0.702
WideResnet50 | 0.783 | 0.630 | 0.683 | 0.763 | 0.763 | 0.683
Efficientnet-b4 | 0.904 | 0.723 | 0.749 | 0.817 | 0.817 | 0.759
Unet++ | Resnet50 | 0.815 | 0.637 | 0.662 | 0.746 | 0.746 | 0.678
Resnext50 | 0.822 | 0.651 | 0.760 | 0.771 | 0.838 | 0.720
WideResnet50 | 0.790 | 0.664 | 0.680 | 0.761 | 0.761 | 0.699
Efficientnet-b4 | 0.847 | 0.650 | 0.769 | 0.795 | 0.837 | 0.725
FPN | Resnet50 | 0.688 | 0.627 | 0.606 | 0.601 | 0.668 | 0.626
Resnext50 | 0.745 | 0.652 | 0.714 | 0.752 | 0.794 | 0.703
WideResnet50 | 0.758 | 0.547 | 0.710 | 0.749 | 0.790 | 0.648
Efficientnet-b4 | 0.796 | 0.680 | 0.741 | 0.776 | 0.817 | 0.729
DeepLabV3+ | Resnet50 | 0.809 | 0.676 | 0.715 | 0.769 | 0.794 | 0.718
Resnext50 | 0.726 | 0.651 | 0.745 | 0.781 | 0.823 | 0.717
WideResnet50 | 0.809 | 0.610 | 0.722 | 0.797 | 0.797 | 0.691
Efficientnet-b4 | 0.892 | 0.694 | 0.798 | 0.813 | 0.864 | 0.760

####  4.0.2 Overall Results

Figure 2: F1-score and Overall score of segmentation and classification task

For the classification task, EfficientNet-B4 consistently outperforms nearly
all other backbone models across different architectures. It takes the top
position with DeeplabV3+, FPN, and Unet while taking the second position with
Unet++. Remarkably, EfficientNet-B4 maintains competitive performance despite
utilizing fewer parameters when employed as the segmentation head. Resnet50
and Resnext50 alternate between the second and third positions.

Regarding segmentation results, EfficientNet-B4 stands out for its consistent
and competitive performance across segmentation models, demonstrating its
efficacy. EfficientNet-B4 consistently achieves top rankings for IoU, Dice,
and F1-score, showcasing high performance across many architectures like
DeeplabV3+, FPN, Unet, and Unet++.

In summary, the combination of EfficientNet-B4 and Unet demonstrates the best
performance in both methods, encompassing most of the classification metrics
and Dice for segmentation. Meanwhile, the pairing of EfficientNet-B4 with
DeepLabV3++ achieves the highest IoU and F1-score for segmentation, with a
Dice result closely approaching the top rank. In conclusion, EfficientNet-B4
exhibits effectiveness across all segmentation methods.

####  4.0.3 Visualization

Figure 3: Visualization of prediction results of the naive approach and our
multi-task framework. (a) Breast ultrasound image; (b) Ground truth masks; (c)
Naive segmentation approach output; (d) Our multi-task framework output.

Qualitative predictions of the best-performing model are displayed in Fig. 3.
The segmentation predictions of the multi-task framework look strong, with
small gaps from the ground truth. In particular, the middle visualization
shows our proposed method, in contrast to the naive approach, can identify the
position of a very small tumor with a boundary quite well. Additionally, in
the bottom illustration, the tumor attribute is also precisely predicted, and
the segmentation result is extremely close to the ground truth. This
demonstrates how information from segmentation and classification tasks
complements each other and helps to enhance boundary and tumor attribute
prediction.

##  5 Conclusion

In this work, we used Breast Ultrasound Images to examine the efficacy of
multi-task learning. Our network with a shared backbone and two heads was
proposed to effectively conduct two tasks. Furthermore, we empirically
demonstrate the efficacy of the hyperparameter $\lambda$ weight in the loss
function to determine the potency of which each task can effectively
contribute to the others. We intend to extend this straightforward methodology
to tackle the gradient projection among different goal functions and the
dependency problem in multi-task learning.

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