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Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surface E. V. Nozdrinova, O. V. Pochinka, E. V. Tsaplina National Research University – Higher School of Economics in Nizhny Novgorod
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    § 1. Introduction and formulation of the resultLet $f\colon M^n\to M^n$ be a Morse–Smale diffeomorphism on a closed connected $n$-manifold. Let $\Omega^0_f$, $\Omega^1_f$, and $\Omega^2_f$ be, respectively, the sets of sinks, saddles and sources of the diffeomorphism $f$. For any (possibly empty) $f$-invariant set $\Sigma\subset\Omega^1_f$, let $W^{\mathrm{u}}_\Sigma$ denote the union of unstable manifolds of all points from $\Sigma$. For a set $\Sigma$ such that $\operatorname{cl}(W^{\mathrm{u}}_\Sigma)\setminus W^{\mathrm{u}}_\Sigma\subset\Omega^0_f$, we put
$$
\begin{equation*}
A_\Sigma=\Omega^0_f\cup W^{\mathrm{u}}_\Sigma,\qquad R_\Sigma=\Omega^2_f\cup W^{\mathrm{s}}_{\Omega^1_f\setminus\Sigma}.
\end{equation*}
\notag
$$
According to [1], the sets $A_\Sigma$ and $R_\Sigma$ are, respectively, the dual attractor and the dual repeller. In [2], the set is called the characteristic space, and the orbit space $\widehat V_\Sigma=V_\Sigma/f$ of the action of $f$ on $V_\Sigma$ is called the characteristic orbit space. Note that, depending on the choice of an attractor $A_\Sigma$ and a repeller $R_\Sigma$, the characteristic space of the orbits $\widehat{V}_\Sigma$ can be either connected or disconnected. An orientation-preserving gradient-like Morse–Smale diffeomorphism on a two-dimensional sphere $\mathbb{S}^2$ is shown in Fig. 1. In the first case, $\Sigma=\{\varnothing\}$. In this case, $\widehat V_\Sigma$ is homeomorphic to the disjoint union of two tori. In the second case, $\Sigma=\{\sigma\}$, and the characteristic space of orbits is homeomorphic to a torus. There are a number of examples in which by a reasonable choice of a dual pair one can obtain a complete topological classification of some subset of Morse–Smale dynamical systems (see, for example, [3]–[7] and the survey [8]). In the majority of cases, the problem of finding complete topological invariants is based on the existence of a connected characteristic space of orbits for the class of considered systems. For example, according to [3], for any 3-Morse–Smale diffeomorphism, the characteristic space of orbits constructed for the set $\Sigma$ consisting of saddle points with one-dimensional unstable manifold is connected. This fact played a key role in obtaining a complete topological classification of such diffeomorphisms in [3]. According to [1], any Morse–Smale diffeomorphism on a manifold of dimension $n>3$ also has a connected characteristic space of orbits. For Morse–Smale diffeomorphisms on a surface, this is not true in general. So, Fig. 2 shows the phase portrait of a Morse–Smale diffeomorphism on the 2-sphere $f\colon \mathbb S^2\to\mathbb S^2$ which is not gradient-like and does not have a connected characteristic space of orbits. Indeed, the non-wandering set $\Omega_f$ consists of six fixed points (two sinks $\omega_1$, $\omega_2$, two sources $\alpha_1$, $\alpha_2$, and two saddles $\sigma_1$, $\sigma_2$), which leads to three possible different variants of the set $\Sigma$: 1) $\Sigma=\varnothing$, $A_\Sigma=\omega_1\sqcup\omega_2$; 2) $\Sigma=\{\sigma_2\}$, $A_\Sigma=\omega_1\sqcup (W^{\mathrm{u}}_{\sigma_2}\cup\omega_2)$; 3) $\Sigma=\{\sigma_1,\sigma_2\}$, $R_\Sigma=\alpha_1\sqcup\alpha_2$. In all cases, either $A_\Sigma$ or $R_\Sigma$ consists of two connected components, each of which is $f$-invariant. We have $V_\Sigma=W^{\mathrm{s}}_{A_\Sigma}\setminus A_\Sigma= W^{\mathrm{u}}_{R_\Sigma}\setminus R_\Sigma$, and hence $V_\Sigma$ consists of two connected components of which each componet is $f$-invariant. Hence the set $\widehat V_f$ is disconnected. In this paper, we study of the existence of a connected characteristic space of orbits for gradient-like diffeomorphisms on surfaces. Let $f\colon M^2\to M^2$ be such a diffeomorphism. According to [1], the set $A_{\Omega_f^1}$ is a connected attractor. Hence of the graph $\Gamma_f$ whose edges correspond to the unstable saddle separatrices and whose vertices correspond to the points of the set $\Omega_f^0\cup\Omega_f^1$ is also connected. The existence of a connected characteristic space in a diffeomorphism $f$ depends on the existence of a special connected subgraph in the graph $\Gamma_f$. We need the following definitions for the description of this graph. The orientation type of a saddle point $\sigma$ of period $m_\sigma$ is the pair $\varsigma_\sigma=(\nu_\sigma, \lambda_\sigma)$, where $\nu_\sigma=+1\ (-1)$ if $f^{m_\sigma}|_{W^{\mathrm{s}}_{\sigma}}$ preserves (reverses) orientation; $\lambda_\sigma=+1\ (-1)$ if $f^{m_\sigma}|_{W^{\mathrm{u}}_{\sigma}}$ preserves (reverses) orientation. Let ${\overline\Omega}^1_f$ be the set of saddle points with orientation type $(-1,+1)$ such that there is no sink $\omega$ such that $\operatorname{cl}(W^{\mathrm{u}}_\sigma)\setminus W^{\mathrm{u}}_\sigma \subset \mathcal{O}_{\omega}$. Let $\overline{\Omega}^{\,0}_f$ be the set of $\omega$ sinks such that $f^{m_\omega}|_{W^{\mathrm{s}}_{\omega}}$ reverses orientation, and there is no saddle point $\sigma\in\overline{\Omega}^{\,1}_f$ such that $\operatorname{cl}(W^{\mathrm{u}}_\sigma)\setminus W^{\mathrm{u}}_\sigma \subset \mathcal{O}_{\omega}$. The subgraph $\overline\Gamma_f\subset\Gamma_f$ is called a special subgraph if all its vertices belong to the set $\overline{\Omega}^{\,0}_f\cup\overline{\Omega}^{\,1}_f$ and contain exactly one point from each orbit of the set $\overline{\Omega}^{\,0}_f$. If $\overline{\Omega}^{\,0}_f=\varnothing$, then we put $\overline\Gamma_f=\varnothing$. The main result of the present paper is as follows. Theorem 1. A gradient-like diffeomorphism $f\colon M^2\to M^2$ admits a connected characteristic space of orbits if and only if its graph $\Gamma_f$ has a connected special subgraph $\overline\Gamma_f$. In this case, if $\overline\Gamma_f=\varnothing$, then the orbit space is homeomorphic to the torus; otherwise, it is homeomorphic to the Klein bottle. As a direct corollary to Theorem 1, we have Corollary 1. Any gradient-like diffeomorphism $f\colon M^2\to M^2$ such that $\overline{\Omega}^{\,0}_f$ consists of at most one orbit admits a connected characteristic space of orbits. In particular, any orientation-preserving diffeomorphism admits a connected characteristic space of orbits,1 since it has no sinks of negative orientation type. The following result shows that any surface admits a gradient-like diffeomorphism without connected characteristic space of orbits. Theorem 2. 1) On any orientable surface of genus, there is an orientation-changing gradient-like diffeomorphism that does not admit a connected characteristic space. 2) A gradient-like diffeomorphism without connected characteristic space exists on a non-orientable surface of any genus. § 2. Morse–Smale diffeomorphismsLet $M^n$ be a smooth closed orientable manifold and let $f$ be a diffeomorphism on $M^n$. For a diffeomorphism $f$, we say that $x \in X$ is a wandering point if there exists an open neighbourhood $U_x$ of $x$ such that $f^{n} (U_x)\cap U_x=\varnothing $ for all $n\in \mathbb{N}$. Otherwise, $x$ is a non-wandering point. It is immediate from the definition that any point in the neighbourhood $U_x$ of a wandering point $x$ is wandering itself, and therefore, the set of wandering points is open, while the set of non-wandering points is closed. The set of all non-wandering points of a diffeomorphism $f$ is called the non-wandering set (and is usually denoted by $\Omega_f$). As simplest examples of hyperbolic sets, we mention the hyperbolic fixed points of a diffeomorphism, which can be classified as follows. Let $f\colon X\to X$ be a diffeomorphism and $f(p)=p$. A point $p$ is hyperbolic if the absolute value of each eigenvalue of the Jacobi matrix $(\partial f/\partial x)|_p$ is not 1. If the absolute values of each eigenvalue is smaller than 1, then $p$ is an attracting point (a sink point) or a sink; if the absolute values of each eigenvalue is greater than 1, then $p$ is a repelling point (a source point) or a source. A point is a node if it is either attracting or repelling. A hyperbolic fixed point that is not a node is called a saddle point or a saddle. If a point $p$ is a periodic point of $f$ with period $\operatorname{per}(p)$, then by applying the previous construction to the diffeomorphism $f^{\operatorname{per}(p)}$, we obtain a classification of hyperbolic periodic points similar to that of fixed hyperbolic points. For a hyperbolic periodic point $p$, we define the stable manifold
$$
\begin{equation*}
W^{\mathrm{s}}_p=\Bigl\{x\in M^n\colon \lim_{k\to +\infty}d(f^{k\operatorname{per}(p)}(x),p)\to 0\Bigr\}
\end{equation*}
\notag
$$
and the unstable manifold
$$
\begin{equation*}
W^{\mathrm{u}}_p=\Bigl\{x\in M^n\colon\lim_{k\to +\infty}d(f^{-k\operatorname{per}(p)}(x),p)\to 0\Bigr\},
\end{equation*}
\notag
$$
which are smooth embeddings of $\mathbb R^{n-q_p}$ and $\mathbb R^{q_p}$, respectively. Here, $q_p$ is the number of eigenvalues of the Jacobian matrix $(\partial f^{\operatorname{per}(p)}/\partial x)|_p$ whose absolute values exceed $1$. For a hyperbolic fixed or periodic point $p$, the stable and unstable manifolds are called the invariant manifolds of this point; a connected component of the set $W^{\mathrm{u}}_p\setminus p$ (respectively, $W^{\mathrm{s}}_p\setminus p$) is called an unstable (stable) separatrix. A closed $f$-invariant set $A\subset M^n$ is called an attractor of a discrete dynamical system $f$ if it has a compact neighbourhood $U_A$ such that $f(U_A)\subset \operatorname{int} U_A$ and $A=\bigcap_{k\geqslant 0}f^k(U_A)$. This neighbourhood of $U_A$ is a basin of attraction. A repeller is an attractor for $f^{-1}$. An attractor and a repeller are called dual if the complement of the basin of attraction of the attractor is a basin of attraction of the repeller. A diffeomorphism $f\colon M^n\to M^n$ is called a Morse–Smale diffeomorphism if 1) the non-wandering set $\Omega_f$ consists of a finite number of hyperbolic orbits; 2) the manifolds $W^{\mathrm{s}}_p$, $W^{\mathrm{u}}_q$ intersect transversally for all non-wandering points $p$ and $q$. A Morse–Smale diffeomorphism is a gradient-like diffeomorphism if the condition $W^{\mathrm{s}}_{\sigma_1} \cap W^{\mathrm{u}}_{\sigma_2} \neq \varnothing $ for different points $\sigma_1, \sigma_2 \in \Omega_f$ implies that $\operatorname{dim}W^{\mathrm{u}}_{\sigma_1}< \operatorname{dim} W^{\mathrm{u}}_{\sigma_2}$. In the dimension $n=2$, the set of gradient-like diffeomorphisms coincides with that of Morse–Smale diffeomorphisms with disjoint saddle separatrices. If $M^n$ is an orientable manifold, a diffeomorphism $f \colon M^n \to M^n$ is said to be an orientation-preserving diffeomorphism if $f$ has positive Jacobian at least at one point; otherwise $f$ is an orientation-changing diffeomorphism. § 3. Orbit spaces of invariant subsets of gradient-like diffeomorphisms of surfacesLet $f\colon M^2\to M^2$ be a gradient-like diffeomorphism on a closed surface $M^2$. Let $\omega$ be an $m_\omega$-periodic sink of of the diffeomorphism $f$. By Theorem 5.5 in [10], in some neighbourhood of $\omega$, the diffeomorphism $f^{m_\omega}$ is topologically conjugate to the linear diffeomorphism of the plane given by the matrix $\left(\begin{smallmatrix} 1/2 &0\\ 0 &\varsigma_\omega\cdot 1/2 \end{smallmatrix}\right)$, where $\varsigma_\omega=+1\ (-1)$ if $f^{m_\omega}|_{W^{\mathrm{s}}_{\omega}}$ preserves (reverses) orientation. We will say that the sink $\omega$ has positive orientation type if $\varsigma_\omega=+1$; otherwise, it has negative orientation type. Let $\mathcal O_\omega$ be the orbit of the point $\omega$. We put $V_\omega=W^{\mathrm{s}}_{\mathcal O_{\omega}}\setminus\mathcal O_{\omega}$. Let $\widehat V_{\omega}=V_{\omega}/f$ be the space of orbits of the action of the group $F=\{f^k, \, k\in\mathbb Z\}\cong\mathbb Z$ on $V_{\omega}$, and let $p_{\omega}\colon V_{\omega}\to\widehat V_{\omega}$ be the natural projection. The group $F$ acts freely and discontinuously2 on $V_\omega$, and hence the projection $p_{\omega}$ is a covering3 if 1) $p$ is surjective; 2) for any $x\in X$, there exists a neighbourhood $U$ such that $p^{-1}(U)=\bigcup_{j\in J}\overline U_j$ for some family $\{\overline U_j,\, j\in J\}$ of open subsets $\overline X$ satisfying $\overline U_j\cap \overline U_k=\varnothing$ for $j\neq k$, and $p|_{\overline U_j}\colon \overline U_j\to U$ is a homeomorphism for all $j\in J$, which induces the structure of a smooth 2-manifold on $\widehat V_{\omega}$ and the map $\eta_{\omega}$ is composed of homomorphisms4 into the group $\mathbb Z$ from the fundamental group of each connected component of the space $\widehat V_{\omega}$ (see, for example, [11]). Proposition 1 (see Proposition in [12]). The manifold $\widehat V_{\omega}$ is diffeomorphic to a two-dimensional torus if $\varsigma_\omega=+1$ and is diffeomorphic to a Klein bottle if $\varsigma_\omega = -1$. In addition, $\eta_{\omega}(\pi_1(\widehat V_{\omega}))= m_\omega\mathbb Z$ (here, $m_\omega\mathbb Z$ is a group of integer multiples of $m_\omega$). Proposition 2 (see Proposition 2 in [12]). Let $l\in V_\omega$ be an $m_l$-periodic separatrix of a saddle point of a diffeomorphism $f$. Then the set $\widehat{l}=p_{\omega}(l)$ is a circle smoothly embedded in $\widehat{V}_\omega$ such that $\eta_{\omega}([\widehat{l}])=m_l$. In a similar way, one defines the orientation type $\varsigma_\alpha$ for a periodic source $\alpha$ of a diffeomorphism $f$, the orbit space $\widehat V_\alpha$, and the projection of the stable separatrix of a saddle point into the orbit space. Let $\sigma$ be an $m_\sigma$-periodic saddle point of a diffeomorphism $f$. By Theorem 5.5 in [10], in some neighbourhood of $\sigma$, the diffeomorphism $f^{m_\sigma}$ is topologically conjugate to a linear diffeomorphism of the plane defined by the matrix $\left(\begin{smallmatrix} \nu_\sigma\cdot 1/2 &0\\ 0&\lambda_\sigma\cdot 2 \end{smallmatrix}\right)$, where $\nu_\sigma=+1\ (-1)$ if $f|_{W^{\mathrm{s}}_p}$ preserves (reverses) orientation, and $\lambda_\sigma=+1\ (-1)$ if $f|_{W^{\mathrm{u}}_p}$ preserves (reverses) orientation. The pair $\varsigma_\sigma=(\nu_\sigma, \lambda_\sigma)$ will be called the orientation type of a saddle point $\sigma$; the corresponding linear diffeomorphism is denoted by $a_{\varsigma_\sigma }\colon \mathbb R^2\to\mathbb R^2$. If $\nu_\sigma>0$ and $\lambda_\sigma>0$, then the orientation type is called positive; otherwise it is called negative. The diffeomorphism $a_{\varsigma_\sigma }\colon \mathbb R^2\to\mathbb R^2$ has a unique fixed saddle point at the origin $O$ with stable manifold $W^{\mathrm{s}}_O=Ox_1$ and unstable manifold $W^{\mathrm{u}}_O=Ox_{2}$. Setting $\mathcal N=\{(x_1,x_2)\in\mathbb{R}^2\colon |x_1x_{2}|\leqslant 1\}$, we note that the set $\mathcal N$ is invariant under the the canonical diffeomorphism $a_{\varsigma_\sigma}$. We say that $\mathcal N_\sigma$ of a point $\sigma$ is a linearizing neighbourhood if there is a homeomorphism ${\mu}_\sigma\colon \mathcal N_\sigma\to \mathcal{N}$ which conjugates the diffeomorphism $f^{m_{\sigma}}|_{\mathcal N_\sigma}$ with $a_{\varsigma_\sigma}|_{\mathcal{N}}$. The neighbourhood $\mathcal N_{\mathcal O_\sigma}=\bigcup_{j=0}^{m_{\sigma}-1}f^j(N_\sigma)$ of the orbit ${\mathcal O_\sigma}=\bigcup_{j=0}^{m_{\sigma}-1}f^j(\sigma)$ equipped with the map $\mu_{\mathcal O_\sigma}$ composed of the homeomorphisms $\mu_\sigma f^{-j}\colon f^j(\mathcal N_\sigma)\to \mathcal N$, $j=0,\dots,m_{\sigma}-1$, is the linearizing neighbourhood of the orbit $\mathcal O_\sigma$. Proposition 3 (see Theorem 2.2 in [2]). Any saddle point (orbit) of a gradient-like diffeomorphism $f\colon M^2\to M^2$ has a linearizing neighbourhood. Let $N^{\mathrm{u}}_{\sigma}=N_{\mathcal O_\sigma}\setminus W^{\mathrm{s}}_{\mathcal O_\sigma}$. Then the group $F$ acts freely and discontinuously on $N^{\mathrm{u}}_{\sigma}$. Hence it generates the space of orbits $\widehat N^{\mathrm{u}}_{\sigma}=N^{\mathrm{u}}_{\sigma}/f$, the natural projection of $p_{\sigma}^{\mathrm{u}}\colon N^{\mathrm{u}}_{\sigma} \to\widehat N^{\mathrm{u}}_{\sigma}$, and the map $\eta^{\mathrm{u}}_{\sigma}$ composed of the homomorphisms into the group $\mathbb Z$ from the fundamental group of each connected component of the space $\widehat N^{\mathrm{u}}_{\sigma}$. Proposition 4 (see Proposition 5 in [12]). The manifold $\widehat N^{\mathrm{u}}_{\sigma}$ has the following topological type (see Fig. 3) depending on $\varsigma_{\sigma}$: 1) if $\varsigma_{\sigma}=(+1,+1)$, then the space $\widehat N^{\mathrm{u}}_{\sigma}$ consists of two connected components $\widehat N^{\mathrm{u}1}_{\sigma}$ and $\widehat N^{\mathrm{u}2}_{\sigma}$ each of which is diffeomorphic to an annulus, and $\eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}1}_{\sigma}))= \eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}2}_{\sigma}))= m_\sigma\mathbb Z$; 2) if $\varsigma_{\sigma}=(-1,+1)$, then the space $\widehat N^{\mathrm{u}}_{\sigma}$ consists of two connected components $\widehat N^{\mathrm{u}1}_{\sigma}$ and $\widehat N^{\mathrm{u}2}_{\sigma}$ each of which is diffeomorphic to a Mobius strip, and $\eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}1}_{\sigma}))= \eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}2}_{\sigma}))= m_\sigma\mathbb Z$; 3) if $\varsigma_{\sigma}=(+1,-1)$, then the space $\widehat N^{\mathrm{u}}_{\sigma}$ consists of one connected component $\widehat N^{\mathrm{u}}_{\sigma}$, which is diffeomorphic to an annulus, and $\eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}}_{\sigma}))= 2m_\sigma\mathbb Z$; 4) if $\varsigma_{\sigma}=(-1,-1)$, then the space $\widehat N^{\mathrm{u}}_{\sigma}$ consists of one connected component $\widehat N^{\mathrm{u}}_{\sigma}$, which is diffeomorphic to an annulus, and $\eta^{\mathrm{u}}_{\sigma}(\pi_1(\widehat N^{\mathrm{u}}_{\sigma}))= 2m_\sigma\mathbb Z$. In Fig. 3, the fundamental domains5 of actions of the group $F$ on $N^{\mathrm{u}}_{\sigma}$ for various types of $\varsigma_{\sigma}$ are darkened. In the cases $\varsigma_{\sigma}=(+1,+1)$ and $\varsigma_{\sigma}=(-1,+1)$, the fundamental domain consists of two disjoint curvilinear trapezoids, and for $\varsigma_{\sigma}=(+1,-1)$ and $\varsigma_{\sigma}=(-1,-1)$, the fundamental domain can be chosen as a single curvilinear trapezoid. The space of orbits $\widehat {N}^{\mathrm{u}}_\sigma$ is obtained from the corresponding curvilinear trapezoids by identifying the points on the horizontal segments of the boundary. In a similar way, one can introduce the space of orbits $\widehat N^{\mathrm{s}}_{\sigma}=N^{\mathrm{s}}_{\sigma}/f$ of actions of the group $F$ on $N^{\mathrm{s}}_{\sigma}=N_{\mathcal O_{\sigma}}\setminus W^{\mathrm{u}}_{\mathcal O_{\sigma}}$, the covering $p_{\sigma}^{\mathrm{s}}\colon N^{\mathrm{s}}_{\sigma}\to \widehat N^{\mathrm{s}}_{\sigma}$, and the map $\eta^{\mathrm{s}}_{\sigma}$ composed of the homomorphisms into the group $\mathbb Z$ from the fundamental group of each connected component of the space $\widehat N^{\mathrm{s}}_{\sigma}$. In addition, the rearrangement map
$$
\begin{equation*}
\widehat\psi_\sigma=p_{\sigma}^{\mathrm{s}} (p^{\mathrm{u}}_\sigma)^{-1} \colon \partial \widehat{N}^{\mathrm{u}}_{\sigma}\to \partial \widehat{N}^{\mathrm{s}}_{\sigma},
\end{equation*}
\notag
$$
is well defined (see Fig. 4). Let $\Omega^0_f$, $\Omega^1_f$, $\Omega^2_f$ be the sets of sinks, saddles, and sources of a diffeomorphism $f$. For any (possibly empty) $f$-invariant set $\Sigma\subset\Omega^1_f$ such that $\operatorname{cl}(W^{\mathrm{u}}_\Sigma)\setminus W^{\mathrm{u}}_\Sigma\subset\Omega^0_f$, we define
$$
\begin{equation*}
A_\Sigma=\Omega^0_f\cup W^{\mathrm{u}}_\Sigma,\qquad R_\Sigma=\Omega^2_f\cup W^{\mathrm{s}}_{\Omega^1_f\setminus\Sigma}.
\end{equation*}
\notag
$$
The set is known as the characteristic space. The group $F$ acts freely and discontinuously on $V_\Sigma$. The quotient space is called the characteristic space of orbits. The natural projection $p_{\Sigma}\colon V_\Sigma\to\widehat V_\Sigma$ induces the map $\eta_{\Sigma}$ composed of homomorphisms into the group $\mathbb Z$ from the fundamental group of each connected component of the space $\widehat V_{\Sigma}$. In general, the characteristic space of orbits is not connected. Let $\widehat V_\Sigma^1,\dots,\widehat V_\Sigma^k$ be the connected components of the space $\widehat V_{\Sigma}$. Let $V_\Sigma^1=p_{\Sigma}^{-1}(\widehat V_\Sigma^1),\dots,V_\Sigma^k = p_{\Sigma}^{-1}(\widehat V_\Sigma^k)$, and let $m_1,\dots,m_k$ be the number of connected components of $V_\Sigma^1,\dots, V_\Sigma^k$, respectively.Proposition 5 (see6 Proposition 1 in [13]). 1) Each connected component of $V_\Sigma^i$, $i\in\{1,\dots,k\}$, is diffeomorphic to $\mathbb S^1\times\mathbb R$. 2) Each connected component of $\widehat V_{\Sigma}^i$ is diffeomorphic to a two-dimensional torus if the diffeomorphism $f^{m_i}|_{V_\Sigma^i}$ preserves orientation; otherwise, it is diffeomorphic to the Klein bottle. 3) $\eta_{\Sigma}(\pi_1(\widehat V_{\Sigma}^i))= m_i\mathbb Z$. 4) If $l\in V_\Sigma$ is an $m_l$-periodic separatrix of a saddle point of a diffeomorphism $f$, then the set $\widehat l=p_{\Sigma}(l)$ is a circle smoothly embedded in $\widehat{V}_\Sigma$ and such that $\eta_{\Sigma}([\widehat{l}])=m_l$. The homology group $H_1(\widehat V_{\Sigma}^i)$ is isomorphic to $\mathbb Z^2$ if $\widehat V_{\Sigma}^i$ is a torus, and is isomorphic to $\mathbb{Z} \times \mathbb{Z}_2$ if $\widehat V_{\Sigma}^i$ is a Klein bottle. In both cases, the homology class of a loop $c\subset\widehat V_{\Sigma}^i$ is a pair of numbers $(\alpha,\beta)$, where $\eta_{\Sigma}([c])=\alpha m_i$. If the curve $c$ is a node, then $(\alpha,\beta)$ are coprime. In particular, for the node $\widehat l$ which is the projection of a separatrix, we have $\alpha=m_l/m_i\neq 0$, that is, $\widehat l$ is an essential node. Both a tubular neighbourhood of such a node on the torus $\widehat V_{\Sigma}^i$ and its complement is an annulus. The nodes (up to a change of orientation) on the Klein bottle are realized in the following homology classes: $(0,0)$, $(0,1)$, $(2,0)$, $(1,0)$, $(1,1)$. Note that the node $\widehat l$ can only lie in the last three classes. If the node $\widehat l$ belongs to the homology class $(2,0)$, then its tubular neighbourhood on the Klein bottle $\widehat V_{\Sigma}^i$ is an annulus, and its complement consists of two Mobius strips. Otherwise, both a tubular neighbourhood of the node $\widehat l$ and its complement is a Mobius strip. § 4. Rearrangement of characteristic spaces of orbitsIn this section, we study how the characteristic space of orbits changes if the set $\Sigma$ is augmented with a single saddle orbit. Lemma 1. Let $\Sigma'=\Sigma\cup\mathcal O_\sigma$ for some saddle orbit $\mathcal O_\sigma$ and let $\widehat v$, $\widehat v^{\,\prime}$ be the disjoint union of connected components of the spaces $\widehat V_{\Sigma}$ and $\widehat V_{\Sigma'}$ which have non-empty intersection with $\widehat N^{\mathrm{u}}_\sigma$ and $\widehat N^{\mathrm{s}}_\sigma$, respectively. Then In addition, where, for various types of orientation, the following possibilities hold. 1. If $\varsigma_\sigma=(+1,+1)$, then
2. If $\varsigma_\sigma=(-1,+1)$, then 3. If $\varsigma_\sigma=(+1,-1)$, then 4. If $\varsigma_\sigma=(-1,-1)$, then Proof. We have $\Sigma'=\Sigma \cup \mathcal O_{ \sigma}$, and hence $A_{\Sigma'}=A_\Sigma\cup W^{\mathrm{u}}_{\mathcal O_\sigma}$, $R_{\Sigma'}=R_\Sigma\setminus W^{\mathrm{s}}_{\mathcal O_\sigma}$, $V_{\Sigma}=M^2\setminus(A_{\Sigma}\cup R_{\Sigma})$, and $V_{\Sigma'}=M^2\setminus(A_{\Sigma'}\cup R_{\Sigma'})$. Next, we have $N^{\mathrm{u}}_{\sigma}=N_{\mathcal O_\sigma}\setminus W^{\mathrm{s}}_{\mathcal O_\sigma}$, $N^{\mathrm{u}}_\sigma\subset V_\Sigma$, $\widehat N^{\mathrm{u}}_\sigma\cong p_{\Sigma}(N^{\mathrm{u}}_\sigma) \subset\widehat V_\Sigma$ and $N^{\mathrm{s}}_{\sigma}= N_{\mathcal O_\sigma}\setminus W^{\mathrm{u}}_{\mathcal O_\sigma}$, $N^{\mathrm{s}}_\sigma\subset V_{\Sigma'}$, $\widehat N^{\mathrm{s}}_\sigma \cong p_{\Sigma'}(N^{\mathrm{s}}_\sigma)\subset\widehat V_{\Sigma'}$. Hence, the boundaries of the sets $\widehat V_\Sigma\setminus \operatorname{int}\widehat N^{\mathrm{u}}_\sigma$, $\widehat V_{\Sigma'} \setminus \operatorname{int}\widehat N^{\mathrm{s}}_\sigma$ are homeomorphic by the rearrangement homeomorphism $\widehat\psi_\sigma$, that is,
So, to get the space $\widehat V_{\Sigma'}$, we should delete the set $\widehat N^{\mathrm{u}}_{{ \sigma}}$ from the space $\widehat V_\Sigma$, and glue the set $\widehat N^{\mathrm{s}}_{\sigma}$ to the boundary of the resulting set via $\widehat\psi_\sigma$. Let us verify that in the formula
$$
\begin{equation*}
\widehat V_{\Sigma'}\cong(\widehat V_\Sigma\setminus \widehat v)\sqcup\widehat v^{\,\prime}
\end{equation*}
\notag
$$
only the cases listed in the lemma are realized.
1. Let $\varsigma_\sigma=(+1,+1)$, then by Proposition 4, the sets $\widehat N^{\mathrm{u}}_\sigma$ and $\widehat N^{\mathrm{s}}_\sigma$ are homeomorphic to pairs of annuli. Hence the connected components (thee are one or two of such components) of the set $\widehat v$ are either tori and Klein bottles. There are several cases to consider. a) Let $\widehat v$ be a disjoint union of two tori $\widehat v_1$ and $\widehat v_2$. Then the annulus $\widehat N^{\mathrm{u}1}_\sigma$ lies in $\widehat v_1$ and the annulus $\widehat N^{\mathrm{u}2}_\sigma $ lies in $\widehat v_2$, and, by Proposition 4, each of them is embedded in the torus $\widehat v_i$, so that $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma))\ne\{0\}$. Therefore, the set $\widehat v_i\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to the annulus. Now the torus $\widehat v^{\,\prime}$ is obtained by gluing the pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ to the pair of annuli $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map (see Fig. 5). b) Let $\widehat v$ be a disjoint union of the torus $\widehat v_1$ and the Klein bottle $\widehat v_2$. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v_1$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v_2$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}1}$ is embedded in the torus $\widehat v_1$, and so $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}1}_\sigma))\ne\{0\}$. Hence the set $\widehat v_1\setminus \widehat N^{\mathrm{u}1}_\sigma$ is homeomorphic to the annulus. Similarly, the annulus $\widehat N^{\mathrm{u}2}_\sigma$ is embedded in the Klein bottle $\widehat v_2$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}2}_\sigma)) = 2\mathbb{Z}$, and is included in the homology class $(2,0)$. Hence the set $\widehat v_2\setminus \widehat N^{\mathrm{u}2}_\sigma$ is homeomorphic to two Mobius strips. So, if a pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ is glued to the annulus $\widehat v_1\setminus \widehat N^{\mathrm{u}1}_\sigma$ and to the pair of Mobius strips $\widehat v_2\setminus \widehat N^{\mathrm{u}2}_\sigma$ via the rearrangement map, then we get the Klein bottle $\widehat v^{\,\prime}$ (see Fig. 6). c) Let $\widehat v$ be the disjoint union of two Klein bottles $\widehat v_1$ и $\widehat v_2$. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v_1$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v_2$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the Klein bottle $\widehat v_i$ so that $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma)) = 2\mathbb{Z}$, is included in the homology class $(2,0)$. Hence, the set $\widehat v_i\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to two Mobius strips. So, if a pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ is glued to two Mobius strips $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get a disjoint union of two Klein bottles $\widehat v^{\,\prime}$ (see Fig. 7). d) Let $\widehat v$ be a torus. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the torus $\widehat v$, so that $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma))\ne\{0\}$, and, consequently, the set $\widehat v\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to a pair of annuli. By gluing a pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ to the pair of annuli $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get a disjoint union of two tori $\widehat v^{\,\prime}$ (see Fig. 8). e) Let $\widehat v$ be a Klein bottle. Then $\widehat N^{\mathrm{u}1}_\sigma\,{\subset}\, \widehat v$ and $\widehat N^{\mathrm{u}2}_\sigma\,{\subset}\, \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the Klein bottle $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma)) = 2\mathbb{Z}$, and and is included in the homology class $(2,0)$. Hence the set $\widehat v\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to a pair of Mobius strips and an annulus. So, by gluing the pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ to two pairs of Mobius strips and the annulus $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get a disjoint union of the torus and the Klein bottle $\widehat v^{\,\prime}$ (see Fig. 9). f) Let $\widehat v$ be a torus. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the torus $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma))\ne\{0\}$. Consequently, $\widehat v\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to a pair of annuli. If $M^2$ is a non-orientable surface, by gluing the pair of annuli $\widehat N^{\mathrm{s}}_{\sigma}$ to the pair of annuli $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get the torus $\widehat v^{\,\prime}$ (see Fig. 10). 2. Let $\varsigma_\sigma=(-1,+1)$. By Proposition 4, the sets $\widehat N^{\mathrm{u}}_\sigma$ and $\widehat N^{\mathrm{s}}_\sigma$ are homeomorphic, respectively, to a pair of Mobius strip and an annulus. Hence the connected components that contain $\widehat N^{\mathrm{u}}_\sigma$ can only be one or two Klein bottles, and $\widehat N^{\mathrm{s}}_\sigma$ is a torus or a Klein bottle. There are several cases to consider. a) Let $\widehat v$ be a disjoint union of two Klein bottles $\widehat v_1$ and $\widehat v_2$. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v_1$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v_2$. By Proposition 4, the Mobius strip $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the Klein bottle $\widehat v_i$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma)) = 1\mathbb{Z}$, and is included in the homology class $(1,0)$. Hence the set $\widehat v_i\setminus \widehat N^{\mathrm{u}i}_\sigma$ is homeomorphic to the Mobius strip. So, by gluing the annulus $\widehat N^{\mathrm{s}}_\sigma$ to the pair of Mobius strips $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$, via the rearrangement map, we get the Klein bottle $\widehat v^{\,\prime}$ (see Fig. 11). b) Let $\widehat v$ be a Klein bottle. Then $\widehat N^{\mathrm{u}1}_\sigma\subset \widehat v$ and $\widehat N^{\mathrm{u}2}_\sigma\subset \widehat v$. By Proposition 4, the Mobius strip $\widehat N^{\mathrm{u}i}_\sigma$, $i=1,2$, is embedded in the Klein bottle $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}i}_\sigma)) = 1\mathbb{Z}$, and is included in the homology class $(1,0)$. Hence the set $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to the annulus. By gluing the annulus $\widehat N^{\mathrm{s}}_\sigma$ to the annulus $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get the torus $\widehat v^{\,\prime}$ (see Fig. 12). 3. Let $\varsigma_\sigma=(+1,-1)$. By Proposition 4, the set $\widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to the annulus, and $\widehat N^{\mathrm{s}}_\sigma$ is homeomorphic to a pair of Mobius strips. Hence if a connected component contains $\widehat N^{\mathrm{u}}_\sigma$, then it is either a torus or a Klein bottle, and if a connected component contains $\widehat N^{\mathrm{s}}_\sigma$, then it is only a Klein bottle. a) Let $\widehat v$ be a torus. Then $\widehat N^{\mathrm{u}}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}}_\sigma$ is embedded in the torus $\widehat v$, so that $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}1}_\sigma))\ne\{0\}$. Hence the set $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to the annulus. By gluing a pair of Mobius strips $\widehat N^{\mathrm{s}}_\sigma$ to the annulus $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get the Klein bottle $\widehat v^{\,\prime}$ (see Fig. 13). b) Let $\widehat v$ be a Klein bottle. Then $\widehat N^{\mathrm{u}}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}}_\sigma$ is embedded in the Klein bottle $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}}_\sigma)) = 2\mathbb{Z}$, and is included in the homology class $(2,0)$. Hence the set $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to a pair of Mobius strips. By gluing a pair of Mobius strips $\widehat N^{\mathrm{s}}_\sigma$ to the pair of Mobius strips $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get a disjoint union of two Klein bottles $\widehat v^{\,\prime}$ (see Fig. 14). 4. Let $\varsigma_\sigma=(-1,-1)$. By Proposition 4, the sets $\widehat N^{\mathrm{u}}_\sigma$ and $\widehat N^{\mathrm{s}}_\sigma$ are homeomorphic to pairs of annuli. Hence the connected component of the set $\widehat v$ is either a torus or a Klein bottle. There are several cases to consider. a) Let $\widehat v$ be a Klein bottle. Then $\widehat N^{\mathrm{u}}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}}_\sigma$ is embedded in the Klein bottle $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}}_\sigma)) = 2\mathbb{Z}$, and is included in the homology class $(2,0)$. Hence the set $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to a pair of Mobius strips. By gluing the annulus $\widehat N^{\mathrm{s}}_\sigma$ to the pair of Mobius strips $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get the Klein bottle $\widehat v^{\,\prime}$ (see Fig. 15). b) Let $\widehat v$ be a torus. Then $\widehat N^{\mathrm{u}}_\sigma\subset \widehat v$. By Proposition 4, the annulus $\widehat N^{\mathrm{u}}_\sigma$ is embedded in the torus $\widehat v$, $\eta_{\Sigma}(\pi_1(\widehat N^{\mathrm{u}}_\sigma))\ne\{0\}$. Hence (see, for example, [14]), the set $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ is homeomorphic to the annulus. By gluing the annulus $\widehat N^{\mathrm{s}}_\sigma$ to the annulus $\widehat v\setminus \widehat N^{\mathrm{u}}_\sigma$ via the rearrangement map, we get the torus $\widehat v^{\,\prime}$ (see Fig. 16). This proves the lemma. § 5. Criterion for the existence of a connected characteristic space of orbits in a gradient-like diffeomorphism of a surfaceIn this section, we will prove Theorem 1. Namely, we will show that a gradient-like diffeomorphism $f\colon M^2\to M^2$ has a connected characteristic space of orbits if and only if its graph $\Gamma_f$ has a connected special subgraph $\overline\Gamma_f$. In this case, the orbit space is homeomorphic to the torus if $\overline\Gamma_f=\varnothing$; otherwise, it is homeomorphic to the Klein bottle. Proof. Necessity. Let $f\colon M^2\to M^2$ be a gradient-like diffeomorphism with connected characteristic space of orbits $\widehat V_\Sigma$. Let $\Gamma_{\Sigma} \subset \Gamma_f$ be the subgraph which coincides (geometrically) with the attractor $A_{\Sigma}$. We will show that the graph $\Gamma_{\Sigma}$ has connected special subgraph.
Suppose on the contrary that the graph $\Gamma_{\Sigma}$ does not have a connected special subgraph. Then there exist different orbits of sinks $\mathcal O_{\omega_1}, \mathcal O_{\omega_2} \subset\overline\Omega^{\,0}_f$ such that any path connecting point of the orbit $\mathcal O_{\omega_1}$ with point of the orbit $\mathcal O_{\omega_2}$ in the graph $\Gamma_{\Sigma}$ passes through vertices that do not belong to the set $\overline\Omega^{\,0}_f\cup\overline\Omega^{\,1}_f$. We let $\widetilde\Sigma\subset\Sigma$ denote the union of all saddle orbits that have non-empty intersection with these paths. Let $A_{\widetilde\Sigma}\subset A_\Sigma$ be the corresponding attractor. By definition, the space of orbits $\widehat V_\varnothing$ of sink basins consists of tori and Klein bottles, and the characteristic space $\widehat V_{\widetilde\Sigma}$ is obtained from it by rearranging along the projections of unstable separatrices of saddle points of the set $\widetilde\Sigma$. By the definition of the set $\widetilde\Sigma$ and Lemma 1, the space $\widehat V_{\widetilde\Sigma}$ is disconnected. The characteristic space $\widehat V_{\Sigma}$ is obtained from the space $\widehat V_{\widetilde\Sigma}$ by rearranging along the projections of unstable separatrices of saddle points of the set $\Sigma\setminus\widetilde\Sigma$. Since the closures of these separatrices do not connect points of the orbit $\mathcal O_{\omega_1}$ with points of the orbit $\mathcal O_{\omega_2}$, the space of orbits $\widehat V_{\Sigma}$ is not connected, which contradicts the assumption that it is connected. Sufficiency. Let $f\colon M^2\to M^2$ be a gradient-like diffeomorphism such that its graph $\Gamma_f$ has connected special subgraph $\overline\Gamma_f$. We construct a connected characteristic space of orbits $\widehat V_\Sigma$ for the diffeomorphism $f$. We set $\Sigma_0=\varnothing$. This means that there are no saddle points and their orbits in the attractor. Let $\widehat V_1,\dots,\widehat V_l$ be the connected components of the space $\widehat V_{\Sigma_0}$. If $l=1$, then $\Sigma=\Sigma_0$. Otherwise, by Proposition 1, each connected component $\widehat V_i$ is a torus or a Klein bottle. Let $\widehat V_1,\dots,\widehat V_{l_1}$ be the Klein bottles not corresponding to the sinks of the set $\overline{\Omega}^{\,0}_f$, let $\widehat V_{l_1+1},\dots,\widehat V_{l_1+l_2}$ be the Klein bottles corresponding to the sinks of the set $\overline{\Omega}^{\,0}_f$, and let $\widehat V_{l_1+l_2+1},\dots,\widehat V_{l_1+l_2+l_3}$ be the tori. We add to the set $\Sigma_0$ the saddle points $\sigma$ of orientation type $\varsigma_\sigma=(-1,+1)$ whose unstable manifolds form loops at the sinks to which there correspond the Klein bottles. This gives us the set $\Sigma_1$. By Lemma 1, the space $\widehat V_{\Sigma_1}$ also consists of $l$ connected components $\widetilde V_1,\dots, \widetilde V_{l_1},\widehat V_{l_1+1},\dots,\widehat V_{l_1+l_2}, \widehat V_{l_1+l_2+1},\dots,\widehat V_{l_1+l_2+l_3}$, of which the first ones $l_1$ are tori. Adding the saddle points of the special subgraph to the set $\Sigma_1$, we get the set $\Sigma_2$ and the space $\widehat V_{\Sigma_2}$, which, by Lemma 1, consists of $l_1+1+ l_3$ connected components $\widetilde V_1,\dots,\widetilde V_{l_1},\widetilde V_*$, $ \widehat V_{l_1+l_2+1},\dots,\widehat V_{l_1+l_2+l_3}$, where $\widetilde V_*$ is a Klein bottle. In the connected graph $\Gamma_f$, consider a path connecting two connected components of the orbit space $\widehat V_{\Sigma_2}$. By Lemma 1, this path may consist of saddle points of orientation type $(+1,+1)$ only; they connect $l_1{+}\,1\,{+}\,l_3$ connected components of the sets $p^{-1}_{\Sigma_2}(\widetilde V_1), \dots,p^{-1}_{\Sigma_2}(\widetilde V_{l_1}),p^{-1}_{\Sigma_2}(\widetilde V_*), p^{-1}_{\Sigma_2}(\widehat V_{l_1+l_2+1}),\dots, p^{-1}_{\Sigma_2}(\widehat V_{l_1+l_2+l_3})$, where each component is taken from each set. Adding these saddles to the set $\Sigma_2$, we get the set $\Sigma$ and the connected space $\widehat V_{\Sigma}$, which by Lemma 1 is a torus if $l_2=0$ and a Klein bottle, otherwise. This proves Theorem 1. § 6. Constructive proof of Theorem 2In this section, we construct, on an arbitrary surface, a gradient-like diffeomorphism with various sinks of negative orientation type connected by unstable manifolds of saddle points of only positive orientation type. By Theorem 1, this implies the absence of a connected characteristic space of orbits for such a diffeomorphism. The required diffeomorphism will be constructed as the connected sum of elementary diffeomorphisms on surfaces of genus not exceeding 1. Below, we will describe the operation of taking the connected sum of diffeomorphisms, and the construction of elementary maps. 6.1. Connected sum of diffeomorphismsLet $f_i\colon M^2_i\,{\to}\, M^2_i$, $i\,{=}\,1,2$, be a gradient-like diffeomorphism on a closed surface $M^2_i$ of genus $g_i$. We set $\dot M^2_1=M^2_1\setminus \omega$ and $\dot M^2_2=M^2_2\setminus \alpha$. Let $\omega$ be the fixed sink of the diffeomorphism $f_1$, and $\alpha$ be the fixed source of the diffeomorphism $f_2$ such that $(W^{\mathrm{s}}_\omega\setminus\omega)/f_1 \cong (W^{\mathrm{u}}_\alpha\setminus\alpha)/f_2$. Then there exists a diffeomorphism $\nu\colon W^{\mathrm{s}}_\omega\setminus\omega\to W^{\mathrm{u}}_\alpha\setminus\alpha$ which conjugates the diffeomorphisms $f_1$ and $f_2$. We set $M^2=\dot M^2_1\cup_\nu \dot M^2_2$ and denote by $\zeta\colon \dot M^2_1\sqcup \dot M^2_2\to M^2$ the natural projection. By the construction, $M^2\cong M^2_1\# M^2_2$. Consider the diffeomorphism $f\colon M^2\to M^2$ defined by
$$
\begin{equation*}
f=\begin{cases} \zeta f_1 \zeta^{-1}|_{\zeta(\dot M^2_1)} & \text{on }\zeta(\dot M^2_1), \\ \zeta f_2 \zeta^{-1}|_{\zeta(\dot M^2_2)} & \text{on }\zeta(\dot M^2_2). \end{cases}
\end{equation*}
\notag
$$
This diffeomorphism $f=f_1\#f_2$ will be referred to as the connected sum of the diffeomorphisms $f_1$ and $f_2$ along the sink $\omega$ and the source $\alpha$; $\nu$ is the binding map. 6.2. Construction of elementary diffeomorphisms6.2.1. Diffeomorphism $\phi$ on a sphere $\mathbb S^2$On the plane $\mathbb R^2$, consider the polar coordinates $(r,\varphi)$. Let $\varrho(r)$ be the function specified by the graph in Fig. 17. On the plane $\mathbb R^2$, consider the vector field defined the system of differential equations
$$
\begin{equation*}
\begin{aligned} \, \dot r &=\begin{cases} -r(r-1), &0\leqslant r \leqslant 1, \\ 1-r, &r>1, \end{cases} \\ \dot\varphi &= -\varrho(r)\sin 2\varphi. \end{aligned}
\end{equation*}
\notag
$$
Let $\chi^t$ be the flow induced by this vector field, and let $\chi$ be the diffeomorphism defined as the shift of the flow $\chi^t$ per time unit. The resulting diffeomorphism has a hyperbolic source at the origin $O$, hyperbolic saddles at points $A_1$, $A_3$, and hyperbolic sinks at points $A_0$, $A_2$ (see Fig. 18). Let the diffeomorphism $\theta\colon \mathbb R^2 \to \mathbb R^2$ be defined by $\theta (r,\varphi)=(r, -\varphi)$, and let the diffeomorphism $\overline \phi\colon \mathbb R^2 \to \mathbb R^2$ be defined by By the construction, the non-wandering set of the diffeomorphism $\overline \phi$ coincides with that of the diffeomorphism $\chi$. Consider the standard two-dimensional sphere
$$
\begin{equation*}
\mathbb S^2=\{(x_1,x_2,x_3)\in\mathbb R^3\colon x_1^2+x^2_2+x_3^2=1\}.
\end{equation*}
\notag
$$
Let $S(0,0,-1)$ be its south pole, and let the stereographic projection $\vartheta\colon \mathbb {S}^2\setminus\{S\}\to\mathbb{R}^2$ be defined by
$$
\begin{equation*}
\vartheta(x_1,x_2,x_3)=\biggl(\frac{x_1}{1+x_3},\frac{x_2}{1+x_3}\biggr).
\end{equation*}
\notag
$$
Next, let the diffeomorphism $\phi\colon \mathbb S^2 \to\mathbb S^2$ be defined by
$$
\begin{equation*}
\phi(s)=\begin{cases} \vartheta^{-1}\circ \overline \phi \circ \vartheta(s), &s\in S^2\setminus\{S\}, \\ S, &s=S. \end{cases}
\end{equation*}
\notag
$$
By the construction, the diffeomorphism $\phi$ is a gradient-like diffeomorphism of the 2-sphere with the following non-wandering set (see Fig. 19):
For the diffeomorphism $\phi$, there are two possibilities for choosing the set $\Sigma$ (see Fig. 19). By the construction, the saddle orbit has positive orientation type, and so, by Lemma 1 (case 1, c)), in both cases the characteristic space of the orbits $\widehat V_{\Sigma}$, consists of two connected components, of which each is homeomorphic to the Klein bottle.  Figure 19.Two possibilities for $\Sigma$: (a) $\Sigma=\varnothing$; (b) $\Sigma=\mathcal O_{\sigma}$ 6.2.2. Diffeomorphism $\psi_1$ on a torus $\mathbb T^2$Let us construct a diffeomorphism $\psi_1$ on the two-dimensional torus $\mathbb T^2$ as the Cartesian product of two orientation-preserving source–sink diffeomorphisms on the circle $\mathbb S^1$. Consider the function $\overline{F}\colon \mathbb R\to\mathbb R$ (see Fig. 20) defined by .Let the projection $\pi\colon \mathbb{R}\to \mathbb{S}^1$, be defined by $\pi(x)=e^{2\pi i x}$. The function $\overline{F}$ is strictly monotone increasing and satisfies $\overline{F}(x+1)=\overline{F}(x)+1$. Hence it admits a projection onto the circle as the diffeomorphism $F\colon \mathbb S^1\to\mathbb S^1$ defined by
$$
\begin{equation*}
F(z)=\pi\overline F \pi^{-1}(z),\qquad z\in\mathbb S^1.
\end{equation*}
\notag
$$
By the construction, the diffeomorphism $F$ has a fixed hyperbolic sink and a fixed source, and it is an orientation-preserving source–sink diffeomorphism. Let the diffeomorphism $F_1\colon \mathbb T^2\to\mathbb T^2$ be defined by It is clear that the diffeomorphism $F_1$ preserves orientation, and its non-wandering set consists of four fixed points: the source $\alpha$ of positive orientation type $(\varsigma_{\alpha}=+1)$, the sink $\omega$ of positive orientation type $(\varsigma_{\omega}=+1)$, and two saddles $\sigma_1$, $\sigma_2$ of positive orientation type (see Fig. 21).We represent the two-dimensional torus $\mathbb{T}^2$ as the quotient group of the group $\mathbb{R}^2$ over the integer lattice $\mathbb{Z}^2:\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$. Consider the matrix $A=\left( \begin{smallmatrix} 0& 1\\ 1& 0 \end{smallmatrix}\right)\in \mathrm{GL}(2,\mathbb{Z})$ and the corresponding algebraic automorphism $\widehat A\colon \mathbb T^2\to\mathbb T^2$ of the torus given by We set
$$
\begin{equation*}
\psi_1=\widehat A\circ F_1\colon \mathbb{T}^2\to\mathbb T^2.
\end{equation*}
\notag
$$
By the construction, the diffeomorphism $\psi_1$ is an orientation-changing gradient-like diffeomorphism whose non-wandering set consists of a source $\alpha$ and a sink $\omega$ of negative orientation types $(\varsigma_{\alpha}=\varsigma_{\omega}=-1)$, as well as a periodic saddle orbit $\mathcal O_{\sigma_1}=\{\sigma_1,\psi_1(\sigma_1)\}$ of period $2$ and orientation type $\varsigma_{\sigma_1}=(+1,+1)$ (see Fig. 22):
$$
\begin{equation*}
\Omega_{\psi_1}=\{\alpha,\omega,\sigma_1,\psi_1(\sigma_1)\}.
\end{equation*}
\notag
$$
6.2.3. Diffeomorphism $\widetilde\psi_1$ on the projective plane $\mathbb R P^2$Consider the diffeomorphism $\phi\colon \mathbb S^2 \to \mathbb S^2$ defined in § 6.2.1, and consider the group $\mathbb Z_2=\{ +1, -1\} $ defined on the two-dimensional sphere $\mathbb S^2=\{(x_1,x_2,x_3)\in\mathbb R^3\colon x_1^2+x^2_2+x_3^2=1\}$ by
$$
\begin{equation*}
\pm{1} \cdot x=\pm{x},\qquad x=(x_1, x_2, x_3)\in \mathbb S^2.
\end{equation*}
\notag
$$
The space of orbits of the action of the group $\mathbb Z_2$ on $\mathbb S^2$ is the projective plane $\mathbb R P^2$. Let $p\colon \mathbb S^2 \to \mathbb S^2/ \mathbb Z_2$ be the natural projection. Since $\phi(-x)=-\phi(x)$, the formula
$$
\begin{equation*}
\widetilde\psi_1(x)=p \circ \phi \circ p^{-1},\qquad x\in \mathbb R P^2,
\end{equation*}
\notag
$$
correctly defines the diffeomorphism $\widetilde\psi_1\colon \mathbb R P^2 \to \mathbb R P^2$ with non-wandering set $\Omega_{\widetilde\psi_1}=\{\widetilde\alpha, \widetilde \omega, \widetilde \sigma_1\}$ (see Fig. 23). 6.3. Construction of diffeomorphisms on a surface of any genus6.3.1. Diffeomorphisms on orientable surfacesOn any orientable surface $M^2$ of genus $g$, we construct an orientation-changing diffeomorphism $f_g$ that does not have a connected characteristic space of orbits. Consider the diffeomorphisms $\phi$, $\psi_1$ constructed in the previous section. Let $f_1=\phi\sharp \psi_1\colon \mathbb S^2 \sharp \mathbb T^2 \to \mathbb S^2 \sharp \mathbb T^2$ be the connected sum along the sink $\omega_0$ of the diffeomorphism $\phi$ and the source $\alpha$ of the diffeomorphism $\psi_1$. Since the saddle separatrices $L_{\omega_0}$ in the basin of sink $\omega_0$ are 2-periodic, like the saddle separatrices $L_{\alpha}$ in the source basin $\alpha$, the binding diffeomorphism $\nu\colon W^{\mathrm{s}}_{\omega_0}\setminus\omega_0\to W^{\mathrm{u}}_{\alpha} \setminus \alpha$ can be chosen so that $\nu(L_{\omega_0})\cap L_{\alpha}=\varnothing$. Thus, $f_1\colon \mathbb T^2\to\mathbb T^2$ is an orientation-changing gradient-like diffeomorphism with exactly two sinks $\omega_1$, $\omega$. At the same time, each of the orbit spaces $\widehat V_{\omega_1}$, $\widehat V_{\omega}$ is homeomorphic to the Klein bottle, the saddle $\sigma$ has orientation type $\varsigma_\sigma=(+1,+1)$, and the unstable separatrix of the saddles $\sigma$, $f_1(\sigma)$ are the only unstable separatrices lying in the sink basin $\omega_1$ (see Fig. 24). The diffeomorphism $f_g$ is constructed by induction on an orientable surface of genus $g\geqslant 2$ as the connected sum of the diffeomorphisms $\psi_g$ and $\phi$ ($f_g=\psi_g\sharp \phi$) along the sink $\omega_0$ of the diffeomorphism $\phi$ and the source $\alpha$ of the diffeomorphism $\psi_g$ (see Fig. 25). Similarly, since the saddle separatrices $L_{\omega_0}$ in the sink basin $\omega_0$ are 2-periodic, like the saddle separatrices $L_{\alpha}$ in the source basin $\alpha$, the binding diffeomorphism $\nu\colon W^{\mathrm{s}}_{\omega_0}\setminus\omega_0\to W^{\mathrm{u}}_{\alpha} \setminus \alpha$ can be chosen so that $\nu(L_{\omega_0})\cap L_{\alpha}=\varnothing$. Thus, $f_g$ is an orientation-changing gradient-like diffeomorphism with exactly two sinks $\omega_1$, $\omega$ of negative orientation type. Let us show that the diffeomorphism $f_g$ does not have a connected characteristic space of orbits. To this end, consider $\Gamma_{f_g}$ (see Fig. 26). Since the set of saddle points is $\overline{\Omega}^{\,1}_f =\varnothing$, and since $\overline{\Omega}^{\,0}_f=\omega \cup \omega_1$, the special subgraph $\overline\Gamma_{f_g}$ consists of exactly two vertices (which denote the sink points $\omega$, $\omega_1$). Thus, the graph $\overline\Gamma_{f_g}$ is not connected, and hence by Theorem 1 the characteristic space of orbits is also disconnected. 6.3.2. Diffeomorphisms on non-orientable surfacesOn any non-orientable surface $M^2$ of genus $q$, we construct a gradient-like diffeomorphism $\widetilde f_q$ which does not admit a connected characteristic space of orbits. Let us define the diffeomorphism $\widetilde f_1\colon \mathbb R P^2 \to \mathbb R P^2$ as the connected sum of diffeomorphisms $\phi$ and $\widetilde \psi_1$ ($\widetilde f_1=\phi\sharp \widetilde \psi_1$) along the sink $\omega_0$ of the diffeomorphism $\phi$ and the source $\widetilde\alpha$ of the diffeomorphism $\widetilde \psi_1$ (see Fig. 27). The saddle separatrices $L_{\omega_0}$ in the basin of sink $\omega_0$ is 2-periodic, and the saddle separatrices $L_{\widetilde\alpha}$ in the source basin $\widetilde\alpha$ is 1-periodic. Hence the binding diffeomorphism $\nu\colon W^{\mathrm{s}}_{\omega_0}\setminus\omega_0\to W^{\mathrm{u}}_{\alpha} \setminus \alpha$ can be chosen so that $\nu(L_{\omega_0})\cap L_{\widetilde\alpha}=\varnothing$. By induction, the diffeomorphism $\widetilde f_q$ is constructed on a non-orientable surface of genus $q$ as a smooth connected sum of diffeomorphisms $\phi$ and $\widetilde\psi_q$ ($\widetilde f_q=\widetilde\psi_q\sharp \phi$) along the sink $\omega_0$ of the diffeomorphism $\phi$ and the source $\widetilde\alpha$ of the diffeomorphism $\widetilde\psi_q$. By arguing as in § 6.3.1, it can be shown that the diffeomorphism $\widetilde f_q$ does not admit a connected characteristic space of orbits. 
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