UDC: 517.938.5

MSC 2010: 37D15

Elena V. Nozdrinova and Olga V. Pochinka

Stable arcs connecting polar cascades on a torus

E.V. Nozdrinova<EMAIL_ADDRESS>HSE, B. Pecherskaya street, 25/12, Nizhny
Novgorod 603150, Russia; ORCID: 0000-0001-5209-377X
O.V. Pochinka<EMAIL_ADDRESS>HSE, B. Pecherskaya street, 25/12,
Nizhny Novgorod 603150, Russia; ORCID: 0000-0002-6587-5305
Abstract In this paper, we obtain a solution to the 33rd Palis-Pugh problem
for polar gradient-like diffeomorphisms on a two-dimensional torus, under the
assumption that all non-wandering points are fixed and have a positive
orientation type.

Keywords: stable arc, flip, saddle-node, gradient-like diffeomorphism, two-
dimensional torus;

Acknowledgements. Study of the dynamics of diffeomorphisms of the class under
consideration is supported by RSF (Grant No. 17-11-01041), the construction of
a stable arc is supported by Laboratory of Dynamical Systems and Applications
NRU HSE, of the Ministry of science and higher education of the RF grant ag.
No 075-15-2019-1931

##  1 Introduction and formulation of results

The problem of the existence of an arc with no more than a countable (finite)
number of bifurcations connecting structurally stable systems (Morse-Smale
systems) on manifolds is on the list of fifty Palis-Pugh problems [27] under
number 33.

In 1976, S. Newhouse, J. Palis, F. Takens [20] introduced the concept of a
stable arc connecting two structurally stable systems on a manifold. Such an
arc does not change its quality properties with a small perturbation. In the
same year, S. Newhouse and M. Peixoto [22] proved the existence of a simple
arc (containing only elementary bifurcations) between any two Morse-Smale
flows. It follows from the result of G. Fleitas [8] that a simple arc
constructed by Newhouse and Peixoto can always be replaced by a stable one
[21]. For Morse-Smale diffeomorphisms given on manifolds of any dimension,
examples of systems that cannot be connected by a stable arc are known.

Obstruction appear already for orientation-preserving diffeomorphisms of the
circle $S^{1}$, which are connected by a stable arc only if the rotation
numbers coincide [23].

Beginning with dimension two, additional obstruction appear to the existence
of stable arcs between isotopic diffeomorphisms. They are associated with the
existence of periodic points [4], [25], heteroclinic intersections [17], wild
embeddings of separatrices [7], etc.

On the 6-dimensional sphere, examples of source-sink diffeomorphisms are known
that are not connected by any smooth arc [6], which, in fact, became the
source for constructing different smooth structures on a sphere of dimension
7. For $n=2,3$ the non-trivial fact of the existence of an arc without
bifurcations between two source-sink diffeomorphisms was established in [6],
[24].

A natural generalization of source-sink systems are polar diffeomorphisms –
gradient-like diffeomorphisms with a unique source and a unique sink. It
follows from Morse theory that such diffeomorphisms exist on any manifolds.

In this paper, we consider the class $G$ of polar gradient-like
diffeomorphisms on the two-dimensional torus $\mathbb{T}^{2}$, under the
assumption that all non-wandering points are fixed and have positive
orientation type. In chapter 2 it is established that any diffeomorphism $f\in
G$ has exactly two saddle points and is isotopic to the identity. Moreover,
all diffeomorphisms of the class under consideration are pairwise
topologically conjugate (see, for example, [5], [11]). Moreover, the closures
of stable (unstable) manifolds of saddle points of different diffeomorphisms
can belong to different homotopy classes of closed curves on the torus.
Therefore that in the general case there is no arc without bifurcations
connecting two diffeomorphisms of the class under consideration.

The main result of this work is the proof of the following theorem.

######  Theorem 1.

Any diffeomorphisms $f,f^{\prime}\in G$ can be connected by a stable arc with
a finite number of saddle-node bifurcations.

##  2 Diffeomorphisms of class $G$

###  2.1 On gradient-like flows on surfaces

In this section, we establish the basic dynamical properties of
diffeomorphisms $f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ from the class $G$.

Recall that a diffeomorphism $f$ is gradient-like if its non-wandering set
$\Omega_{f}$ consists of a finite number of hyperbolic points and the
invariant manifolds of different saddle points do not intersect. A
diffeomorphism $f$ is called polar if the set $\Omega_{f}$ contains exactly
two nodal points, namely, one sink and one source.

Fix system of generators of fundamental group of torus
$\mathbb{T}^{2}=\mathbb{S}^{1}\times\mathbb{S}^{1}$:

$a=\mathbb{S}^{1}\times\\{0\\}=<1,0>;\,\,b=\\{0\\}\times\mathbb{S}^{1}=<0,1>.$

Recall that the algebraic torus automorphism
$\widehat{L}:\mathbb{T}^{2}\to\mathbb{T}^{2}$,
$\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ is called the diffeomorphism
defined by the matrix $\,{L}=\left(\begin{array}[]{cc}\alpha&\beta\\\
\gamma&\delta\end{array}\right)$, belonging to the set $GL(2,\mathbb{Z})$
unimodular matrices – integer matrices with determinant $\pm 1$. That is

$\widehat{L}(x,y)=(\alpha x+\beta y,\gamma x+\delta y)\pmod{1}.$

######  Theorem 2.

Any diffeomorphism $f\in G$ has the following properties:

  1. 1.

Non-wandering set $\Omega_{f}$ of the diffeomorphism $f$ consists of exactly
four fixed hyperbolic points: the sink $\omega_{f}$, the source $\alpha_{f}$,
and the saddles $\sigma_{f}^{1},\sigma_{f}^{2}$, the closures of invariant
manifolds of which are closed curves:

$c^{s1}_{f}=cl\,W^{s}_{\sigma_{f}^{1}}=W^{s}_{\sigma_{f}^{1}}\cup\alpha_{f},\,\,c^{u1}_{f}=cl\,W^{u}_{\sigma_{f}^{1}}=W^{u}_{\sigma_{f}^{1}}\cup\omega_{f},$
$c^{s2}_{f}=cl\,W^{s}_{\sigma_{f}^{2}}=W^{s}_{\sigma_{f}^{2}}\cup\alpha_{f},\,\,c^{u2}_{f}=cl\,W^{u}_{\sigma_{f}^{2}}=W^{u}_{\sigma_{f}^{2}}\cup\omega_{f}.$

  2. 2.

There is only one choice of saddle points numbering
$\sigma_{f}^{1},\,\sigma_{f}^{2}$ and the orientation of the closures of their
invariant manifolds such that the curves $c^{s1}_{f},\,c^{u2}_{f}$ are of
homotopy type $<\mu_{f}^{1},\nu_{f}^{1}>$ and the curves
$c^{s2}_{f},\,c^{u1}_{f}$ are of homotopy type $<\mu_{f}^{2},\nu_{f}^{2}>$ in
the basis $a,b$, also $J_{f}=\begin{pmatrix}\mu^{1}_{f}&\mu_{f}^{2}\\\
\nu_{f}^{1}&\nu_{f}^{2}\end{pmatrix}$ is unimodular matrix with the following
properties:

a) $\mu^{1}_{f}\geq\mu^{2}_{f}\geq 0$,

b) $\nu^{1}_{f}>\nu^{2}_{f}$, if $\mu^{1}_{f}=\mu^{2}_{f}$,

c) $\nu^{2}_{f}=1$, if $\mu_{f}^{2}=0$.

  3. 3.

The diffeomorphism $f$ is isotopic to the identity map.

Proof:  Let $f\in G$. We prove all the points of the theorem sequentially.

1\. We denote by $k_{f}^{0},k_{f}^{1},k_{f}^{2}$ the number of sinks, saddles
and sources of the diffeomorphism $f$ respectively. According to [28], there
is a Morse function on the torus $\mathbb{T}^{2}$, the set of critical points
of which coincides with the set $\Omega_{f}$ and the indices of the critical
points coincide with the dimensions of the unstable manifolds of non-wandering
points of the diffeomorphism $f$. Then it follows from Morse inequalities
(see, for example, [18]) that

$k_{f}^{0}-k_{f}^{1}+k_{f}^{2}=0.$

Since the diffeomorphism $f$ is polar, then $k_{f}^{0}=k_{f}^{2}=1$ and
therefore $k_{f}^{1}=2$. Thus, the non-wandering set $\Omega_{f}$ of the
diffeomorphism $f$ consists of exactly four fixed hyperbolic points: the sink
$\omega_{f}$, the source $\alpha_{f}$ and the saddles
$\sigma_{f}^{1},\sigma_{f}^{2}$.

Since the invariant manifolds of different saddle points of the diffeomorphism
$f$ do not intersect, then, according to [13, proposition 2.1.3],

$cl(W^{u}_{\sigma_{f}^{1}})\setminus
W^{u}_{\sigma_{f}^{1}}=cl(W^{u}_{\sigma_{f}^{2}})\setminus
W^{u}_{\sigma_{f}^{2}}=\omega_{f},$ $cl(W^{s}_{\sigma_{f}^{1}})\setminus
W^{s}_{\sigma_{f}^{1}}=cl(W^{s}_{\sigma_{f}^{2}})\setminus
W^{s}_{\sigma_{f}^{2}}=\alpha_{f}$

and sets
$c^{u1}_{f}=cl\,W^{u}_{\sigma_{f}^{1}},\,\,c^{s1}_{f}=cl\,W^{s}_{\sigma_{f}^{1}},\,\,c^{u2}_{f}=cl\,W^{u}_{\sigma_{f}^{2}},\,\,c^{s2}_{f}=cl\,W^{s}_{\sigma_{f}^{2}}$
are homeomorphic to circles.

2\. Denote by $<\mu_{f}^{1},\nu_{f}^{1}>,\,\,<\mu_{f}^{2},\nu_{f}^{2}>$
homotopy types of oriented curves $c^{s1}_{f},\,c^{u1}_{f}$, respectively, in
the basis $a,b$. Let $J_{f}=\begin{pmatrix}\mu^{1}_{f}&\mu_{f}^{2}\\\
\nu_{f}^{1}&\nu_{f}^{2}\end{pmatrix}.$ The previous item implies, that the
closed curves $c^{s1}_{f},\,c^{u1}_{2}$ have a unique transverse intersection
point $\sigma_{f}^{1}$. Then the modulus of the intersection index of these
curves is one. Since this index coincides with the determinant of the matrix
$J_{f}$ (see, for example, [29, exercise 7, pp. 28]), the matrix $J_{f}$ is
unimodular. onsequently, the curves $c^{s1}_{f},\,c^{u1}_{f}$ are not
homotopic to zero. Without loss of generality, suppose the orientation on the
curves was chosen so that $\mu^{i}_{f}\geq 0$ and $\nu^{i}_{f}=1$, if
$\mu^{i}_{f}=0$ (these conditions uniquely orient the curves).

Since the diffeomorphism $f$ is gradient-like, then the curves
$c^{u1}_{f},\,c^{s2}_{f}$ and $c^{u2}_{f},\,c^{s1}_{f}$ are pairwise disjoint.
Then the oriented curves $c^{u2}_{f},\,c^{s2}_{f}$ also have homotopy types
$<\mu_{f}^{1},\nu_{f}^{1}>$ and $<\mu_{f}^{2},\nu_{f}^{2}>$, respectively, in
the basis $a,b$ (see, for example, [29, Theorem 13, pp. 25]). Without loss of
generality, suppose the numbering of saddle points is chosen so that
$\mu^{1}_{f}\geq\mu^{2}_{f}$ and, if $\mu^{1}_{f}=\mu^{2}_{f}$, then
$\nu^{1}_{f}>\nu^{2}_{f}$ (these conditions determine the numbering of saddle
points uniquely).

3\. The diffeomorphism $f$ induces an isomorphism
$f_{*}:\pi_{1}(\mathbb{T}^{2})\to\pi_{1}(\mathbb{T}^{2})$ in the fundamental
group $\pi_{1}(\mathbb{T}^{2})$ of the torus, isomorphic to the abelian group
$\mathbb{Z}^{2}$. Then the isomorphism $f_{*}$ is determined by the unimodular
integer matrix uniquely $L_{f}=\begin{pmatrix}\alpha&\beta\\\
\gamma&\delta\end{pmatrix},$ taking basis $a,b$ into basis
$<\alpha,\gamma>,\,<\beta,\delta>$. Diffeomorphism $f$ isotopic to the
identical iff $L_{f}=E$, where $E=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
(see, for example, [29, lemma 3, pp. 26]). Let us show that $L_{f}=E$ for
$f\in G$.

Let $h=\widehat{J}^{-1}_{f}f\widehat{J}_{f}:\mathbb{T}^{2}\to\mathbb{T}^{2}$.
By construction, the diffeomorphism $h$ is smoothly conjugate to the
diffeomorphism $f$. Moreover, the curves $c^{s1}_{h},c^{u2}_{h}$ and
$c^{s2}_{h},c^{u1}_{h}$ are of homotopy types $<1,0>$ and $<0,1>$,
respectively. Since all these curves are $h$-invariant, then $L_{h}=E$. Since
$f=\widehat{J}_{f}h\widehat{J}_{f}^{-1}$, then
$L_{f}=J_{f}L_{h}J_{f}^{-1}=J_{f}EJ_{f}^{-1}=E$. $\square$

##  3 Construction of a stable arc between diffeomorphisms of the class $G$

In this section, we outline the proof of theorem 1 with references to the
statements that will be proved in the following sections. Let us first give
the necessary definitions.

Consider a 1-parametric family of diffeomorphisms (an arc) $\varphi_{t}:M\to
M,t\in[0,1]$. An arc $\varphi_{t}$ is called smooth, if map $F:M\times[0,1]\to
M$, defined by the formula $F(x,t)=\varphi_{t}(x)$ is smooth.

The smooth arc $\varphi_{t}$ is called a smooth product of the smooth arcs
$\varphi^{1}_{t}$ and $\varphi^{2}_{t}$ such that
$\varphi^{1}_{1}=\varphi^{2}_{0}$, if
$\varphi_{t}=\begin{cases}\varphi^{1}_{2\tau(t)},~{}0\leqslant
t\leqslant\frac{1}{2},\\\ \varphi^{2}_{2\tau(t)-1},\frac{1}{2}\leqslant
t\leqslant 1,\end{cases}$ where $\tau:[0,1]\to[0,1]$ is a smooth monotone map
such that $\tau(t)=0$ for $0\leqslant t\leqslant\frac{1}{3}$ and $\tau(t)=1$
for $\frac{2}{3}\leqslant t\leqslant 1$. We will write
$\varphi_{t}=\varphi^{1}_{t}*\varphi^{2}_{t}$.

Following [21], an arc $\varphi_{t}$ is called stable if it is an inner point
of the equivalence class with respect to the following relation: two arcs
$\varphi_{t}$, $\varphi^{\prime}_{t}$ are called conjugate if there are
homeomorphisms $h:[0,1]\to[0,1],\,H_{t}:M\to M$ such that
$H_{t}\varphi_{t}=\varphi^{\prime}_{{h(t)}}H_{t},t\in[0,1]$ and $H_{t}$
continuously depend on $t$.

In [21] also established that the arc $\\{\varphi_{t}\\}$, consisting of
diffeomorphisms with a finite limit set, is stable iff all its points are
structurally stable diffeomorphisms with the exception of a finite number of
bifurcation points, $\varphi_{b_{i}},i=1,\dots,q$ such that $\varphi_{b_{i}}$:

1) has no cycles;

2) has a unique non-hyperbolic periodic orbit, which is a non-critical saddle-
node or flip;

3) the invariant manifolds of all periodic points of the
diffeomorphism$\varphi_{b_{i}}$ intersect transversally;

4) the transition through $\varphi_{b_{i}}$ is a generically unfolded saddle-
node or period doubling bifurcation, wherein the saddle-node point is non-
critical.

Recall the definition of unfolding generically arc $\varphi_{t}$ through the
saddle-node or flip. We give the definition for a fixed non-hyperbolic point,
in the case when it has a period $k>1$, a similar definition is given for the
arc $\varphi^{k}_{t}$.

An arc $\\{\varphi_{t}\\}\in\mathcal{Q}$ unfolds generically through a saddle-
node bifurcation $\varphi_{b_{i}}$ (Fig. 1), if in some neighborhood of the
nonhyperbolic point $(p,b_{i})$ the arc $\varphi_{t}$ is conjugate to

$\tilde{\varphi}_{\tilde{t}}(x_{1},x_{2},\dots,x_{1+n_{u}},x_{2+n_{u}},\dots,x_{n})=$
$\left(x_{1}+\frac{x_{1}^{2}}{2}+\tilde{t},\pm 2x_{2},\dots,\pm
2x_{1+n_{u}},\frac{\pm x_{2+n_{u}}}{2},\dots,\frac{\pm x_{n}}{2}\right),$

where $(x_{1},\dots,x_{n})\in\mathbb{R}^{n},\,|x_{i}|<1/2,\,|\tilde{t}|<1/10$.

Figure 1: Saddle-node bifurcation

In the local coordinates $(x_{1},\dots,x_{n},\tilde{t})$ the bifurcation
occurs at time $\tilde{t}=0$ and the origin $O\in\mathbb{R}^{n}$ is a saddle-
node point. The axis $Ox_{1}$ is called a central manifold $W^{c}_{O}$, the
half-space $\\{(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}:\,x_{1}\geq
0,\,x_{2+n_{u}}=\dots=x_{n}=0\\}$ is the unstable manifold $W^{u}_{O}$, half-
space $\\{(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}:\,x_{1}\leq
0,\,x_{2}=\dots=x_{1+n_{u}}=0\\}$ is the stable manifold $W^{s}_{O}$ of the
point $O$.

If $p$ is a saddle-nodal point of the diffeomorphism $\varphi_{b_{i}}$, then
there exists a unique $\varphi_{b_{i}}$\- invariant foliation $F^{ss}_{p}$
with smooth leaves such that $\partial W^{s}_{p}$ is a leave of this foliation
[15]. $F^{ss}_{p}$ is called a strongly stable foliation (Fig. 2). A similar
strongly unstable foliation is denoted by $F^{uu}_{p}$. A point $p$ is called
$s$-critical, if there exists some hyperbolic periodic point $q$ such that
$W^{u}_{q}$ non-transversally intersect some leaf of the foliation
$F^{ss}_{p}$; $u$-criticality is defined similarly. Point $p$ is called

\- semi-critical if it is either $s$\- or $u$-critical;

\- bi-critical if it is $s$\- and $u$-critical;

\- non-critical if it is not semi-critical111For the first time, the effect of
arc instability in a neighborhood of a non-critical saddle- was discovered in
1974 by V. Afraimovich and L. Shilnikov [1], [2]. The existence of invariant
foliations $F^{ss}_{p},\,F^{uu}_{p}$ was also proved earlier in the works of
V. Lukyanov and L. Shilnikov[16]..

Figure 2: Strongly stable and unstable foliations

Let ${f},f^{\prime}\in{G}$. Let us prove that the diffeomorphisms
$f,f^{\prime}$ are connected by a stable arc
$\varphi_{t}:\mathbb{T}^{2}\rightarrow\mathbb{T}^{2},t\in[0,1]$, whose
diffeomorphisms are gradient-like except for a finite number of generically
unfolding non-critical saddle-node bifurcations.

Proof:  In section 4 for any unimodular matrix
$J=\begin{pmatrix}\mu^{1}&\mu^{2}\\\ \nu^{1}&\nu^{2}\end{pmatrix}$ such that
$\mu^{1}\geq\mu^{2}\geq 0$ and $\nu^{1}>\nu^{2}$, if $\mu^{1}=\mu^{2}$, we
construct the a model diffeomorphism ${f}_{J}\in{G}$, for which $J_{f_{J}}=J$.
According to lemma 5.1, every diffeomorphism $f\in G$ can be connected with
the model diffeomorphism $f_{J_{f}}$ by an arc without bifurcations $H_{f,t}$.
According to lemma 7.2, the diffeomorphism $f_{J}$ can be joined by an arc
$H_{J,t}$ with a finite number of generically unfolding non-critical saddle-
node bifurcations with diffeomorphism $f_{0}$. Then the desired arc
$\varphi_{t}$ has the form

$\varphi_{t}=H_{f,t}*H_{J_{f},t}*H_{J_{f^{\prime}},1-t}*H_{f^{\prime},1-t}.$

$\square$

##  4 Construction of model diffeomorphisms in the class $G$

In this section for any unimodular matrix $J=\begin{pmatrix}\mu^{1}&\mu^{2}\\\
\nu^{1}&\nu^{2}\end{pmatrix}$ such that $\mu^{1}\geq\mu^{2}\geq 0$ and
$\nu^{1}>\nu^{2}$ if $\mu^{1}=\mu^{2}$ we construct a model diffeomorphism
${f}_{J}\in{G}$, for which $J_{f_{J}}=J$.

The simplest example of a diffeomorphism from the class $G$ is the direct
product of two copies of a source-sink diffeomorphism on the circle
$\mathbb{S}^{1}$, denote it by $f_{0}$. First, we construct a source-sink
diffeomorphism on the circle. In order to do this consider the map
$\bar{F}_{0}:\mathbb{R}\to\mathbb{R}$, given by the formula:

$\bar{F}_{0}(x)=x-\frac{1}{4\pi}sin\left(2\pi\left(x-\frac{1}{4}\right)\right).$

By construction $x=\frac{1}{4}$ and $x=\frac{3}{4}$ – are fixed points of the
map $\bar{F}_{0}$ on the segment $[0,1]$ (Fig. 3).

Consider the projection $\pi:\mathbb{R}\to\mathbb{S}^{1}$ given by the formula
$\pi(x)=e^{2\pi ix}$. As $\bar{F}_{0}$ is strictly increasing and satisfy the
condition $\bar{F}_{0}(x+1)=\bar{F}_{0}(x)+1$, there is a diffeomorphism
projecting it to the circle

$F_{0}=\pi\bar{F}_{0}\pi^{-1}:\mathbb{S}^{1}\to\mathbb{S}^{1}.$

Figure 3: Graph of the map $\bar{F}_{0}$

By construction, the diffeomorphism $F_{0}$ has a fixed hyperbolic sink at the
point $N=\pi\left(\frac{1}{4}\right)$ and a fixed hyperbolic source at the
point $S=\pi\left(\frac{3}{4}\right)$.

Define the diffeomorphism $f_{0}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ by the
formula (Fig. 4)

$f_{0}(z,w)=(F_{0}(z),F_{0}(w)),\,z,w\in\mathbb{S}^{1}.$

Figure 4: Cartesian square of the diffeomorphism $F_{0}$

By construction the diffeomorphism $f_{0}$ contain a fixed hyperbolic sink at
the point $\omega=(N,N)$, hyperbolic source $\alpha=(S,S)$ and has two saddle
points $\sigma_{1}=(N,S),\sigma_{2}=(S,N)$ (Fig. 5). Moreover, the closures of
their invariant manifolds lie in the classes of generators $a,b$. Exactly,

$c^{s1}_{f_{0}}=cl\,W^{s}_{\sigma_{1}}=\mathbb{S}^{1}\times\\{S\\},\,c^{u1}_{f_{0}}=cl\,W^{u}_{\sigma_{1}}=\\{N\\}\times\mathbb{S}^{1},$
$c^{s2}_{f_{0}}=cl\,W^{s}_{\sigma_{2}}=\\{S\\}\times\mathbb{S}^{1},\,c^{u2}_{f_{0}}=cl\,W^{u}_{\sigma_{2}}=\mathbb{S}^{1}\times\\{N\\}.$

Figure 5: Diffeomorphism $f_{0}$.

Let $f_{J}=\widehat{J}f_{0}\widehat{J}^{-1}$. We will call the diffeomorphism
$f_{J}$ model diffeomorphism. By construction $f_{E}=f_{0}$.

##  5 Construction the arc $H_{f,t}$

###  5.1 Outline of the construction

######  Lemma 5.1.

Every diffeomorphism $f\in G$ is connected by an arc without bifurcations
$H_{f,t}$ with the diffeomorphism $f_{J_{f}}$.

Proof:  We give a scheme of the proof of Lemma 5.1 with links to statements
that will be proved below.

Consider two possible cases: 1) $J_{f}=E$, 2) $J_{f}\neq E$.

1) According to the lemma 5.2, we assume that the diffeomorphism $f$ in a
neighborhood of the sink $\omega_{f}$ has a local chart
$(U_{\omega_{f}},\psi_{\omega_{f}}),\psi_{\omega_{f}}:U_{\omega_{f}}\to\mathbb{R}^{2}$
such that the diffeomorphism
$g=\psi_{\omega_{f}}f\psi_{\omega_{f}}^{-1}:\mathbb{R}^{2}\to\mathbb{R}^{2}$
has the form $g(x,y)=\left(\frac{x}{2},\frac{y}{2}\right)$. Moreover,
according to the lemma 6.1, we can assume that

$\psi_{\omega_{f}}(c^{u2}_{f}\cap U_{\omega_{f}})\subset
Ox,\,\psi_{\omega_{f}}(c^{u1}_{f}\cap U_{\omega_{f}})\subset Oy,$

that is, the curves $c^{u2}_{f},\,c^{u1}_{f}$ are smooth. Since they lie in
the same homotopy class with curves
$c^{u2}_{f_{0}}=\mathbb{S}^{1}\times\\{N\\},\,c^{u1}_{f_{0}}=\\{N\\}\times\mathbb{S}^{1}$,
then, by Proposition 6.1, there exists a diffeomorphism
$\xi:\mathbb{T}^{2}\to\mathbb{T}^{2}$ smoothly isotopic to the identity such
that

$\xi(c^{u1}_{f})=c^{u1}_{f_{0}},\,\xi(c^{u2}_{f})=c^{u2}_{f_{0}}.$

Let $\xi_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ – be a smooth isotopy such that
$\xi_{0}=id$ and $\xi_{1}=\xi$. Then the arc
$\xi_{t}f\xi_{t}^{-1}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ connects the
diffeomorphism $f$ with the diffeomorphism $f_{1}=\xi f\xi^{-1}\in G$ such
that

$c^{u1}_{f_{1}}=c^{u1}_{f_{0}},\,c^{u2}_{f_{1}}=c^{u2}_{f_{0}}.$

Since the diffeomorphisms $f_{1}$ and $f_{0}$ are topologically conjugate on
the closures of unstable saddle manifolds, according to the lemma 6.2, there
exists an arc without biffurcations connecting $f_{1}$ with the diffeomorphism
$f_{2}\in G$, which coincides with $f_{0}$ in some neighborhoods
$K_{1}^{u},\,K^{u}_{2}$ of curves $c^{u1}_{f_{0}},\,c^{u2}_{f_{0}}$.

The set $A=c^{u1}_{f_{0}}\cup c^{u2}_{f_{0}}$ an attractor of the
diffeomorphisms $f_{2}$, $f_{0}$ and the set $U_{A}=K_{1}^{u}\cup K^{u}_{2}$
is its neighborhood. By construction, the space $\mathbb{T}^{2}\setminus
U_{A}$ is homeomorphic to a two-dimensional disk and contains in its interior
the points $\alpha$ and $\alpha_{f_{2}}$. According to [14, Theorem 3.2,
chapter 8], there exists a diffeomorphism smoothly isotopic to the identity
$\eta:\mathbb{T}^{2}\to\mathbb{T}^{2}$ such that

$\eta(\alpha_{f})=\alpha,\,\eta|_{U_{A}}=id.$

According to the isotopy extension theorem (see, for example, [18, Theorem
5.8]), there exists an arc $\eta_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ such
that $\eta_{0}=id$, $\eta_{1}=\eta$ and $\eta_{t}|_{U_{A}}=id$. Then the arc
$\eta_{t}f_{2}\eta_{t}^{-1}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ connects the
diffeomorphism $f_{2}$ with the diffeomorphism $f_{3}=\eta f_{2}\eta^{-1}\in
G$ such that $f_{2}|_{U_{A}\cup\alpha}=f_{0}|_{U_{A}\cup\alpha}$. Moreover,
according to Lemma 5.2, we can assume that the diffeomorphism $f_{3}$
coincides with the diffeomorphism $f_{0}$ in a neighborhood of the source
$\alpha$. Moreover, according to Lemma 6.3, exists an arc without
biffurcations connecting the diffeomorphism $f_{3}$ with the diffeomorphism
$f_{0}$.

2) For a diffeomorphism $f\in G$ such that $J_{f}\neq E$ let
$h=\widehat{J}^{-1}_{f}f\widehat{J}_{f}:\mathbb{T}^{2}\to\mathbb{T}^{2}$. Then
the diffeomorphism $h$ belongs to the class $G$ and $J_{h}=E$. According to
item 1) there exists an arc without bifurcations
$\zeta_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ such that $\zeta_{0}=h$ and
$\zeta_{1}=f_{0}$. Then

$H_{f,t}=\widehat{J}_{f}\zeta_{t}\widehat{J}^{-1}_{f}:\mathbb{T}^{2}\to\mathbb{T}^{2}$

– is the required isotopy connecting the diffeomorphism $H_{f,0}=f$ with
diffeomorphism $H_{f,1}=\widehat{J}_{f}f_{0}\widehat{J}^{-1}_{f}=f_{J_{f}}$.
$\square$

###  5.2 Reduction of a structurally stable diffeomorphism to a linear
diffeomorphism in neighborhoods of hyperbolic periodic points

Let $p$ be a hyperbolic fixed point of a diffeomorphism $f:M^{n}\to M^{n}$. A
type of point $p$ is the set of parameters $(q_{p},\nu_{p},\mu_{p})$, where
$q_{p}=\dim~{}W^{u}_{p}$, $\nu_{p}=+1~{}(-1)$, if $f|_{W^{u}_{p}}$ preserves
(reverses) orientation and $\mu_{p}=+1~{}(-1)$, if $f|_{W^{s}_{p}}$ preserves
(reverses) orientation. According to [26, Theorem 5.5], the diffeomorphism $f$
in some neighborhood of a point $p$ of type $(q_{p},\nu_{p},\mu_{p})$ is
topologically conjugate to a linear diffeomorphism of the space
$\mathbb{R}^{n}$, defined by

$A_{p}=\begin{pmatrix}\nu_{p}\cdot 2&0&\dots&0&0&0&\dots&0\cr
0&2&\dots&0&0&0&\dots&0\cr&&\ddots\cr 0&0&\dots&2&0&0&\dots&0\cr
0&0&\dots&0&\mu_{p}\cdot 1/2&0&\dots&0\cr
0&0&\dots&0&0&1/2&\dots&0\cr&&&&&&\ddots\cr
0&0&\dots&0&0&0&\dots&1/2\cr\end{pmatrix},$

the number of the rows of $A_{p}$, containing $2$ (including $\nu_{p}\cdot
2$), equals $q_{p}$. Denote by $\bar{A}_{p}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ a
linear diffeomorphism defined by $A_{p}$. Let $\mathbb{R}^{u}=Ox_{1}\dots
x_{q_{p}},\,\mathbb{R}^{s}=Ox_{q_{p}+1}\dots x_{n}$,
$\bar{A}^{u}_{p}=\bar{A}_{p}|_{\mathbb{R}^{u}}$ and
$\bar{A}^{s}_{p}=\bar{A}_{p}|_{\mathbb{R}^{s}}$. Then in local coordinates
$x^{u}=(x_{1},\dots,x_{q_{p}})\in\mathbb{R}^{u},\,x^{s}=(x_{q_{p}+1},\dots,x_{n})\in\mathbb{R}^{s}$
diffeomorphism $\bar{A}_{p}$ has a form

$\bar{A}_{p}(x^{u},x^{s})=(\bar{A}^{u}_{p}(x^{u}),\bar{A}^{s}_{p}(x^{s})).$

######  Lemma 5.2.

Let a structurally stable diffeomorphism $\varphi_{0}:M^{n}\rightarrow M^{n}$
has an isolated hyperbolic fixed point $p$, let $(U_{0},\psi_{0})$ be a local
manifold chart of $M^{n}$ such that $p\in U_{0}$, $\psi_{0}(p)=O$ and $U_{0}$
does not contain non-wandering points of the diffeomorphism $\varphi_{0}$
other than $p$. Therefore, there exists neighborhoods $U_{1},U_{2}$ of the
point $p$, $U_{2}\subset U_{1}\subset U_{0}$ and the arc $\varphi_{t}:M^{n}\to
M^{n},t\in[0,1]$ without bifurcations such that:

1) the diffeomorphism $\varphi_{t},\,t\in[0,1]$ coincides with the
diffeomorphism $\varphi_{0}$ outside the set $U_{1}$;

2) $p$ is an isolated hyperbolic point for each $\varphi_{t}$;

3) $W^{s}_{p}(\varphi_{t})=W^{s}_{p}(\varphi_{0})$ and
$W^{u}_{p}(\varphi_{t})=W^{u}_{p}(\varphi_{0})$ outside the set $U_{1}$;

4) the diffeomorphism $\psi_{0}\varphi_{1}\psi_{0}^{-1}$ coincides with the
diffeomorphism $\bar{A}_{p}$ on the set $\psi_{0}(U_{2})$.

Proof:  For $r>0$ let
$B_{r}=\\{(x_{1},\dots,x_{n})\in\mathbb{R}^{n}:\sum\limits_{i=1}^{n}x_{i}^{2}\leqslant
r^{2}\\}$,
$B^{u}_{r}=\\{(x_{1},\dots,x_{q_{p}})\in\mathbb{R}^{u}:\sum\limits_{i=1}^{q_{p}}x_{i}^{2}\leqslant
r^{2}\\}$ and
$B^{s}_{r}=\\{(x_{q_{p}+1},\dots,x_{n})\in\mathbb{R}^{s}:\sum\limits_{i=q_{p}+1}^{n}x_{i}^{2}\leqslant
r^{2}\\}$.

By virtue of the structural stability of the diffeomorphism $\varphi_{0}$ any
diffeomorphism sufficiently close to $\varphi_{0}$ in the $C^{1}$-topology can
be joined with $\varphi_{0}$ by an arc without bifurcations. According to
Franks’ lemma [9] 222In Franks’ lemma, in a neighborhood $U_{p}$ of the fixed
point $p$ of the diffeomorphism $f:M^{n}\to M^{n}$ we consider the local chart
$(U_{p},\psi_{p})$ where $\psi^{-1}_{p}=exp:T_{x}M^{n}\to U_{p}$– exponential
map. Then in these local coordinates the diffeomorphism $f$ has the form
$\hat{f}=exp^{-1}fexp:\mathbb{R}^{n}\to\mathbb{R}^{n}$. The Franks lemma is
that in any neighborhood of a diffeomorphism $f$ there exists a diffeomorphism
$g$ having a fixed point $p$ and a linear local representation
$\hat{g}=exp^{-1}gexp$ if it is close enough to $Df_{p}$. Thus, in any
neighborhood of the diffeomorphism $f$ there exists a diffeomorphism $g$,
having a fixed point $p$ and a linear local representation given by a matrix
all of whose eigenvalues are pairwise different. we can assume that the
diffeomorphism $\bar{\varphi}_{0}=\psi_{0}\varphi_{0}\psi_{0}^{-1}$ in some
ball $B_{r_{0}}\subset\psi_{0}(U_{0})$ coincides with the linear
diffeomorphism $\bar{\Phi}_{p}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ given by a
matrix $\Phi_{p}$ all of whose eigenvalues are pairwise different. Then the
diffeomorphism $\bar{\Phi}_{p}$ is smoothly conjugate to the linear
diffeomorphism $\bar{Q}_{p}$ given by the normal Jordan form $Q_{p}$ of the
matrix $\Phi_{p}$ (see, for example, [10, Chapter 3]). That is, there exists
an orientation-preserving diffeomorphism $\xi:\mathbb{R}^{n}\to\mathbb{R}^{n}$
such that $\bar{Q}_{p}=\xi\bar{\Phi}_{p}\xi^{-1}$. According to [18, section
6, Lemma 2], $\xi$ is isotopic to the identity, which means there is an
isotopy $\xi_{t}$ from $\xi_{0}=id$ to $\xi_{1}=\xi$.

According to the isotopy extension theorem (see, for example, [18, Theorem
5.8]), there is an isotopy $\Xi_{t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ between
the identity map $\Xi_{0}=id$ and the diffeomorphism $\Xi_{t}$ coincides with
$\xi_{t}$ on $B_{r_{2}}$ and is the identity map outside $B_{r_{1}}$ for some
$r_{2}<r_{1}<r_{0}$.

Thus, the arc
$\bar{\eta}_{t}=\Xi_{t}\bar{\Phi}_{p}\Xi_{t}^{-1}:\mathbb{R}^{n}\to\mathbb{R}^{n}$
connects the diffeomorphism $\bar{\eta}_{0}=\bar{\Phi}_{p}$ with a
diffeomorphism $\bar{\eta}_{1}$, coinciding with $\bar{Q}_{p}$ on $B_{r_{2}}$
and with $\bar{\Phi}_{p}$ outside $B_{r_{1}}$. Additionally, $\bar{\eta}_{t}$
is an arc without bifurcations $O$, an isolated hyperbolic point for each
$\bar{\eta}_{t}$ and $W^{s}_{O}(\bar{\eta}_{t})=W^{s}_{O}(\bar{\Phi}_{p})$,
$W^{u}_{O}(\bar{\eta}_{t})=W^{u}_{O}(\bar{\Phi}_{p})$ outside the set
$B_{r_{1}}$.

If $Q_{p}=A_{p}$, then the lemma is proved. Otherwise, due to the fact that
the eigenvalues of the matrix $Q_{p}$ are pairwise different, it has a quasi-
diagonal form with blocks consisting either of eigenvalues or of matrices of
the form $\left(\begin{array}[]{cccc}\alpha&\beta\\\ -\beta&\alpha\\\
\end{array}\right),$ where $0<\alpha^{2}+\beta^{2}<1$ or
$\alpha^{2}+\beta^{2}>1$. Then the diffeomorphism $\bar{Q}_{p}$ has the form

$\bar{Q}_{p}(x^{u},x^{s})=(\bar{Q}^{u}_{p}(x^{u}),\bar{Q}^{s}_{p}(x^{s})),$

where $(\bar{Q}^{u}_{p})^{-1}(B^{u}_{r})\subset int\,B^{u}_{r}$ for every disk
$B^{u}_{r}$ and $\bar{Q}^{s}_{p}(B^{s}_{r})\subset int\,B^{s}_{r}$ for every
disk $B^{s}_{r}$. Choose $r_{3}<r_{2}$ in such a way that
$B^{u}_{r_{3}}\times(\bar{Q}^{s}_{p})^{-1}(B^{s}_{r_{3}})\subset
int\,B_{r_{2}}$. Choose $r^{u}_{4},r^{s}_{4}\in(r_{3}/2,r_{3})$ in such a way
that $(\bar{Q}^{u}_{p})^{-1}(B^{u}_{r_{3}})\subset int\,B^{u}_{r_{4}^{u}}$ and
$\bar{Q}^{s}_{p}(B^{s}_{r_{3}})\subset int\,B_{r_{4}^{s}}$.

In the proof of the proposition 5.4 of monography[26] arcs
$\bar{\tau}^{u}_{t}:\mathbb{R}^{u}\to\mathbb{R}^{u},\,\bar{\tau}^{s}_{t}:\mathbb{R}^{s}\to\mathbb{R}^{s}$
are constructed composing by linear hyperbolic contractions such that

  * •

$(\bar{\tau}^{u}_{t})^{-1}(B^{u}_{r})\subset int\,B^{u}_{r}$ for any disk
$B^{u}_{r}$ and $\bar{\tau}^{s}_{t}(B^{s}_{r})\subset int\,B^{s}_{r}$ for any
disk $B^{s}_{r}$;

  * •

$\bar{\tau}^{u}_{0}=\bar{Q}^{u}_{p},\,\bar{\tau}^{u}_{1}=\bar{A}^{u}_{p}$ and
$\bar{\tau}^{s}_{0}=\bar{Q}^{s}_{p},\,\bar{\tau}^{s}_{1}=\bar{A}^{s}_{p}$.

Consider isotopies
$\bar{\lambda}^{u}_{t}=\bar{Q}^{u}_{p}\bar{\tau}^{u}_{t},\,\bar{\lambda}^{s}_{t}=(\bar{Q}^{s}_{p})^{-1}\bar{\tau}^{s}_{t}$,
which takes the identity maps $\bar{\lambda}^{u}_{0}=\bar{\lambda}^{s}_{0}=id$
to diffeomorphism
$\bar{\lambda}^{u}_{1}=\bar{Q}^{u}_{p}(\bar{A}^{u}_{p})^{-1},\,\bar{\lambda}^{s}_{1}=(\bar{Q}^{s}_{p})^{-1}\bar{A}^{s}_{p}$,
respectively. By construction,
$\bar{\lambda}^{u}_{t}(B^{u}_{r_{3}})\subset\bar{Q}^{u}_{p}(B_{r^{u}_{4}})$
and
$\bar{\lambda}^{s}_{t}(B^{s}_{r_{3}})\subset(\bar{Q}^{s}_{p})^{-1}(B_{r^{s}_{4}})$
for each $t\in[0,1]$. Thus, by virtue of the isotopy extension theorem (see
e.g., [18, Theorem 5.8]), there exist isotopies
$\bar{\Lambda}^{u}_{t}:\mathbb{R}^{u}\to\mathbb{R}^{u},\,\bar{\Lambda}^{s}_{t}:\mathbb{R}^{s}\to\mathbb{R}^{s}$
taking the identity map $\bar{\Lambda}^{u}_{0}=\bar{\Lambda}^{s}_{0}=id$ which
coincide with $\bar{\lambda}^{u}_{t},\,\bar{\lambda}^{s}_{t}$ on
$B^{u}_{r_{3}},\,B^{s}_{r_{3}}$ and are exactly identity outside
$\bar{Q}^{u}_{p}(B_{r^{u}_{4}}),\,(\bar{Q}^{s}_{p})^{-1}(B_{r^{s}_{4}})$,
respectively. Let

$\bar{\Lambda}_{t}(x^{u},x^{s})=((\bar{\Lambda}^{u}_{t})^{-1}\bar{Q}^{u}_{p}(x^{u}),\bar{Q}^{s}_{p}\bar{\Lambda}^{s}_{t}(x^{s})).$

Let us denote an arc coinciding with $\bar{\eta}_{1}$ outside $B_{r_{2}}$ and
with $\bar{\Lambda}_{t}$ on $B_{r_{2}}$ by
$\bar{\zeta}_{t}:\mathbb{R}^{n}\to\mathbb{R}^{n}$. Choose $r_{5}<r_{4}$, such
that $B_{r_{5}}\subset\bar{Q}^{u}_{p}(B^{u}_{r_{3}})\times B^{s}_{r_{3}}$. Let
$\bar{U}_{2}=B_{r_{5}}$, $\bar{U}_{1}=B_{r_{1}}$ and

$\bar{\varphi}_{t}=\bar{\eta}_{t}*\bar{\zeta}_{t}.$

Then $\bar{\varphi}_{t}$ is an arc without bifurcations, which coincide with
$\bar{\varphi}_{0}$ outside $\bar{U}_{1}$, $O$ is an isolated hyperbolic point
for each $\bar{\varphi}_{t}$,
$W^{s}_{O}(\bar{\varphi}_{t})=W^{s}_{O}(\bar{\varphi}_{0})$,
$W^{u}_{O}(\bar{\varphi}_{t})=W^{u}_{O}(\bar{\varphi}_{0})$ outside the set
$\bar{U}_{1}$ and the diffeomorphism $\bar{\varphi}_{1}$ coinsedes with the
diffeomorphism $\bar{A}_{p}$ on $\bar{U}_{2}$. So, the arc
$\varphi_{t}:M^{n}\to M^{n}$ which coincides with
$\psi_{0}^{-1}\bar{\varphi}_{1}\psi_{0}$ on $U_{0}$ and with $\varphi_{0}$
outside $U_{0}$ satisfies all the conditions of the lemma in
$U_{1}=\psi_{0}^{-1}(\bar{U}_{1})$ and $U_{2}=\psi_{0}^{-1}(\bar{U}_{2})$.
$\square$

######  Collary 5.1.

The statement analogous to 5.2 holds, if point $p$ is periodic with period
$m$. To prove that it is sufficient to consider isotopy for the diffeomorphism
$f^{m}$ with fixed point $p$, which exists according to lemma 5.2, and extend
it along the neighbourhood of point $p$ orbit with $f$.

##  6 Straightening of invariant saddle manifolds

The invariant manifold of the saddle point of a Morse-Smale diffeomorphism is
always a smooth submanifold. However, its closure, even in a gradient-like
case, may not even be a topological submanifold (Fig. 6). In this section, we
present statements and facts that allow us to connect any gradient-like
2-diffeomorphism by an arc without bifurcations with a diffeomorphism for
which the closures of the invariant manifolds of all saddle points are smooth
submanifolds.

######  Proposition 6.1.

([29, Theorem 13, p. 25] and [19], [13, Proposition 10.60] and [13, Statement
10.51]) Suppose that on the torus $\mathbb{T}^{2}$ there are
$p\in\mathbb{N}\cup\\{0\\}$ pairwise disjoint closed smooth curves
$c_{1},\dots,c_{p}$, of homotopy type $<1,0>$ and $q\in\mathbb{N}\cup\\{0\\}$
pairwise disjoint closed smooth curves $d_{1},\dots,d_{q}$, of homotopy type
$<0,1>$ such that each pair of curves $c_{i},\,d_{j}$ intersects transversally
at one point. Let $C_{p}$ be a disjunct union of $p$ circles of the form
$\mathbb{S}^{1}\times\\{w\\},\,w\in\mathbb{S}^{1}$ and $D_{q}$ be the disjoint
union of $q$ circles of the form
$\\{z\\}\times\mathbb{S}^{1},\,z\in\mathbb{S}^{1}$.

Then there is an isotopic to identity diffeomorphism
$\xi:\mathbb{T}^{2}\to\mathbb{T}^{2}$ such that $\xi(c_{1}\cup\dots\cup
c_{p})=C_{p}$ and $\xi(d_{1}\cup\dots\cup d_{q})=D_{q}$.

######  Lemma 6.1.

Let a diffeomorphism $\varphi_{0}:M^{2}\to M^{2}$ has a hyperbolic sink
$\omega_{0}$ and hyperbolic saddles $\sigma_{1},\dots,\sigma_{k}$ such that
their unstable separatrices
$\gamma^{1}_{\varphi_{0}},\dots,\gamma^{k}_{\varphi_{0}}$ lies in the sink
basin $W^{s}_{\omega_{0}}$ and has the same period $m$ as the sink
$\omega_{0}$. Let $(U_{0},\psi_{0})$ be a local chart of the manifold $M^{2}$
such that $\omega_{0}\in U_{0}$, $\psi_{0}(\omega_{0})=O$ and
$\varphi_{0}^{m}(U_{0})\subset U_{0}$. Let $L_{k}\subset\mathbb{R}^{2}$ be a
sheaf of rays $l_{1},\dots,l_{k}$, which in polar coordinates $(\rho,\theta)$
has the form $l_{i}=\\{(\rho,\theta)\in\mathbb{R}^{2}:\theta=\theta_{i}\\}$,
$\theta_{i}\in[0,2\pi)$. Then there are neighborhoods $V_{1},V_{2}$ of the
point $\omega_{0}$ such that $V_{2}\subset V_{1}\subset U_{0}$ and the arc
$\varphi_{t}:M^{2}\to M^{2},\,t\in[0,1]$ without bifurcations with the
following properties:

1) the diffeomorphism $\varphi_{t},\,t\in[0,1]$ coincides with the
diffeomorphism $\varphi_{0}$ outside the set
$\bigcup\limits_{k=0}^{m-1}\varphi_{0}^{k}(V_{1})$ and
$\bigcup\limits_{k=0}^{m-1}\varphi_{0}^{k}(\omega_{0})$ is a hyperbolic sink
orbit of period $m$ for all $\varphi_{t}$;

2) $\psi_{0}(\bigcup\limits_{i=0}^{k}\gamma^{i}_{\varphi_{1}})\cap
V_{2})\subset L_{k}$, where $\gamma^{i}_{\varphi_{1}}$ are unstable
separatrices of saddles $\sigma_{i}$ with respect to the diffeomorphism
$\varphi_{1}$.

Proof:

Figure 6: Straightening the separatrix

Let $\phi_{0}=\varphi_{0}^{m}$ and
$\bar{\phi}_{0}=\psi_{0}\phi_{0}\psi_{0}^{-1}:\mathbb{R}^{2}\to\mathbb{R}^{2}$.
Denote by $O(0,0)$ the origin of coordinates on the plane $\mathbb{R}^{2}$.
For any $r>0$ we put $B_{r}=\\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<r^{2}\\}$.
According to lemma 5.2, without loss of generality, we can assume that
$\bar{\phi}_{0}=g,$ where $g(x,y)=\left(\frac{x}{2},\frac{y}{2}\right)$ on
disk $B_{2r_{0}}$ for some $r_{0}>0$. Let $K_{0}=B_{2r_{0}}\setminus
B_{r_{0}}$,$\gamma^{i}_{\bar{\phi}_{0}}=\psi_{0}(\gamma^{i}_{\varphi_{0}})$.

Denote by $E_{g}$ the set of contractions
$\bar{\phi}:\mathbb{R}^{2}\to\mathbb{R}^{2}$, coinciding with $\bar{\phi}_{0}$
outside $B_{2r_{0}}$ and with $g$ with $B_{r_{\bar{\phi}}}$, where
$r_{\bar{\phi}}\leq 2r_{0}$. For any $\bar{\phi}\in E_{g}$ let

$\gamma^{i}_{\bar{\phi}}=\bigcup\limits_{k\in\mathbb{Z}}\bar{\phi}^{k}(\gamma^{i}_{\bar{\phi}_{0}}\cap
K_{0}).$

By construction, the $\bar{\phi}$-invariant curves $\gamma^{i}_{\bar{\phi}}$
coincide with $\bar{\phi}_{0}$-invariant curves $\gamma^{i}_{\bar{\phi}_{0}}$
outside the disk $B_{r_{0}}$. Then, to prove the lemma, it suffices to
construct an arc of contractions
$\bar{\phi}_{t}:\mathbb{R}^{2}\to\mathbb{R}^{2},t\in[0,1]$ such that

$\bar{1})$ the diffeomorphism $\bar{\phi}_{t},t\in[0,1]$ coincides with the
diffeomorphism $\bar{\phi}_{0}$ outside the set $B_{r_{0}}$;

$\bar{2})$ $\psi_{0}(\bigcup\limits_{i=0}^{k}\gamma^{i}_{\varphi_{1}})\cap
V_{2})\subset L_{k}$.

The arc $\varphi_{t}:M^{2}\to M^{2}$ is obtained from the arc $\bar{\phi}_{t}$
as in lemma 5.2, if we put $V_{1}=\psi^{-1}_{0}(B_{r_{0}})$ and
$V_{2}=\psi_{0}^{-1}(B_{r_{\bar{\phi}_{1}}})$.

To construct the arc $\bar{\phi}_{t}$ we introduce the following notations for
any diffeomorphism $\bar{\phi}\in E_{g}$.

Represent the 2-torus $\mathbb{T}^{2}$ as the orbit space of the
diffeomorphism $g$ on the set $\mathbb{R}^{2}\setminus O$ and denote by
$p:\mathbb{R}^{2}\setminus O\to\mathbb{T}^{2}$ the natural projection. Let
$\hat{a}=p(OX^{+})$ and $\hat{b}=p(\mathbb{S}^{1})$ be generators of
$\mathbb{T}^{2}$. Let $\hat{l}_{i}=p(l_{i})$ and
$K_{\bar{\phi}}=B_{r_{\bar{\phi}}}\setminus B_{r_{\bar{\phi}}/2}$,
$\hat{\gamma}^{i}_{\bar{\phi}}=p(\gamma^{i}_{\bar{\phi}}\cap K_{\bar{\phi}})$.
Then the curve $\hat{\gamma}^{i}_{\bar{\phi}}$ is the knot on the torus
$\mathbb{T}^{2}$ with the homotopy type
$<1,-n_{\bar{\phi}}>,\,n_{\bar{\phi}}\in\mathbb{Z}$ in the basis
$\hat{a},\hat{b}$ (see, for example, [13]).

The arc $\bar{\phi}_{t}$ is the smooth product of $\eta_{t}$ and $\zeta_{t}$
where

I) the arc $\eta_{t},t\in[0,1]$ consists of contractions, it coincides with
$\bar{\phi}_{0}$ outside the set $B_{r_{0}}$ and it joins the diffeomorphism
${\eta}_{0}=\bar{\phi}_{0}$ with some diffeomorphism ${\eta}_{1}\in E_{g}$
such that the knot $\hat{\gamma}_{\eta_{1}}$ and $\hat{\xi}_{\eta_{1}}$ has
the homotopy type $<1,0>$ in the basis $\hat{a},\hat{b}$;

II) the arc $\zeta_{t}\in E_{g},t\in[0,1]$ joins the diffeomorphism
$\zeta_{0}=\eta_{1}$ to the diffeomorphism $\zeta_{1}$ such that
$\hat{\gamma}^{i}_{\zeta_{1}}=\hat{l}_{i}$.

Figure 7: Illustration to Lemma 6.1, Part I)

I) If $n_{\bar{\phi}}=0$, then let $\eta_{t}=\bar{\phi}_{0}$ for every
$t\in[0,1]$. Otherwise define the diffeomorphism
$\theta_{t},t\in[0,1]:\mathbb{R}^{2}\to\mathbb{R}^{2}$ in polar coordinates
$\rho,\varphi$ in such a way that $\theta_{t}(O)=O$ and ${\theta}_{t}(\rho
e^{i\varphi})=\rho e^{i(\varphi+\varphi_{t}(\rho))}$ where $\varphi_{t}(\rho)$
is a smooth monotonic function equal to $2n_{\bar{\phi}}\pi t$ for
$\rho\leqslant\frac{1}{2}$ and equal to $0$ for $\rho\geqslant{1}$. Then
$\eta_{t}=\theta_{t}\bar{\phi}_{0}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is the
desired arc (Fig. 7).

II) By construction, the diffeomorphism $\eta_{1}\in E_{g}$ and the knot
$\hat{\gamma}^{i}_{\eta_{1}}$ has the homotopy type $<1,0>$ in the basis
$\hat{a},\hat{b}$. Let $\hat{L}_{k}=p(L_{k})$. According to 6.1, there is a
diffeomorphism $\hat{h}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ which is smoothly
isotopic to the identity and such that
$\hat{h}(\bigcup\limits_{i=1}^{k}\hat{\gamma}^{i}_{\eta_{1}})=\hat{L}_{k}$.
For $r>0$ let $K_{r}=B_{r}\setminus B_{r/2}$. Pick an open cover
$D=\\{D_{1},\dots,D_{q}\\}$ of the torus $\mathbb{T}^{2}$ such that the
connected component $\bar{D}_{i}$ of the set $p^{-1}(D_{i})$ is a subset of
$K_{r_{i}}$ for some $r_{i}<\frac{r_{i-1}}{2}$ and $r_{1}\leq r_{0}/2$.
According to [3, Lemma de fragmentation] there are diffeomorphisms
$\hat{w}_{1},\dots,\hat{w}_{q}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ smoothly
isotopic to the identity, with the following properties:

i) for each $i\in\\{1,\dots,q\\}$ there exists a smooth isotopy
$\\{\hat{w}_{i,t}\\}$ which is the identity outside $D_{i}$ and which joins
the identity and $\hat{w}_{i}$;

ii) $\hat{h}=\hat{w}_{1}\dots\hat{w}_{q}$.

Let ${w}_{i,t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be a diffeomorphism which
coincides with $(p|_{K_{r_{i}}})^{-1}\hat{w}_{i,t}p$ on $K_{r_{i}}$ and which
coincides with the identity outside $K_{r_{i}}$ (Fig. 8). Then the desired arc
is defined by

${\zeta}_{t}=\nu_{1}{w}_{1,t}\dots w_{q,t}.$

Figure 8: Illustration to Lemma 6.1, Part II)

$\square$

###  6.1 Changes in dynamics in the vicinity of the attractor

In this section, we prove that the dynamics of a gradient-like
2-diffeomorphism in a neighborhood of a smoothly embedded closure of an
unstable manifold of a saddle point can be replaced by any topologically
conjugate one by passing along an arc without bifurcations.

######  Lemma 6.2.

Let

  * •

$\phi_{0}:M^{2}\to M^{2}$ be a orientation-preserving Morse-Smale
diffeomorphism defined on an orientable surface $M^{2}$,

  * •

$\sigma_{1},\dots,\sigma_{n}$ be saddle fixed points of the diffeomorphism
$\phi_{0}$, the curve $c=\bigcup\limits_{i=1}^{n}cl\,W^{u}_{\sigma_{i}}$ be
smoothly embedded in $M^{2}$ and $K_{0}$ be a basin of the attractor,

  * •

a diffeomorphism $\phi_{1}$ be topologically conjugate with the diffeomorphism
$\phi_{0}$ on the curve $c$ and in some of its neighborhood.

Then there is an arc without bifurcations $\varphi_{t}:M^{2}\to
M^{2},\,t\in[0,1]$ and a neighborhood $K_{*}\subset K_{0}$ of the curve $c$
with the following properties:

  1. 1.

$\varphi_{0}=\phi_{0}$, $\varphi_{1}$ coincides with $\phi_{1}$ on $K_{*}$ and
coincides with $\phi_{0}$ outside $K_{0}$;

  2. 2.

$\phi_{1}$ coincides with $\phi_{0}$ outside $K_{0}$ and if $\phi_{1}$
coincides with $\phi_{0}$ in some neighborhood of the sink on the curve $c$
then $\varphi_{t}$ coincides with $\phi_{0}$ in this neighborhood.

Proof:  Consider the case when the curve $c$ is closed (in the case when the
curve $c$ is not closed, the proof is similar). Then there is a smooth
embedding $\nu:\mathbb{S}^{1}\times[-1,1]\to M^{2}$ such that
$\nu:\mathbb{S}^{1}\times\\{0\\}=c$ and
$K_{0}=\nu(\mathbb{S}^{1}\times[-1,1])$.

Then the desired arc $\varphi_{t}$ will be the smooth product of arcs without
bifurcations $\mu_{t}$ and $\delta_{t}$ where

I) the arc $\mu_{t},t\in[0,1]$ joins the diffeomorphism $\mu_{0}=\varphi_{0}$
with the diffeomorphism $\mu_{1}$ such that
$\Omega_{\eta_{1}|_{c}}=\Omega_{\phi_{1}|_{c}}$;

II) the arc $\delta_{t},t\in[0,1]$ joins the diffeomorphism
$\delta_{0}=\mu_{1}$ with the desired diffeomorphism $\delta_{1}=\varphi_{1}$.

I) If $\Omega_{\phi_{0}|_{c}}=\Omega_{\phi_{1}|_{c}}$ then $\mu_{t}=\phi_{0}$
for all $t\in[0,1]$. Otherwise, by the condition the diffeomorphisms
$w_{0}={\phi_{0}}|_{c},\,w_{1}={\phi_{1}}|_{c}$ are topologically conjugate to
rough transformations of the circle. According to [17], they have the same
number of hyperbolic attracting and repelling periodic points alternating on a
circle. Then there are three possible cases where these points are located on
the circle $c$: 1) there are periodic points $p_{0}^{0},p_{0}^{1}$ of
diffeomorphisms $w_{0},\,w_{1}$, respectively, which coincide and have the
same stability; 2) there exists an arc $\beta\subset c$ such that $int\,\beta$
contains periodic points $p_{0}^{0},p_{0}^{1}$ of diffeomorphisms
$w_{0},\,w_{1}$, respectively, of the same stability and does not contain
other periodic points of diffeomorphisms $w_{0},\,w_{1}$; 3) there exists an
arc $\beta\subset c$ such that $int\,\beta$ contains all periodic points of
both diffeomorphisms
$p_{0}^{0},p_{1}^{0},p_{0}^{1},p_{1}^{1},\dots,p_{2n-2}^{0},p_{2n-1}^{0},p_{2n-2}^{1},p_{2n-1}^{1}$
located on the arc in the specified order, where $p_{2i}^{0},p_{2i}^{1}$ are
attracting points, $p_{2i-1}^{0},p_{2i-1}^{1}$ are repelling points of
diffeomorphisms $w_{0},\,w_{1}$, respectively, for $i\in\\{0,\dots,n-1\\}$.

In case 1) let $\beta=c\setminus U_{0}$, where $U_{0}$ is the arc of the
circle $c$ containing the point $p_{0}^{0}$ and not containing any other
periodic points of the diffeomorphisms $w_{0},\,w_{1}$. Choose a tubular
neighborhood $N(\beta)\subset K_{0}$ of the arc $\beta$ so that there exists a
diffeomorphism $h:N(\beta)\to\Pi$ where $\Pi=[0,1]\times[-1,1]$. We renumber
$p_{1}^{0},p_{2}^{0},\dots,p_{2n-1}^{0};\,p_{1}^{1},p_{2}^{1},\dots,p_{2n-1}^{1}$
periodic points of diffeomorphisms $w_{0},\,w_{1}$, respectively, on the arc
$\beta$ so that
$0<h(p_{1}^{0})<h(p_{2}^{0})<\dots<h(p_{2n-1}^{0})<1;\,0<h(p_{1}^{1})<h(p_{2}^{1})<\dots<h(p_{2n-1}^{1})<1$.

Figure 9: Graph of the map $r(x)$

Consider a smooth monotonic increasing function $r(x):[0,1]\to[0,1]$ which is
identical in some neighborhoods of the points $0,\,1$ and on the set
$h(\nu(B))$ such that $r(h(p_{i}^{0}))=h(p_{i}^{1})$ (Fig. 9). Let
$r_{t}(x)=tr(x)+(1-t)x,\,x\in[0,1],\,t\in[0,1]$ and
$r_{t,y}(x)=(1-y)r_{t}(x)+yx,x\in[0,1],\,t\in[0,1],\,y\in[0,1]$. We define a
smooth isotopy $R_{t}:\Pi\to\Pi$ by the formula

$R_{t}(x,y)=(r_{t,y^{2}}(x),y).$

Since the isotopy $R_{t}$ is identical on $\partial\Pi$ for all $t\in[0,1]$,
there exists a smooth isotopy $\rho_{t}:M^{2}\to M^{2}$ which is identical
outside $N(\beta)$ and coincides with $h^{-1}R_{t}h$ on $N(\beta)$. Then
$\mu_{t}=\rho_{t}f\rho^{-1}_{t}$ is the required isotopy that superpose the
periodic points of the diffeomorphism $w_{0}$ with the periodic points of the
diffeomorphism $w_{1}$ on the arc $\beta$.

In case 2) using the technique above, the desired isotopy $\mu_{t}$ is
composed of an isotopy that superpose the point $p_{0}^{1}$ on the arc $\beta$
and an isotopy that combines the periodic points of the diffeomorphism $w_{0}$
with periodic points of the diffeomorphism $w_{1}$ on the complementary arc to
$\beta$.

In case 3) the desired isotopy $\mu_{t}$ superpose the periodic points of the
diffeomorphism $w_{0}$ with the periodic points of the diffeomorphism $w_{1}$
while preserving the order on the arc $\beta$.

If $\phi_{1}$ coincides with $\phi_{0}$ in some neighborhood of the sink on
the curve $c$, then, evidently, the arc $\mu_{t}$ can be constructed to
coincide with $\phi_{0}$ in this neighborhood.

II) If ${\mu_{1}}$ coincides with $\phi_{1}$ in some neighborhood of the
circle $c$, then $\delta_{t}=\mu_{1}$ for all $t\in[0,1]$. Otherwise, since
the circle $c$ is an attractor for the diffeomorphism $\phi_{1}$, then there
exists a smooth annulus $K\subset int\,K_{0}$, which is the trapping
neighborhood of the attractor $c$ of the diffeomorphism $\phi_{1}$. Choose
$0<\varepsilon_{*}<1$ so that for the annulus
$K_{*}=\nu(\mathbb{S}^{1}\times[-\varepsilon_{*},\varepsilon_{*}])$ satisfies
the inclusion $K_{*}\subset int\,(\phi^{2}_{1}(K)\cap\mu^{2}_{1}(K_{0}))$.
From the annulus theorem (see, for example, [29]), there exists the
diffeomorphism $\gamma:K\to K_{0}$ such that
$\gamma|_{K_{\varepsilon_{*}}}=id$ and
$\gamma(\phi^{i}_{1}(K))=\mu^{i}_{1}(K_{0}),\,i=1,2$. Define on
$\mathbb{S}^{1}\times[-1,1]$ the diffeomorphisms
$\tilde{\phi}_{1}=\nu^{-1}\gamma\phi_{1}\gamma^{-1}\nu$ and
$\tilde{\mu}_{1}=\nu^{-1}\mu_{1}\nu$. Let

$\tilde{\xi}_{t}=(1-t)\tilde{\mu}_{1}+t\tilde{\phi}_{1}.$

Then the arc $\tilde{q}_{t}=\tilde{\mu}_{1}^{-1}\tilde{\xi}_{t}$ joins the
identity map $\tilde{q}_{0}=id$ and the diffeomorphism
$\tilde{q}_{1}=\tilde{\mu}_{1}^{-1}\tilde{\phi}_{1}$ and
$\tilde{q}_{t}(\tilde{\mu}_{1}(\mathbb{S}^{1}\times[-1,1]))\subset\mathbb{S}^{1}\times(-1,1)$.
According to isotopy extension theorem (see, for example, [18, Theorem 5.8]),
there exists an isotopy
$\tilde{Q}_{t}:\mathbb{S}^{1}\times[-1,1]\to\mathbb{S}^{1}\times[-1,1]$,
coincides with $\tilde{q}_{t}$ on
$\tilde{\mu}_{1}(\mathbb{S}^{1}\times[-1,1])$ and is identical on
$\mathbb{S}^{1}\times\\{-1,1\\}$. The desired isotopy which
$\delta_{t}:M^{2}\to M^{2}$, coincides with $\mu_{1}$ outside $K_{0}$ and
coincides with $\mu_{1}\nu{\tilde{Q}}_{t}\nu^{-1}$ on $K_{0}$.

If $\phi_{1}$ coincides with $\mu_{1}$ in some neighborhood of the sink on the
curve $c$, then evidently the annulus $K$ can be chosen to coincide with the
annulus $K_{0}$ and the diffeomorphism $\gamma$ is identical in this
neighborhood. $\square$

###  6.2 Changing dynamics in a wandering set

The closures of unstable saddle manifolds of any gradient-like
2-diffeomorphism form a connected attractor of this diffeomorphism, and the
repeller dual to it consists of all sources (see, for example, [12]). In this
section, we will prove that any two diffeomorphisms that coincide in some
neighborhoods of such an attractor and a repeller can be connected by an arc
without bifurcations.

######  Lemma 6.3.

Let the gradient-like 2-diffeomorphisms $\varphi_{0},\,\varphi_{1}:M^{2}\to
M^{2}$ coincide on the closures of unstable saddle manifolds and in some of
their neighborhoods, as well as in neighborhoods of sources. Then there is a
bifurcation free arc connecting them.

Proof:  Let $A$ be an attractor of diffeomorphisms
$\varphi_{0},\,\varphi_{1}$, formed by the closures of the unstable manifolds
of all saddle points, and $R$ is its dual repeller consisting of sources. Let
$K_{A}$ be a trapping neighborhood of attractor $A$. Then each connected
component $M^{2}\setminus K_{A}$ is a two-dimensional disk lying in the basin
of some source from the repeller $R$. We pick in the repeller the sources
$\alpha_{1},\dots,\alpha_{k}$ with pairwise disjoint orbits of periods
$m_{1},\dots,m_{k}$, respectively, so that
$R=\bigcup\limits_{i=1}^{k}\bigcup\limits_{j=0}^{m_{i}-1}\varphi_{0}^{j}(\alpha_{i})$.

We construct an arc $H_{\varphi_{0},\alpha_{1},t}$ without bifurcations with
the following properties:

\- $H_{\varphi_{0},\alpha_{1},t}$ connects the diffeomorphism $\varphi_{0}$
with the diffeomorphism $\varphi_{\alpha_{1}}=H_{\varphi_{0},\alpha_{1},1}$
such that $\varphi_{\alpha_{1}}=\varphi_{1}$ on
$K_{A}\cup\bigcup\limits_{j=0}^{m_{i}-1}\varphi_{0}^{j}(W^{u}_{\alpha_{1}})$;

\- $H_{\varphi_{0},\alpha_{1},t}=\varphi_{0}$ outside
$K_{A}\cup\bigcup\limits_{j=0}^{m_{i}-1}\varphi_{0}^{j}(W^{u}_{\alpha_{1}})$.

Analogously one constructs the arc
$H_{\varphi_{\alpha_{1}},\alpha_{2},t},\dots,H_{\varphi_{\alpha_{k-1}},\alpha_{k},t}$.

Then the desired arc $\varphi_{t}$ will be the smooth product of these arcs,
that is

$\varphi_{t}=H_{\varphi_{0},\alpha_{1},t}*\dots*H_{\varphi_{\alpha_{k-1}},\alpha_{k},t}$

Construction of the arc $H_{\varphi_{0},\alpha_{1},t}$. For definiteness, we
assume that the source $\alpha_{1}$ is fixed.

In the case when the point $\alpha_{1}$ is periodic with period $m_{1}$
suffices to construct an isotopy for the diffeomorphism $f^{m_{1}}$ with a
fixed point $\alpha_{1}$ and to extend it along the neighborhood of the orbit
of the point $\alpha_{1}$ with diffeomorphism $\varphi_{0}$

Denote by $\Phi_{\varphi_{1}}$ the set of gradient-like diffeomorphisms of the
surface $M^{2}$ that coincide with $\varphi_{1}$ on $K_{A}$ and in some
neighborhood of the point $\alpha_{1}$ (in particular, the diffeomorphism
$\varphi_{0}$ belongs to the set $\Phi_{\varphi_{1}}$). Then any
diffeomorphism $\varphi\in\Phi_{\varphi_{1}}$ is smoothly conjugate on
$W_{\alpha_{1}}^{u}\setminus\alpha_{1}$ with $\varphi_{1}$ by the
diffeomorphism

$\rho_{\varphi}(x)=\varphi_{1}^{-k}(\varphi^{k}(x)),$

where $k\in\mathbb{Z}$ such that $\varphi^{k}(x)\in K_{A}$ for $x\in
W_{\alpha_{1}}^{u}\setminus\alpha$. Thus
$\varphi_{1}=\rho_{\varphi}\varphi_{0}\rho_{\varphi}^{-1}$ on
$W_{\alpha_{1}}^{u}\setminus\alpha_{1}$.

For the diffeomorphism $\varphi$ onsider two subcases: I) $\rho_{\varphi}$ be
smoothly extended to the point $\alpha_{1}$ by the condition
$\rho_{\varphi}(\alpha_{1})=\alpha_{1}$, II) $\rho_{\varphi}$ does not
smoothly extended to the point $\alpha_{1}$.

In case I) according to [30] the diffeomorphism $\rho_{\varphi}$ is isotopic
to the identity.

Moreover, according to Thoms isotopy extension theorem, on
$W_{\alpha_{1}}^{u}$ there exists a smooth isotopy $\varrho_{\varphi,t}$ such
that $\varrho_{\varphi,0}=id,\,\varrho_{\varphi,1}=\rho_{\varphi}$ and
$\varrho_{\varphi,t}=id,\,t\in[0,1]$ in some neighborhood $K_{\varphi}\subset
K_{A}$ of attractor $A$. The isotopy $\varrho_{\varphi,t}$ extends to isotopy
of ambient surface $\varrho_{\varphi,t}:M^{2}\to M^{2}$ and coincides with
identity map outside $W^{u}_{\alpha_{1}}$. Then a bifurcation free arc

$H_{\varphi,\alpha_{1},t}=\varrho_{\varphi,t}^{-1}\varphi\varrho_{\varphi,t}$

connects the diffeomorphism $\varphi$ with the diffeomorphism $\varphi_{1}$.

In case II) let us apply the following construction. Let
$(U_{\alpha_{1}},\psi_{\alpha_{1}}),\psi_{\alpha_{1}}:U_{\alpha_{1}}\to\mathbb{R}^{2}$,
$\psi_{\alpha_{1}}=O$ be a local chart of $M^{2}$. We consider on
$\mathbb{R}^{2}$ diffeomorphisms
$\bar{\varphi}=\psi_{\alpha_{1}}\varphi\psi_{\alpha_{1}}^{-1}$,
$\bar{\varphi_{1}}=\psi_{\alpha_{1}}\varphi_{1}\psi_{\alpha_{1}}^{-1}$ and
$\bar{\rho}_{\varphi}=\psi_{\alpha_{1}}\rho_{\varphi}\psi_{\alpha_{1}}^{-1}$,
which are local representations of the diffeomorphisms $\varphi,\varphi_{1}$
and $\rho$, respectively. Since the point $O$ is a hyperbolic source of the
diffeomorphisms $\varphi$ and $\varphi_{1}$, there exists a 2-disk
$B_{\varphi}\ni O$ such that $\bar{\varphi}^{-1}(B_{\varphi})\subset
int\,B_{\varphi}$ and the annulus $K_{\varphi}$ is a fundamental domain of the
restriction of the diffeomorphism $\bar{\varphi}$ to
$int\,B_{\varphi}\setminus\\{O\\}$.

Represent the 2-torus $\mathbb{T}^{2}$ as the orbit space
$(int\,B_{\varphi}\setminus\\{O\\})/{\bar{\varphi}}$. Denote by
$p_{\varphi}:B_{\varphi}\setminus\\{O\\}\to\mathbb{T}^{2}$ the natural
projection. Then the curve $b=p_{\varphi}(\partial B_{\varphi})$ has the
homotopy type <0,1> and it is possible to uniquely define a curve $a$ of
homotopy type <1,0> such that the curves $a,b$ are the generators of the
fundamental group $\pi_{1}(\mathbb{T}^{2})$. Since $\bar{\rho}_{\varphi}$ maps
the orbits $\bar{\varphi}$ into the orbits $\bar{\varphi}_{1}$ and
$K_{\varphi}$ is a common fundamental domain for
$\bar{\varphi},\bar{\varphi}_{1}$ on $int\,\ B_{\varphi}\setminus\\{O\\}$,
then $\bar{\rho}_{\varphi}$ is projected onto $\mathbb{T}^{2}$ by the formula
$\hat{\rho}_{\varphi}=p_{\varphi}\bar{\rho}_{\varphi}p_{\varphi}^{-1}$. Then
the induced isomorphism
$\hat{\rho}_{\varphi*}:\pi_{1}(\mathbb{T}^{2})\to\pi_{1}(\mathbb{T}^{2})$
preserves the homotopy class of the generator $a$ and, therefore, is given by
a matrix

$\left(\begin{array}[]{cccc}1&0\\\ n_{\varphi}&1\\\ \end{array}\right)$

for some integer $n_{\varphi}$.

Consider two subcases: a) $n_{\varphi}=0$; b) $n_{\varphi}\neq 0$.

In the case a) let us construct an arc without bifurcations
$\nu_{\varphi,t}:M^{2}\to M^{2}$ such that $\nu_{\varphi,0}=\varphi$,
$\nu_{\varphi,1}\in\Phi_{\varphi_{1}}$ and the diffeomorphism
$\rho_{\nu_{\varphi,1}}$ smoothly continues to the source $\alpha_{1}$. Then
an arc without bifurcations

$H_{\varphi,\alpha_{1},t}=\nu_{\varphi,t}*(\varrho_{\nu_{\varphi,1},t}^{-1}\nu_{\varphi,1}\varrho_{\nu_{\varphi,1},t})$

connects the diffeomorphism $\varphi$ with the diffeomorphism $\varphi_{1}$.

Let us describe the construction of the arc $\nu_{\varphi,t}$.

We choose a cover $U=\\{U_{1},\dots,U_{q}\\}$ of the torus $\mathbb{T}^{2}$,
consisting of 2-disks such that some connected component of the set
$p_{\varphi}^{-1}(U_{i})$ is a subset of
$K_{i}=B_{i}\setminus\bar{\varphi}^{-1}(B_{i})$, obtained from a 2-disc
$B_{i}$ such that $O\in
B_{i}\subset\bar{\varphi}^{-1}(B_{\varphi}),\,i=1,\dots,q$.

According to the lemma of fragmentation [3] there exist smoothly isotopic to
the identity diffeomorphisms
$\hat{w}_{1},\dots,\hat{w}_{q}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ such that

i) for each $i=\overline{1,q}$ there exists $U_{j(i)}\in U$ such that for each
$t\in[0,1]$ the map $\hat{w}_{i,t}$ is identical outside $U_{j(i)}$, where
$\\{\hat{w}_{i,t}\\}$ is the smooth isotopy between the identity map and
$\hat{w}_{i}$;

ii) $\hat{\rho}_{\varphi}=\hat{w}_{1}\dots\hat{w}_{q}$.

We choose numbers $n_{i}\in\mathbb{N}$ so that
$\bar{\varphi}^{-n_{q}}(B_{j(q)})\subset\dots\subset\bar{\varphi}^{-n_{1}}(B_{j(1)})$.
Let
$\bar{K}_{i}=\bar{\varphi}^{-n_{j(i)}}(B_{j(i)})\setminus\bar{\varphi}^{-n_{j(i)}-1}(B_{j(i)})$.
Denote by $\bar{w}_{i,t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ diffeomorphism
coinciding with $(p_{\varphi}|_{\bar{K}_{i}})^{-1}\hat{w}_{i,t}p_{\varphi}$ on
$\bar{K}_{i}$ and identical outside $\bar{K}_{i}$. Let
$\bar{w}_{t}=\bar{w}_{1,t}\dots,\bar{w}_{q,t}$ and
$\bar{\nu}_{\varphi,t}=\bar{\varphi}\bar{w}^{-1}_{t}.$ Denote by
$\nu_{\varphi,t}:M^{2}\to M^{2}$ a diffeomorphism coinciding with
$\psi_{\alpha_{1}}^{-1}\bar{\nu}_{\varphi,t}\psi_{\alpha_{1}}$ on
$B_{\varphi}$ and coinciding with $\varphi$ outside $B_{\varphi}$. Then
$\nu_{\varphi,1}\in\Phi_{\varphi_{1}}$ since $\bar{\nu}_{\varphi,1}$ coincides
with $\bar{\varphi}_{1}$ on some disk
$B_{\nu_{\varphi,1}}=\bar{\varphi}^{-m}(B_{\varphi})\subset\bar{\varphi}^{-n_{j(q)}-1}(B_{j(q)})$.
By construction
$\bar{\rho}_{\nu_{\varphi,1}}=\bar{\varphi}_{1}^{-m}\bar{\rho}_{\varphi}(\bar{\varphi}\bar{w}^{-1}_{1})^{m}$
and therefore
$\hat{\rho}_{\nu_{\varphi,1}}(\hat{x})=\hat{x},\,\hat{x}\in\mathbb{T}^{2}$.

Thus, the diffeomorphism $\rho_{\nu_{\varphi,1}}$ in a neighborhood of the
point $\alpha_{1}$ coincides with some degree of the diffeomorphism $\varphi$
and, therefore, smoothly extends to the point $\alpha_{1}$.

In subcase b), construct an arc without bifurcations $\mu_{\varphi,t}:M^{2}\to
M^{2}$ such that $\mu_{\varphi,0}=\varphi$,
$\mu_{\varphi,1}\in\Phi_{\varphi_{1}}$ and $n_{\mu_{\varphi,1}}=0$. Then the
bifurcation free arc

$H_{\varphi,\alpha_{1},t}=\mu_{\varphi,t}*\nu_{\mu_{\varphi,1},t}*(\varrho_{\nu_{\mu_{\varphi,1},1},t}^{-1}\nu_{{\mu_{\varphi,1},1}}\varrho_{\nu_{\mu_{\varphi,1},1},t})$

connects the diffeomorphism $\varphi$ with the diffeomorphism $\varphi_{1}$.

We describe the construction of the arc $\mu_{\varphi,t}$.

Introduce a coordinates $r,\phi$ on the disk $B_{\varphi}$ in which the curve
$\bar{\varphi}^{-k}(\partial B_{\varphi}),\,k\in\mathbb{N}$ has the form
$r=2^{-k},\,\phi\in[0,2\pi)$. Define the diffeomorphism
$\bar{\theta}_{t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ so that
$\bar{\theta}_{t}(O)=O$ and
$\bar{\theta}_{t}(r,\phi)=re^{i(\phi+\phi_{t}(r))}$, where $\phi_{t}(r)$ is a
smooth monotone function equal to $-2n_{\varphi}\pi t$ for
$r\leqslant\frac{1}{2}$ and equal to $0$ for $r\geqslant{1}$. Let
$\bar{\mu}_{\varphi,t}=\bar{\varphi}\bar{\theta}_{t}$ and denote by
$\mu_{\varphi,t}:M^{2}\to M^{2}$ diffeomorphism coinciding with
$\psi_{\alpha_{1}}^{-1}\bar{\mu}_{\varphi,t}\psi_{\alpha_{1}}$ on
$B_{\varphi}$ and coinciding with $\varphi$ outside $B_{\varphi}$. Then
$\mu_{\varphi,1}\in\Phi_{\varphi_{1}}$ since $\bar{\mu}_{\varphi,1}$ is the
same as $\bar{\varphi}_{1}$ on disk
$B_{\mu_{\varphi,1}}=\bar{\varphi}^{-1}(B_{\varphi})$. By construction
$\bar{\rho}_{\mu_{\varphi,1}}=\bar{\varphi}_{1}^{-1}\bar{\rho}_{\varphi}\bar{\varphi}\bar{\theta}_{1}$
and therefore $n_{\mu_{\varphi,1}}=0$. $\square$

##  7 Construction an arc $H_{J,t}$

###  7.1 Construction of auxiliary functions

In this section, we construct model functions that will later be used to
construct a stable arc. The construction is based on the principle of gluing
infinitely smooth functions by means of the following sigmoid function.

Let $a<b$ and $\delta_{{}_{a;b}}:\mathbb{R}\to[0,1]$ sigmoid function, defined
by the formula (Fig. 10)

$\delta_{{}_{a;b}}(x)=\begin{cases}0,&x\leqslant
a,\cr\frac{1}{1+\exp\left(\frac{(a+b)/2-x}{{(x-a)}^{2}{(x-b)}^{2}}\right)},&a<x<b,\cr
1,&x\geqslant b.\end{cases}$

Figure 10: Graph of the sigmoid function

Define the function $\bar{\phi}_{1}:\mathbb{R}\to\mathbb{R}$ by the formula
(Fig. 11).

$\bar{\phi}_{1}(x)=x-\frac{1}{12\pi}sin\left(6\pi\left(x-\frac{1}{4}\right)\right).$

Figure 11: Graph of the map $\bar{\phi}_{1}(x)$

Define the function $\bar{g}_{1}:\mathbb{R}\to\mathbb{R}$ by the formula (Fig.
12)

$\bar{g}_{1}(x)=\begin{cases}\bar{\phi}_{0}(x),\,0\leqslant x\leqslant
0,26,\\\
(1-\delta_{{}_{0,26;0,27}}(x))\bar{\phi}_{0}(x)+\delta_{{}_{0,26;0,27}}(x)\bar{\phi}_{1}(x),\,0,26<x<0,27,\\\
\bar{\phi}_{1}(x),\,0,27\leqslant x\leqslant 0,76,\\\
(1-\delta_{{}_{0,76;0,77}}(x))\bar{\phi}_{1}(x)+\delta_{{}_{0,76;0,77}}(x)\bar{\phi}_{0}(x),\,0,76<x<0,77,\\\
\bar{\phi}_{0}(x),\,0,77\leqslant x\leqslant 1\\\ \end{cases}.$

Figure 12: Graph of the map $\bar{g}_{1}(x)$

Define the function $\bar{\phi}_{2}:\mathbb{R}\to\mathbb{R}$ by the formula
(Fig. 13).

$\bar{\phi}_{2}(x)=x+\frac{1}{4\pi}sin\left(\frac{5}{6}\pi\left(x-\frac{5}{12}\right)\right).$

Figure 13: Graph of the map $\bar{\phi}_{2}(x)$

Define the function $\bar{g}_{2}:\mathbb{R}\to\mathbb{R}$ by the formula (Fig.
14)

$\bar{g}_{2}(x)=\begin{cases}\bar{g}_{1}(x),\,0\leqslant x\leqslant 0,42,\\\
(1-\delta_{{}_{0,42;0,43}}(x))\bar{g}_{1}(x)+\delta_{{}_{0,42;0,43}}(x)\bar{\phi}_{2}(x),\,0,42<x<0,43,\\\
\bar{\phi}_{2}(x),\,0,43\leqslant x\leqslant 0,98,\\\
(1-\delta_{{}_{0,98;0,99}}(x))\bar{\phi}_{2}(x)+\delta_{{}_{0,98;0,99}}(x)\bar{g}_{1}(x),\,0,98<x<0,99,\\\
\bar{g}_{1}(x),\,0,99\leqslant x\leqslant 1\\\ \end{cases}.$

Figure 14: Graph of the map $\bar{g}_{2}(x)$

###  7.2 Construction the model arcs

In this section, we will construct arcs, which are the main components that
make up an arc $H_{J,t}$.

For $n\in\mathbb{Z}$ let $J_{n}=\begin{pmatrix}1&0\\\ n&1\end{pmatrix}.$

######  Lemma 7.1.

The diffeomorphism $f_{0}$ is connected with the diffeomorphism $f_{J_{1}}$ by
a stable arc $H_{0,1,t}$ with typically passing non-critical two saddle-node
bifurcations.

Proof:  In this proof, the bar-free mappings are projections on
$\mathbb{S}^{1}$ by $\pi$ of the bar-mappings given on the line $\mathbb{R}$.
The stable arc $H_{0,1,t}$, connecting the diffeomorphism $f_{0}$ with the
diffeomorphism $f_{J_{1}}$ product of the arcs
$\Gamma^{1}_{t},\,\Gamma^{2}_{t}$, constructed in step 1 and step 2 below and
the arc $H_{\Gamma^{2}_{1},t}$.

Step 1. First saddle-node bifurcation.

1\. The birth of a saddle-node point.

We start with the diffeomorphism $f_{0}:\mathbb{T}^{2}\to\mathbb{T}^{2}$,
defined by the formula:

$f_{0}(z,w)=(\phi_{0}(z),\phi_{0}(w)),\,z,w\in\mathbb{S}^{1}.$

Let

$\bar{\eta}^{1}_{t}(x)=(1-t)\bar{\phi}_{0}(x)+t\bar{g}_{1}(x),\,x\in\mathbb{R},\,t\in[0,1]$

and

$\bar{\eta}^{1}_{t,\tau}(x)=(1-\tau)\bar{\eta}^{1}_{t}(x)+\tau\bar{\phi}_{0}(x),\,x\in\mathbb{R},\,t\in[0,1],\,\tau\in[0,1].$

Define a smooth arc $H^{1}_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$, $t\in[0,1]$
by the formula:

$H^{1}_{t}(w,z)=\begin{cases}(\phi_{0}(\pi(x)),\eta^{1}_{t,|8x-2|}(z)),\,x\in\left(\frac{1}{8},\frac{3}{8}\right),\,z\in\mathbb{S}^{1},\\\
f_{0}(\pi(x),z),\,x\in\left(-\frac{5}{8},\frac{1}{8}\right),\,z\in\mathbb{S}^{1}\\\
\end{cases}.$

For $t=\frac{3}{4}$, the diffeomorphism $H^{1}_{\frac{3}{4}}$ has a saddle-
node point $p=(N,\pi(0))$, whose stable manifold is diffeomorphic to a half-
plane whose boundary is arc $\gamma_{p}$ (Fig. 15).

Figure 15: Isotopy $H^{1}_{t}$ on the torus

2\. Rotation of the separatrix of the saddle $\sigma_{2}$.

Consider the fundamental domain
$K=\left[\pi(0),\pi\left(\frac{1}{4\pi}\right)\right]\times\mathbb{S}^{1}$
restriction of the diffeomorphism $f_{0}$ to
$V=\left[\pi\left(-\frac{1}{4}\right),\pi\left(\frac{1}{4}\right)\right]\times\mathbb{S}^{1}$.
Let $\hat{V}=V/f_{0}$. Then $\hat{V}$ is a torus got from $K$ by identifying
the boundaries with the map $f_{0}$. Denote by $q:V\to\hat{V}$ the natural
projection. Let $\hat{\gamma}_{2}=q(W^{u}_{\sigma_{2}}\cap V)$ and
$\hat{\gamma}_{1}=q(W^{s}_{\sigma_{1}}\cap V)$. Since the diffeomorphism
$H_{t}$ for all $t\in[0,1]$ coincides with $f_{0}$ on the annulus
$\left[\pi(-\frac{1}{4}),\pi\left(\frac{1}{8}\right)\right]\times\mathbb{S}^{1}$,
then the circle $\hat{\gamma}_{p}=q(\gamma_{p}\cap K)$ is defined correctly.

Let
$W=\left[\pi\left(-\frac{1}{4}\right);\pi\left(\frac{1}{4}\right)\right]\times\left[\pi\left(-\frac{1}{4}\right);\pi\left(\frac{1}{4}\right)\right]$
and $\hat{W}=p(W)$. By construction, the circle $\hat{\gamma}_{p}$ divides the
annulus $\hat{W}$ into two annulus, the closures of which are denoted
by$\hat{W}_{1},\hat{W}_{2}$, assuming that
$\hat{\gamma}_{1}\subset\hat{W}_{1}$ and $\hat{\gamma}_{2}\subset\hat{W}_{2}$
(Fig. 16).

Choose a circle $\hat{\gamma}\subset int\,\hat{W}_{1}$ that is not homotopic
to zero. According to [29], there exists a diffeomorphism
$\hat{h}_{1}:\hat{V}\to\hat{V}$ smoothly isotopic to the identity such that
$\hat{h}_{1}(\hat{\gamma}_{2})=\hat{\gamma}$.

For $x_{i}\in[-\frac{1}{4};0]$ let
$K_{i}=[\pi(x_{i});(\pi(\bar{\phi}_{0}^{-1}(x_{i}))]\times\mathbb{S}^{1}$.
Choose an open cover $D=\\{D_{1},\dots,D_{k_{1}}\\}$ of the torus
$\mathbb{T}^{2}$ such that the connected component $\bar{D}_{i}$ of the set
$q^{-1}(D_{i})$ is a subset of $K_{i}$ for some
$x_{i}<\bar{\phi}_{0}^{-1}(x_{i-1})$. According to the fragmentation lemma [3]
there exist smoothly isotopic to the identity diffeomorphisms
$\hat{w}_{1},\dots,\hat{w}_{k_{1}}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ with the
following properties:

i) for each $i\in\\{1,\dots,k_{1}\\}$ there exists smooth isotopy
$\\{\hat{w}_{i,t}\\}$ which is the identity outside $D_{i}$ and which joins
the identity and $\hat{w}_{i}$

ii) $\hat{h}_{1}=\hat{w}_{1}\dots\hat{w}_{k_{1}}$.

Let ${w}_{i,t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ diffeomorphism that coincides
with $(q|_{K_{i}})^{-1}\hat{w}_{i,t}q$ $K_{i}$ and coincides with the identity
map outside $K_{i}$. Let

${\zeta}_{t}={w}_{1,t}\dots
w_{k_{1},t}f_{0},\,\,\,\,G^{1}_{t}=\begin{cases}\zeta_{2t},\,0\leqslant
t<\frac{1}{2},\\\ \zeta_{1},\,\frac{1}{2}\leqslant t\leqslant 1\\\
\end{cases}.$

Figure 16: Isotopy ${G}^{1}_{t}$ on the torus

3\. ombining isotopies $H^{1}_{t}$ and $G^{1}_{t}$.

Define a smooth arc $\Gamma^{1}_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$,
$t\in[0,1]$ by the formula (Fig. 17):

$\Gamma^{1}_{t}(z,w)=\begin{cases}H^{1}_{t}(\pi(x),w),\,x\in\left(\frac{1}{8},\frac{3}{8}\right),\,z\in\mathbb{S}^{1},\\\
G^{1}_{t}(\pi(x),w),\,x\in\left(-\frac{1}{4},0\right),\,z\in\mathbb{S}^{1},\\\
f_{0}(\pi(x),w),\,x\in\left[-\frac{5}{8},-\frac{1}{4}\right]\cup[0;\frac{3}{8}],\,z\in\mathbb{S}^{1}\\\
\end{cases}.$

Figure 17: Isotopy $\Gamma^{1}_{t}$ on the torus

Step 2. Second saddle-node bifurcation.

1\. Merging saddle and node points

For all $t\in[0;1]$ let
$\bar{\eta}^{2}_{t}(x)=t\bar{g_{2}}(x)+(1-t)\bar{g_{1}}(x),x\in\mathbb{R}$ and

$\bar{\eta}^{2}_{t,\tau}(x)=(1-\tau)\bar{\eta}^{2}_{t}(x)+\tau\bar{\phi}_{0}(x),\,x\in\mathbb{R},\,t\in[0,1],\,\tau\in[0,1].$

Define a smooth arc $H^{2}_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$, $t\in[0,1]$
by the formula:

$H^{2}_{t}(w,z)=\begin{cases}(\phi_{0}(\pi(x)),\eta^{2}_{t,|8x-2|}(z)),\,x\in\left(\frac{1}{8},\frac{3}{8}\right),\,z\in\mathbb{S}^{1},\\\
\Gamma_{1}(\pi(x),z),\,x\in\left(-\frac{5}{8},\frac{1}{8}\right),\,z\in\mathbb{S}^{1}\\\
\end{cases}.$

The arc $H^{2}_{t}$ realizes the merging of the sink $\tilde{\omega}$ and the
saddle $\sigma_{1}$ into the saddle-node point $\tilde{p}$ and its further
disappearance. Denote by$\beta_{\tilde{p}}$ the boundary of the stable
manifold of a saddle-node $\tilde{p}$.

2\. Rotation of the separatrix of the saddle $\sigma_{2}$.

Since the diffeomorphism $H^{2}_{t}$ for all $t\in[0,1]$ coincides with
$f_{0}$ on the annulus $K$, then the circles
$\hat{\beta}_{2}=q(W^{u}_{\sigma_{2}}\cap K)$,
$\hat{\beta}_{1}=q(W^{s}_{\tilde{\sigma}}\cap K)$ and
$\hat{\beta}_{\tilde{p}}=q(\beta_{\tilde{p}}\cap K)isdefinedcorrectly$.

$\hat{\gamma}_{p}=q(\gamma_{p}\cap K)$ .

Let $\hat{W}_{3}=\hat{V}\setminus\hat{W}$. Choose a circle $\hat{\beta}\subset
int\,\hat{W}_{3}$ that is not homotopic to zero. According to [29], there
exists a diffeomorphism $\hat{h}_{2}:\hat{V}\to\hat{V}$ smoothly isotopic to
the identity such that $\hat{h}_{2}(\hat{\beta}_{2})=\hat{\beta}$ and
$\hat{h}_{2}(\hat{\beta}_{1})=\hat{\beta}_{1}$.

Choose an open cover $U=\\{U_{1},\dots,U_{k_{2}}\\}$ of the torus
$\mathbb{T}^{2}$ such that the connected component $\bar{U}_{i}$ of the set
$q^{-1}(U_{i})$ is a subset of $K_{i}$ for some
$x_{i}<\bar{\phi}_{0}^{-1}(x_{i-1})$. According to the lemma of fragmentation
[3] there exist smoothly isotopic to the identity diffeomorphisms
$\hat{v}_{1},\dots,\hat{v}_{k_{2}}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ with the
following properties:

i) for each $i\in\\{1,\dots,k_{2}\\}$ there exist a smooth isotopy
$\\{\hat{v}_{i,t}\\}$ which is the identity outside $U_{i}$ and which joins
the identity and $\hat{v}_{i}$;

ii) $\hat{h}_{2}=\hat{v}_{1}\dots\hat{v}_{k_{2}}$.

Let ${v}_{i,t}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ diffeomorphism that coincides
with $(q|_{K_{i}})^{-1}\hat{v}_{i,t}q$ on $K_{i}$ and coincides with the
identity map outside $K_{i}$. Let

${\xi}_{t}={v}_{1,t}\dots
v_{k_{2},t}\Gamma_{1},\,\,\,\,G^{2}_{t}=\begin{cases}\xi_{2t},\,0\leqslant
t<\frac{1}{2},\\\ \xi_{1},\,\frac{1}{2}\leqslant t\leqslant 1\\\ \end{cases}.$

3\. ombining isotopies $H^{2}_{t}$ and $G^{2}_{t}$.

Define a smooth arc $\Gamma^{2}_{t}:\mathbb{T}^{2}\to\mathbb{T}^{2}$,
$t\in[0,1]$ by the formula (Fig. 17):

$\Gamma^{2}_{t}(z,w)=\begin{cases}H^{2}_{t}(\pi(x),w),\,x\in\left(\frac{1}{8},\frac{3}{8}\right),\,z\in\mathbb{S}^{1},\\\
G^{2}_{t}(\pi(x),w),\,x\in\left(-\frac{1}{4},0\right),\,z\in\mathbb{S}^{1},\\\
f_{0}(\pi(x),w),\,x\in\left[-\frac{5}{8},-\frac{1}{4}\right]\cup[0;\frac{3}{8}],\,z\in\mathbb{S}^{1}\\\
\end{cases}.$

)$t=0$

)$t=\frac{1}{2}$

)$t=1$

Figure 18: Isotopy $\Gamma^{2}_{t}$ on the torus.

According to lemma 5.1 the diffeomorphism $\Gamma^{2}_{1}$ can be connected by
an arc without bifurcations $H_{\Gamma^{2}_{1},t}$ with the diffeomorphism
$f_{J_{1}}$. $\square$

Denote by $H_{n,n+1,t}$ the arc with two saddle-node bifurcations connecting
the diffeomorphisms $f_{J_{n}},\,f_{J_{n+1}}$ and given by the formula

$H_{n,n+1,t}=\hat{J}_{n}H_{0,1,t}\hat{J}_{n}^{-1}.$

###  7.3 Arc construction algorithm $H_{J,t}$

In this section, using the model arcs constructed above, we will prove the
following lemma.

######  Lemma 7.2.

The diffeomorphism $f_{J}$ is joined by a stable arc $H_{J,t}$ with a finite
number of generically unfolding non-critical saddle-node bifurcations with the
diffeomorphism $f_{0}$.

Proof:  Let $J=\begin{pmatrix}\mu^{1}&\mu^{2}\\\ \nu^{1}&\nu^{2}\end{pmatrix}$
– unimodular matrix such that $\mu^{1}\geq\mu^{2}\geq 0$ and
$\nu^{1}>\nu^{2}$, if $\mu^{1}=\mu^{2}$. Consider the following possibilities
for the matrix $J$: 1) $\mu^{2}=0$; 2) $\mu^{1}=\mu^{2}=1$; 3)
$\mu^{2}>\mu^{1}>0$. Construct the arc $H_{J,t}$ in each case separately.

In case 1) $J=J_{n}$. If $n>0$, then
$H_{J_{n},t}=H_{n-1,n,1-t}*\dots*H_{0,1,1-t}$ is the required arc. If $n<0$,
then $H_{J_{n},t}=\hat{J}_{n}H_{J_{-n},1-t}\hat{J}_{n}^{-1}$ is the required
arc.

In case 2) $H_{J,t}=\hat{J}H_{J_{-1},1-t}\hat{J}^{-1}*H_{J_{\nu^{2},t}}$ is
the required arc.

In case 3) applying Euclid’s algorithm to the pair $\mu_{1},\mu_{2}$ generates
a sequence of natural numbers $n_{1},\dots,n_{m},\,k_{1},\dots,k_{m}$ such
that $\mu^{1}=n_{1}\mu^{2}+k_{1}$, $\mu^{2}=n_{2}k_{1}+k_{2}$,
$k_{1}=n_{3}k_{2}+k_{3},\dots,k_{m-2}=n_{m}k_{m-1}+k_{m}$,
$k_{m-1}=1,\,k_{m}=0$. Let $k_{-1}=\mu^{1},\,k_{0}=\mu^{2}$. Then the sequence
$k_{-1},k_{0},k_{1},\dots,k_{m}$ satisfies the recurrence relation

$k_{i+1}=n_{i+1}k_{i}-k_{i-1},\,i=0,...,m-1.$

Let $l_{-1}=\nu^{1},\,l_{0}=\nu^{2}$ and define the sequence
$l_{-1},l_{0},l_{1},\dots,l_{m}$ by the recurrent relation

$l_{i+1}=n_{i+1}l_{i}-l_{i-1},\,i=0,...,m-1.$

Let $L_{i}=\begin{pmatrix}k_{i-1}&k_{i}\\\
l_{i-1}&l_{i}\end{pmatrix},i=0,...,m$. Then the arc
$F_{i,t}=\hat{L}_{i-1}H_{J_{-n_{i}},t}\hat{L}_{i-1}^{-1},\,i=1,...,m$ joins
diffeomorphisms $f_{L_{i-1}}$ and $f_{L_{i}}$ and contains $2n_{i}$ non-
critical saddle-node bifurcations. Since $f_{L_{m}}=f_{J_{l_{m-1}}}$, then
$H_{J,t}=F_{1,t}*...*F_{m,t}*H_{J_{l_{m-1}}}$ is the required arc. $\square$

## References

  * [1] V.S. Afraimovich, L.P. Shilnikov. On some global bifurcations associated with the disappearance of a saddle-node fixed point. Doc. USSR Academy of Sciences, 219: 6 (1974), 1281-1284.
  * [2] V.S. Afraimovich, L.P. Shilnikov. On small periodic disturbances of autonomous systems. Doc. USSR Academy of Sciences, 214: 4 (1974), 739–742.
  * [3] Banyaga, A.The structure of the group of equivariant diffeomorphism 16, Topology,1977, 279-283.
  * [4] P. Blanchard. Duke Mathematical Journal, (1980), 47:1, 33-46.
  * [5] A.N. Bezdenezhykh and V.Z. Grines. Dynamical Properties and Topological Classification of Gradient-Like Diffeomorphisms on Two-Dimensional Manifolds I. Sel. Math. Sov. 1992. V. 11, 1, 1–11.
  * [6] h. Bonatti, V. Z. Grines, V. S. Medvedev and O. V. Pochinka, Bifurcations of Morse-Smale diffeomorphisms with wildly embedded separatrices, Tr. Mat. Inst. Steklova, 256 (2007), 54-69.
  * [7] V.Z. Grines, O.V. Pochinka On a simple isotopic class of a source-sink diffeomorphism on a 3-sphere // Mathematical Notes. 2013. T. 94. No. 6. S. 828-845.
  * [8] Fleitas G. Replacing tangencies by saddle-nodes. Bol. Soc. Brasil. Mat., 8, No. 1, 47–51 (1977).
  * [9] J. Franks. Necessary conditions for the stability of diffeomorphisms, Trans. A. M. S. 158 (1971), 301308.
  * [10] I. M. Gelfand Lectures on linear algebra. M. Science. 1971. 273 p.
  * [11] V. Z. Grines, S. H. Kapkaeva, O. V. Pochinka, A three-colour graph as a complete topological invariant for gradient-like diffeomorphisms of surfaces, Mat. Sb., 205:10 (2014), 1946; Sb. Math., 205:10 (2014), 13871412
  * [12] V. Z. Grines, E. V. Zhuzhoma, V. S. Medvedev and O. V. Pochinka, Global attractor and repeller of MorseSmale diffeomorphisms, Proc. Steklov Inst. Math., 271 (2010), 103-124.
  * [13] V. Grines V., T. Medvedev, O. Pochinka, Dynamical Systems on 2- and 3-Manifolds. Switzerland, Springer International Publishing. (2016).
  * [14] M. Hirsch, Differential Topology, Springer 1976, reprint 1997.
  * [15] M. W. Hirsh, C. C. Pugh, M. Shub. Invariant manifolds. Springer Lecture Notes in Math., 583 (1977).
  * [16] V.I. Lukyanov, L.P. Shilnikov. On some bifurcations of dynamical systems with homoclinic structures. Doc. USSR Academy of Sciences, 1978, 243: 1, 26–29.
  * [17] S. Matsumoto. There are two isotopic Morse-Smale diffeomorphisms which cannot be joined by simple arcs, Inventiones mathematical, vol. 51, pp. 1-7, (1979).
  * [18] J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, Princeton, NJ, 1965.
  * [19] Munkres, J.:Obstructions to the smoothing of piecewise-differentiable homeomorphisms.Ann. 2110 Math. 72(3), 521–554 (1960).
  * [20] S. Newhouse, J. Palis, and F. Takens. Stable arcs of diffeomorphisms, Bull. Amer. Math. Soc., 82, No. 3, 499–502 (1976).
  * [21] S.E. Newhouse, J. Palis, F. Takens. Bifurcations and stability of families of diffeomorphisms, Publications mathematiques de l’ I.H.E.S, (1983), 57, 5-71.
  * [22] S. Newhouse, M. Peixoto. There is a simple arc joining any two Morse-Smale fows, Asterisque, 31, 15–41 (1976).
  * [23] E. Nozdrinova. Rotation number as a complete topological invariant of a simple isotopic class of rough transformations of a circle. Russian Journal of Nonlinear Dynamics, (2018), 14:4, 543-551.
  * [24] Nozdrinova E., Pochinka O. On the existence of a smooth arc without bifurcations joining source-sink diffeomorphisms on the 2-sphere // Journal of Physics: Conference Series. 2018. Vol. 990. No. 1. P. 1-7.
  * [25] Nozdrinova E., Pochinka O. Solution of the 33rd Palis-Pugh problem for gradient-like diffeomorphisms of a two-dimensional sphere. DCDS. 2020.
  * [26] J. Palis, V. de Melo Geometric theory of dynamical systems. Mir. 1998. 301 p.
  * [27] J. Palis, C. Pugh. Lecture Notes in Math. (1975), 468, 345-353.
  * [28] D. Pixton. Wild unstable manifolds // Topology. 1977. V. 16. N. 2. 167–172.
  * [29] Rolfsen D. Knots and links. 1977. vol 83, 5 (Bull. Amer. Math. Soc.) pp 931–35
  * [30] Smale S. Diffeomorphisms of the 2-sphere. Proc. AMS. (1959), Vol. 10, pp. 621–26.