M. K. Barinova, O. A. Kolchurina, E. I. Yakovlev, “On 3-diffeomorphisms with generalized Plykin attractors”, Sb. Math., 215:9 (2024), 1135

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Sbornik: Mathematics, 2024, Volume 215, Issue 9, Pages 1135–1158
DOI: https://doi.org/10.4213/sm10048e (Mi sm10048)

On 3-diffeomorphisms with generalized Plykin attractors

M. K. Barinova, O. A. Kolchurina, E. I. Yakovlev

National Research University Higher School of Economics, Nizhny Novgorod, Russia

English version PDF (816 kB) HTML full-text Russian version article

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DOI: https://doi.org/10.4213/sm10048e

Abstract: It is known that a nontrivial attractor coexists with trivial basic sets in the nonwandering set of an

$\Omega$ -stable 3-diffeomorphism if and only if it is either nonorientable one-dimensional or (orientable or not) expanding and two-dimensional. Examples of such diffeomorphisms were constructed previously, with the exception of the case of a nonorientable two-dimensional attractor. The paper fills this gap. In addition, it is constructively shown that the diffeomorphism obtained has an energy function, which extends thereby the class of cascades with global Lyapunov function whose set of critical points coincides with the nonwandering set of the dynamical system.
Bibliography: 20 titles.

Keywords: basic set,

$\Omega$ -stability, expanding attractor, generalized Plykin attractor, energy function.

Funding agency	Grant number
Russian Science Foundation	22-11-00027
HSE Basic Research Program
With the exception of § 5, this research was supported by the Russian Science Foundation under grant no. 22-11-00027, https://rscf.ru/en/project/22-11-00027/. The results in § 5 were obtained in the framework of the HSE University Basic Research Program.

Received: 17.12.2023 and 27.04.2024

Bibliographic databases:

Document Type: Article

MSC: 37C05, 37C70

Language: English

Original paper language: Russian

§ 1. Introduction

In this paper we consider $\Omega$ -stable¹ diffeomorphisms defined on smooth closed $n$ -manifolds. A spectral decomposition theorem [1] holds for such diffeomorphisms: the nonwandering set of an $\Omega$ -stable diffeomorphism is a disjoint finite union of so-called basis subsets, each of which is compact, invariant and topologically transitive. If a basic set is a periodic orbit, then it is said to be trivial, otherwise it is nontrivial.

Let $f\colon M^n\!\to\! M^n$ be an $\Omega$ -stable diffeomorphism of a manifold $M^n$ with metric $d$ , and let $\Lambda$ be a basic set of $f$ . Each point $x$ in $\Lambda$ has so-called invariant manifolds, a stable one $W^{\mathrm s}_x$ and an unstable one $W^{\mathrm u}_x$ . They are defined by

$\begin{equation*} W^{\mathrm s}_x=\Bigl\{y\in M^n\Bigm| \lim_{n\to+\infty}d(f^{n}(x),f^{n}(y))=0\Bigr\} \end{equation*} \notag$

and

$\begin{equation*} W^{\mathrm u}_x=\Bigl\{y\in M^n\Bigm| \lim_{n\to+\infty}d(f^{-n}(x),f^{-n}(y))=0\Bigr\}. \end{equation*} \notag$

By the generalized stable manifold theorem [1] the invariant manifolds

$W^{\mathrm s}_x$ and

$W^{\mathrm u}_x$ are the images of Euclidean spaces

$\mathbb R^k$ and

$\mathbb R^{n-k}$ for some

$k\in\{0,1,\dots,n\}$ under injective immersions²

$J^{\mathrm s}_x\colon \mathbb R^k\to M^n$ and

$J^{\mathrm u}_x\colon \mathbb R^{n-k}\to M^n$ , respectively. Hence for each

$r>0$ the sets

$W^{\mathrm s}_{x,r}=J^{\mathrm s}_x(D^k_r)$ and

$W^{\mathrm u}_{x,r}=J^{\mathrm u}_x(D^{n-k}_r)$ , where

$D^k_r=\{x\in\mathbb R^k\colon \|x\|\leqslant r\}$ and

$D^{n-k}_r=\{x\in\mathbb R^{n-k}\colon \|x\|\leqslant r\}$ , are smoothly embedded discs.

If at least one of the discs $W^{\mathrm s}_{x,r}$ and $W^{\mathrm u}_{x,r}$ has dimension one, that is, $k$ is equal to $1$ or $n-1$ , then we can speak about the intersection index of these submanifolds. For simplicity let $W^{\mathrm s}_{x,r}$ be one-dimensional. Let $U$ be a tubular neighbourhood of $W^{\mathrm u}_{x,r}$ , which is the image of an embedding in $M^n$ of the total space of a one-dimensional vector bundle on $W^{\mathrm u}_{x,r}$ ([2], Ch. 4, § 5). Since the ball $D^{n-1}_r$ is contractible, this bundle is trivial and $U\setminus{W^{\mathrm u}_{x,r}}$ consists of two connected components $U_+$ and $U_-$ . Hence we can define a function $\nu\colon U_+\cup U_-\to\mathbb{Z}$ such that $\nu(x)=1$ for $x\in U_+$ , and $\nu(x)=0$ for $x\in U_-$ . If the manifolds $W^{\mathrm s}_{x,r}$ and $W^{\mathrm u}_{x,r}$ intersect transversally at a point $y=J^{\mathrm s}_x(t)$ , $t\in D^1_r$ , then there exists $\delta>0$ such that $J^{\mathrm s}_x(t-2\delta,t+2\delta)\subset U$ .

The integer

$\begin{equation*} \operatorname{Ind}_y(W^{\mathrm s}_{x,r},W^{\mathrm u}_{x,r})=\nu(t+\delta)-\nu(t-\delta) \end{equation*} \notag$

is called the intersection index of

$W^{\mathrm s}_{x,r}$ and

$W^{\mathrm u}_{x,r}$ at

$y$ . Note that this index takes values in

$\{1,-1\}$ . A basic set is said to be orientable if one of its invariant manifolds is one-dimensional and the index of intersection of

$W^{\mathrm s}_{x,r}$ and

$W^{\mathrm u}_{x,r}$ is the same at all points for each

$r > 0$ . Otherwise the set is said to be nonorientable. Note that this definition of the orientability of a basic set can be used for diffeomorphisms of orientable and nonorientable manifolds alike.

A compact invariant set $\mathcal A$ is called an attractor of the diffeomorphism $f$ if it has a compact neighbourhood $U_{\mathcal A}$ such that $f(U_{\mathcal A})\subset \operatorname{Int}U_{\mathcal A}$ and $\bigcap_{n\in\mathbb{N}}f^n(U_{\mathcal A})={\mathcal A}$ . If an attractor is a basic set and its topological dimension is equal to the dimension of the unstable manifolds of points in it, then this attractor is said to be expanding.

One important example of a two-dimensional diffeomorphism with an expanding attractor is the so-called DA-diffeomorphism (Derived from Anosov), originally constructed by Smale [1]. It is obtained from an Anosov diffeomorphism³ on a 2-torus by using a Smale surgery⁴ in a neighbourhood of a fixed point. The nonwandering set of a DA-diffeomorphism consists of an expanding orientable one-dimensional attractor and a fixed source. By making Smale surgeries in neighbourhoods of several periodic orbits we can obtain generalized DA-diffeomorphisms with a unique nontrivial attractor and periodic sources. A similar operation can be performed for an Anosov diffeomorphism on an $n$ -torus, provided that the stable manifolds of points on the torus are one-dimensional. Such diffeomorphisms are also called $\mathrm{DA}$ -diffeomorphisms if a single source arises, and they are called generalized $\mathrm{DA}$ -diffeomorphisms if there are several sources.

In 1974 Plykin [3] presented a geometric construction of a structurally stable diffeomorphism of a 2-sphere with nonorientable one-dimensional attractor and four fixed points. Subsequently, it was shown (for instance, see [4], § 17.2) that we can obtain such a diffeomorphism from a generalized DA-diffeomorphism with four sources (one of which is fixed and three have period 3), by taking the quotient of the torus by an involution. We call the attractor obtained by a similar construction on an $n$ -torus a generalized Plykin attractor. By construction it is an expanding nonorientable attractor of dimension $n-1$ . Note that, as follows from [5], a 3-manifold admitting an $\Omega$ -stable diffeomorphism with a generalized Plykin attractor is nonorientable, while the diffeomorphism itself is not structurally stable. In the same paper a diffeomorphism with such an attractor was constructed for which the nontrivial repeller dual to this attractor is a nonwandering set.

It is known from the recent results of Barinova, Pochinka and Yakovlev [6] that if all nontrivial basic sets of an $\Omega$ -stable 3-diffeomorphism are attractors, then they can belong to one of the following three types: a nonorientable one-dimensional attractor, an expanding orientable two-dimensional attractor, or an expanding nonorientable two-dimensional attractor.

One example of such a diffeomorphism with an expanding orientable two-dimensional attractor is a DA-diffeomorphism of a 3-torus. An example of a diffeomorphism with a nonorientable one-dimensional attractor is the cascade on a 3-sphere obtained by compactifying the Cartesian product of a Plykin diffeomorphism of a 2-sphere and a hyperbolic contraction on a line (Figure 1).

Figure 1.A diffeomorphism of the 3-sphere with a one-dimensional Plykin attractor and six isolated periodic points.

In the paper [6] mentioned above no obstructions were found to the simultaneous existence of a generalized Plykin attractor and trivial basic sets in the nonwandering set of an $\Omega$ -stable 3-diffeomorphism, but no examples of such diffeomorphisms have been known so far. In the two-dimensional case, to obtain a cascade on a closed manifold with a single basic set which is a Plykin attractor, it is sufficient to add four fixed hyperbolic sources. We cannot complement similarly the construction of the system in three dimensions because of the structure of the basin of a generalized Plykin attractor.

Our main result in this paper is a constructive proof of the following theorem.

Theorem 1. There exists an $\Omega$ -stable $3$ -diffeomorphism whose nonwandering set consists of a generalized Plykin attractor and 12 isolated periodic points.⁵

For the proof we present a construction of such diffeomorphisms in §§ 2 and 3. Let $\mathcal P$ denote the class of diffeomorphisms constructed in this way. Their supporting manifolds are obtained by gluing four cylinders over the real projective plane to the boundary of a trapping neighbourhood of a nontrivial attractor. In § 4 we calculate the homology of the supporting manifolds in our examples. It turns out that in this way we can obtain diffeomorphisms which are not topologically conjugate to one another because the fundamental group of the supporting manifold depends essentially on the way we glue the cylinders.

Theorem 2. Let $f\colon M^3\to M^3$ be a diffeomorphism in the class $\mathcal P$ . Then the homology groups of the manifold $M^3$ are as follows:

$\begin{equation} H_3(M^3)=0, \qquad H_2(M^3)\cong\mathbb{Z}^3\times\mathbb{Z}_2\quad\textit{and} \quad H_1(M^3)\cong\mathbb{Z}^4\times\mathbb{Z}_2^m, \end{equation} \tag{1.1}$

where

$m\in\{1,2,3\}$ , and each of these values of

$m$ can be realized.

Another result in our paper is the proof that diffeomorphism in the class $\mathcal P$ have energy functions.

Recall that a continuous function $\varphi\colon M^n\to\mathbb{R}$ is called a Lyapunov function for an $\Omega$ -stable diffeomorphism $f$ if $\varphi(x)>\varphi(f(x))$ when $x$ is a wandering point and $\varphi(x)=\varphi(y)$ when $x$ and $y$ lie in the same basic set. If the Lyapunov function is smooth and its critical point set coincides with the nonwandering set of the diffeomorphism, then it is called an energy function.

In contrast to flows, a cascade does not necessarily have an energy function (for instance, see [8]). The first example of such a cascade on a 3-sphere is due to Pixton [9] (1977): it is a Morse-Smale diffeomorphism⁶ with four fixed points. In the same paper Pixton proved that Morse–Smale diffeomorphisms of surfaces have Morse energy functions. In [10]–[13] Grines, Pochinka and Barinova distinguished several classes of 2- and 3-diffeomorphisms with energy functions. In 2022 a class of dynamical systems on surfaces without energy functions was discovered [14].

In this paper we present a constructive proof of the following result.

Theorem 3. Diffeomorphisms in the class $\mathcal P$ have energy functions.

§ 2. Generalized DA-diffeomorphism

In this section we present the construction of a 3-diffeomorphism obtained from an Anosov diffeomorphism by a Smale surgery. We prove that this diffeomorphism is $\Omega$ -stable and its nonwandering set consists of an expanding orientable two-dimensional attractor and eight fixed sources. Our construction is similar to the two-dimensional case described by Katok and Hasselblatt in their book (see [4], §§ 1.8 and 17.2).

2.1. The construction

Let $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ be a 3-torus and $p\colon \mathbb{R}^3\to\mathbb{T}^3$ be the natural projection. We begin with a hyperbolic automorphism $\widehat{L}_A\colon \mathbb{T}^3\to\mathbb{T}^3$ which is induced by a linear map $L_A$ of $\mathbb{R}^3$ with hyperbolic matrix $A\in \operatorname{GL}(3,\mathbb{Z})$ whose eigenvalues $\lambda_1$ , $\lambda_2$ and $\lambda_3$ satisfy $0<|\lambda_1|<1<|\lambda_2|\leqslant|\lambda_3|$ . For instance, we can define $L_A$ by

$\begin{equation*} \begin{pmatrix} \overline{x}\\ \overline{y}\\ \overline{z} \end{pmatrix}=\begin{pmatrix} 3 & 3 & 2\\ 1 & 3 & 1\\ 3 & 2 & 2 \end{pmatrix}\begin{pmatrix} x\\ y\\ z \end{pmatrix}. \end{equation*} \notag$

Then $\widehat{L}_A=p L_A p^{-1}\colon \mathbb T^3\to\mathbb T^3$ is an Anosov diffeomorphism. The one-dimensional stable two-dimensional unstable manifolds of points on the torus form two transversal invariant foliations, each leaf in which is dense on the torus. Each point with rational coordinates is periodic for $\widehat{L}_A$ , so the set of periodic points is also dense in $\mathbb{T}^3$ .

We describe the action of $\Gamma=\{1,-1\}$ on the torus $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ by setting $\gamma\cdot((x,y,z)+\mathbb{Z}^3)=\gamma (x,y,z)+\mathbb{Z}^3$ for all $(x,y,z)\in\mathbb{R}^3$ and $\gamma\in \Gamma$ . Then $\gamma=1$ corresponding to the identity map and $\gamma=-1$ to an involution $J$ with the eight fixed points

$\begin{equation*} v_i=p(\overline v_i), \qquad i=1,\dots,8, \end{equation*} \notag$

where

$\begin{equation*} \begin{gathered} \, \overline v_1=(0, 0, 0), \qquad \overline v_2=\biggl(0, 0, \frac{1}{2}\biggr), \qquad \overline v_3=\biggl(0, \frac{1}{2}, 0\biggr), \qquad \overline v_4=\biggl(0, \frac{1}{2}, \frac{1}{2}\biggr), \\ \overline v_5=\biggl(\frac{1}{2}, 0, 0\biggr), \qquad \overline v_6=\biggl(\frac{1}{2}, 0, \frac{1}{2}\biggr), \qquad \overline v_7=\biggl(\frac{1}{2}, \frac{1}{2}, 0\biggr), \qquad \overline v_8=\biggl(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\biggr). \end{gathered} \end{equation*} \notag$

These are periodic points of

$\widehat{L}_A$ , and therefore there exists

$k\in\mathbb{N}$ such that all these points are fixed by

$\widehat{L}_A^k$ . For

$\widehat{L}_A$ as above

$k=3$ , and the matrix

$A^k$ has the form

$\begin{equation*} A^k=\begin{pmatrix} 115 & 146 & 84\\ 62 & 83 & 46\\ 106 & 132 & 77 \end{pmatrix}. \end{equation*} \notag$

The eigenvalues of $A^k$ are

$\begin{equation*} \lambda^{\mathrm s}=(\lambda_1)^k \approx 0.001\,{<}\, 1, \quad \lambda^{\mathrm u}_1=(\lambda_2)^k \approx 2.802 \,{>}\, 1\quad\text{and} \quad \lambda^{\mathrm u}_2=(\lambda_3)^k \approx 272.197 \,{>}\, 1. \end{equation*} \notag$

The corresponding eigenvectors

$\begin{equation*} \mathbf{e}^{\mathrm s} \approx \begin{pmatrix} -0.456\\ -0.146\\ 0.878 \end{pmatrix}, \qquad \mathbf{e}^{\mathrm u}_1 \approx \begin{pmatrix} 0.278\\ -0.631\\ 0.725 \end{pmatrix}\quad\text{and} \quad \mathbf{e}^{\mathrm u}_2 \approx \begin{pmatrix} 0.684\\ 0.376\\ 0.626 \end{pmatrix} \end{equation*} \notag$

define two invariant foliations on the torus: the one-dimensional stable one

$W^{\mathrm s}$ and the two-dimensional unstable one

$W^{\mathrm u}$ .

We take $r > 0$ to be sufficiently small so that the $\lambda^{\mathrm u}_2 r$ -balls with centres $\overline v_i$ , $i=1,\dots,8$ , are disjoint. Setting $r_1= \frac{\lambda^{\mathrm s}}{16}r$ , $r_2=\frac{r}{2}$ and $t=\sqrt{y^2+z^2}$ we fix $\varepsilon\ll r_1$ .

Lemma 1. There exists a diffeomorphism $\Delta\colon \mathbb R^3\to\mathbb R^3$ of the form

$\begin{equation*} \Delta(x,y,z)=(\delta(x,y,z),y,z) \end{equation*} \notag$

with the following properties:

$\bullet$ if $|x|\geqslant r_2+\varepsilon$ or $t\geqslant r_2$ , then $\Delta$ is the identity map;
$\bullet$ if $|x|\leqslant r_1-\varepsilon$ and $t\leqslant r_2/2$ , then $\Delta(x,y,z)=({2x}/{\lambda^{\mathrm s}},y,z)$ ;
$\bullet$ the diffeomorphism $\Delta$ commutes with the involution $(x,y,z)\,{\mapsto}\,(-x,-y,-z)$ , that is, $\Delta(-x,-y,-z)=-\Delta(x,y,z)$ ;
$\bullet$ partial derivatives of $\delta$ have the estimates
$\begin{equation*} \delta'_x>0, \qquad |\delta'_y|\leqslant 2(2-\lambda^{\mathrm s})\quad\textit{and} \quad |\delta'_z|\leqslant 2(2-\lambda^{\mathrm s}); \end{equation*} \notag$
in addition, $\delta'_x\leqslant 1$ for $|x|\geqslant r_1+\varepsilon$ or $t\geqslant r_2$ .

Proof. Consider a smooth odd function

$\delta_0(x)\colon \mathbb{R}\to\mathbb{R}$ obtained from a piecewise linear function by smoothing it by means of arcs of circles.⁷^[x]⁷Note that

$\delta_0$ can be made arbitrarily smooth; we smoothen it by arcs of circles for geometric transparency. For

$x\geqslant 0$ we show

$\delta_0(x)$ in Figure 2; it is expressed by the formula

$\begin{equation*} \delta_0(x)=\begin{cases} \dfrac{2x}{\lambda^{\mathrm s}} & \text{for } 0\leqslant x < r_1-\varepsilon_1, \\ \sqrt{\rho^2_{1}-(x-x_{O_1})^2}+y_{O_1} & \text{for } r_1-\varepsilon_1\leqslant x < r_1+\varepsilon, \\ \dfrac{2}{4-\lambda^{\mathrm s}}x+\dfrac{2 -\lambda^{\mathrm s}}{4-\lambda^{\mathrm s}}r_2 & \text{for } r_1+\varepsilon \leqslant x \leqslant r_2-\varepsilon, \\ - \sqrt{\rho^2_2-(x-x_{O_2})^2}+y_{O_2} & \text{for } r_2-\varepsilon < x \leqslant r_2+\varepsilon_2, \\ x & \text{for } x > r_2+\varepsilon_2, \end{cases} \end{equation*} \notag$

where

$(r_1-\varepsilon_1,r_1+\varepsilon)$ and

$(r_2-\varepsilon,r_2+\varepsilon_2)$ are the smoothing intervals and we have

$\varepsilon_1<\varepsilon$ and

$\varepsilon_2<\varepsilon$ , and where

$(x_{O_1},y_{O_1})$ and

$(x_{O_2},y_{O_2})$ are the coordinates of the centres of the circles used for smoothing and

$\rho_1$ and

$\rho_2$ are their radii.

Figure 2.Constructing $\delta_0(x)$ for ${x\geqslant 0}$ .

Consider the sigmoid function $\sigma\colon \mathbb R \to [0,1]$ defined by

$\begin{equation*} \sigma(t)= \begin{cases} 1 & \text{for }t > r_2, \\ \dfrac{1}{1+\exp\biggl(\dfrac{(3r_2-4t)r_2^3}{8(t-r_2)^2(2t-r_2)^2}\biggr)} &\text{for }\dfrac{r_2}2 \leqslant t \leqslant r_, \\ 0 &\text{for }t < \dfrac{r_2}2 \end{cases} \end{equation*} \notag$

(Figure 3). This function is monotone, infinitely smooth, and all of its derivatives vanish for

$t\leqslant r_2/2$ and

$t\geqslant r_2$ .

Figure 3.The graph of $\sigma(t)$ .

We define a one-parameter family of functions $\delta_t(x)\colon \mathbb R\to\mathbb R$ by

$\begin{equation*} \delta_t(x)=\delta_0(x)(1-\sigma(t))+x\sigma(t) \end{equation*} \notag$

for

$x\geqslant 0$ .

We extend $\delta_t(x)$ to the negative values of $x$ to have an odd function, that is, $\delta_t(x)=-\delta_t(-x)$ for $x<0$ . We show the graphs of $\delta_t(x)$ for various $t$ in Figure 4. Then $\delta(x,y,z)=\delta_{\sqrt{y^2+z^2}}(x)$ is the required function.

Figure 4.The graphs of $\delta_t$ for different values of $t$ : continuous line for ${t\leqslant r_2/2}$ , dashed line for $r_2/2<t<r_2$ , dash-dotted line for $t\geqslant r_2$ .

The function $\delta(x,y,z)$ is $C^1$ -smooth in the whole domain of definition because $\delta_0(x)$ is $C^1$ -smooth and $\sigma(t)$ is $C^\infty$ -smooth. Now we show that the diffeomorphism $\Delta=(\delta(x,y,z),y,z)$ we have constructed satisfies all conditions in the lemma. In fact,

$\bullet$ if $|x|\geqslant r_2+\varepsilon$ or $t=\sqrt{y^2+z^2}\geqslant r_2$ , then $\delta_t(x)=x$ , and therefore $\Delta$ is the identity map for $\sqrt{x^2+y^2+z^2}>r$ ;
$\bullet$ for $|x| \leqslant r_1-\varepsilon$ and $t \leqslant r_2/2$ we have $\delta_t(x)={2x}/{\lambda^{\mathrm s}}$ , that is, $\Delta(x,y,z)=({2x}/{\lambda^{\mathrm s}},y,z)$ ;
$\bullet$ the equality $\Delta(-x,-y,-z)=-\Delta(x,y,z)$ is a direct consequence of the equality $\delta_{\sqrt{(-y)^2+(-z)^2}}(-x)=-\delta_{\sqrt{y^2+z^2}}(x)$ ;
$\bullet$ partial derivatives of $\delta(x,y,z)$ have the form
$\begin{equation*} \begin{gathered} \, \delta'_x=\delta_0'(x)(1-\sigma(t))+\sigma(t), \\ \delta'_y=(x-\delta_0(x))\sigma'_t\frac{y}{t}\quad\text{and}\quad \delta'_z=(x-\delta_0(x))\sigma'_t\frac{z}{t}. \end{gathered} \end{equation*} \notag$

Since $\delta_0'(x)>0$ and $0\leqslant\sigma\leqslant 1$ , it follows that $\delta'_x>0$ .

Note that the odd function $(\delta_0(x)-x)$ is nonnegative for $x>0$ and attains its maximum on the interval $(r_1-\varepsilon,r_1+\varepsilon)$ . For estimates of the derivatives with respect to $y$ and $z$ we look at the following chains of inequalities bearing in mind that $r_1=\frac{\lambda^{\mathrm s}}{8}r_2$ , $r_1+\varepsilon<2r_1$ and $\delta_0\leqslant 2x/\lambda^{\mathrm s}$ for $x>0$ :

$\begin{equation*} |x-\delta_0(x)|\leqslant\frac{2}{\lambda^{\mathrm s}}2r_1-2r_1=\frac{r_2}{4}(2-\lambda^{\mathrm s}) \end{equation*} \notag$

and

$\begin{equation*} 0\leqslant(\sigma\circ\beta)'_t\leqslant \frac{2\sigma'(r_2/2)}{r_2}=\frac{8}{r_2}. \end{equation*} \notag$

Then

$\begin{equation} |\delta_y|\leqslant 2(2-\lambda^{\mathrm s})\quad\text{and} \quad |\delta_z|\leqslant 2(2-\lambda^{\mathrm s}). \end{equation} \tag{2.1}$

If $|x|\geqslant r_1+\varepsilon$ , then $\delta'_0$ is nondecreasing and $\delta'_0(x)\leqslant 1$ . The range of $\sigma(t)$ is the interval $[0,1]$ . Hence $\delta'_x\leqslant 1$ if $|x|\geqslant r_1+\varepsilon$ or $t\geqslant r_2$ .

Lemma 1 is proved.

Let $\overline D_i\subset \mathbb R^3$ be the ball neighbourhood of $\overline v_i$ of radius $r$ , and let $D_i=p(\overline D_i)$ . In the neighbourhoods $\overline D_i$ of the points $\overline v_i$ we introduce new normalized coordinates so that the new $x$ -axis corresponds to the vector $\mathbf e^{\mathrm s}$ , while the $y$ - and $z$ -axes correspond to $\mathbf e^{\mathrm u}_1$ and $\mathbf e^{\mathrm u}_2$ . Then $T=\begin{pmatrix} \mathbf e^{\mathrm s} & \mathbf e^{\mathrm u}_1 & \mathbf e^{\mathrm u}_2 \end{pmatrix}$ is the transition matrix from the basis $\begin{Bmatrix}\mathbf e^{\mathrm s}, \mathbf e^{\mathrm u}_1, \mathbf e^{\mathrm u}_2\end{Bmatrix}$ to

$\begin{equation*} \begin{Bmatrix} \begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}\end{Bmatrix} \end{equation*} \notag$

and

$T^{-1}$ is the transition matrix from

$\begin{equation*} \begin{Bmatrix} \begin{pmatrix}1\\0\\0\end{pmatrix}, \begin{pmatrix}0\\1\\0\end{pmatrix}, \begin{pmatrix}0\\0\\1\end{pmatrix}\end{Bmatrix} \end{equation*} \notag$

$\begin{Bmatrix}\mathbf e^{\mathrm s}, \mathbf e^{\mathrm u}_1, \mathbf e^{\mathrm u}_2\end{Bmatrix}$ . We denote the linear maps with matrices

$T$ and

$T^{-1}$ by

$L_T$ and

$L_{T^{-1}}$ . Then in the neighbourhood

$D_i$ of the point

$v_i$ ,

$i \in \{1, 2,\dots,8\}$ , the transition map

$\psi_i\colon D_i\to\mathbb R^3$ looks as follows:⁸^[x]⁸Below we need the inverse map of

$\psi$ in a domain wider than

$\psi(D_i)$ . We set

$\psi^{-1}(x,y,z)=p(L_{T}(x,y,z)+\overline v_i)$ for all

$(x,y,z)\in\mathbb R^3$ .

$\begin{equation*} \forall\, a \in D_i \subset \mathbb T^3\quad \psi_i=L_{T^{-1}}((p|_{\overline D_i})^{-1}(a)-\overline v_i). \end{equation*} \notag$

Consider the diffeomorphism

${\mathrm{DA}}\colon \mathbb{T}^3\to\mathbb{T}^3$ defined by

$\begin{equation*} f_{\mathrm{DA}}(a)= \begin{cases} \widehat{L}_A^k(a) & \text{for } a \notin D_i, \\ \widehat{L}_A^k \circ \psi^{-1}_i \circ \Delta \circ \psi_i(a) & \text{for } a \in D_i. \end{cases} \end{equation*} \notag$

Defined in this way,

$f_{\mathrm{DA}}$ commutes with the involution

$J$ , there are fixed hyperbolic sources at the fixed points of the involution, and the one-dimensional foliation

$W^{\mathrm s}$ for the original Anosov diffeomorphism remains invariant under

$f_{\mathrm{DA}}$ .

2.2. $\Omega$ -stability

Theorem 4. The diffeomorphism $f_{\mathrm{DA}}$ is $\Omega$ -stable.

Proof. It follows from [15], Theorems 4.1, 5.7 and 9.1, that when the chain-recurrent set

$\mathrm{CR}(f_{\mathrm{DA}})$ of the diffeomorphism

$f_{\mathrm{DA}}$ is hyperbolic, then this diffeomorphism is

$\Omega$ -stable.

Let $\mathcal A$ denote the set $\mathrm{CR}(f_{\mathrm{DA}})\setminus\bigcup_{i=1}^8\{v_i\}$ . First of all, we prove a lemma on the position of $\mathcal A$ . Let $K_i$ be subsets of the $D_i$ such that their images $\overline K_i$ in the local charts $(D_i,\psi_i)$ are cylinders

$\begin{equation*} \overline K_i=\{(x,y,z)\in\mathbb R^3\mid |x|<r_1+\varepsilon,\,y^2+z^2<r_2^2\}. \end{equation*} \notag$

Lemma 2. The domain $K_i$ is a subset of the basin $W^{\mathrm u}_{v_i}$ of the source $v_i$ .

Proof. Let

$\overline A_i=\{(x,y,z)\in\mathbb R^3\mid |x|<r_1-\varepsilon,\,y^2+z^2<(r_2/2)^2\}$ . In the domain

$A_i=\psi^{-1}(\overline A_i)$ the map

$f_{\mathrm{DA}}$ has the form

$(2x,\lambda^{\mathrm u}_1 y,\lambda^{\mathrm u}_2 z)$ in the local charts under consideration, so that

$A_i$ is a subset of the basin of

$v_i$ . In these charts the image of

$A_i$ has the form

$\begin{equation*} \biggl\{(x,y,z)\in\mathbb R^3\biggm| |x|<2r_1-2\varepsilon,\,\biggl(\frac{y}{\lambda^{\mathrm u}_1}\biggr)^2+ \biggl(\frac{z}{\lambda^{\mathrm u}_2}\biggr)^2<\biggl(\frac{r_2}2\biggr)^2\biggr\} \end{equation*} \notag$

because

$\lambda^{\mathrm u}_2>\lambda^{\mathrm u}_1>2$ . Since

$\varepsilon$ is small, we have

$K_i\subset f(A_i)$ , so that

$K_i\subset W^{\mathrm u}_{v_i}$ .

Thus, the set $\mathcal A$ lies outside the domains $K_i$ , $i=1,\dots,8$ . Now we show that $\mathcal A$ is a hyperbolic set, that is, the tangent subbundle $T_{\mathcal A}(\mathbb T^3)$ has a continuous decomposition into a direct sum $E^{\mathrm s}_{\mathcal A}\oplus E^{\mathrm u}_{\mathcal A}=\bigcup_{x\in\mathcal A}E^{\mathrm s}_x\oplus E^{\mathrm u}_x$ , which is invariant under the differential $Df_{\mathrm{DA}}$ , and there exists constants $c>0$ and $\mu\in (0,1)$ such that

$\begin{equation*} \|Df_{\mathrm{DA}}^n(\mathbf{v})\|\leqslant c\mu^n\|\mathbf{v}\|,\qquad \mathbf{v}\in E^{\mathrm s}_{\mathcal A}, \end{equation*} \notag$

and

$\begin{equation*} \|Df_{\mathrm{DA}}^{-n}(\mathbf{v})\|\leqslant c\mu^n\|\mathbf{v}\|, \qquad \mathbf{v}\in E^{\mathrm u}_{\mathcal A}. \end{equation*} \notag$

Now we find the differential $Df_{\mathrm{DA}}$ of $f_{\mathrm{DA}}$ . If we choose local charts so that the eigenvectors of $A$ are the unit coordinate vectors, similarly to the charts $(D_i,\psi_i)$ , then this differential has the form

$\begin{equation*} Df_{\mathrm{DA}}\begin{pmatrix} x\\ y\\ z \end{pmatrix}=\begin{pmatrix} \lambda^{\mathrm s} \delta'_x & \delta'_y & \delta'_z\\ 0 & \lambda_1^{\mathrm u} & 0\\ 0 & 0 & \lambda_2^{\mathrm u} \end{pmatrix} \begin{pmatrix} x\\ y\\ z \end{pmatrix}, \end{equation*} \notag$

and we have

$\delta'_x=1$ and

$\delta'_y=\delta'_z=0$ if the point lies outside the domains

$D_i$ .

Since the surgery described above preserved the foliation $W^{\mathrm s}$ , as the $Df_{\mathrm{DA}}$ -invariant summand $E^{\mathrm s}_{x}$ , $x\in\mathcal A$ , we take the linear span of $\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix}$ . Let $\mathbf{v}=\begin{pmatrix} x\\ 0\\ 0 \end{pmatrix}\in E^{\mathrm s}_{\mathcal A}$ ; then

$\begin{equation*} \|Df_{\mathrm{DA}} (\mathbf{v})\|=|\lambda^{\mathrm s}\delta_x x|\leqslant \lambda^{\mathrm s}\|\mathbf{v}\|, \end{equation*} \notag$

so that the required conditions on

$E^{\mathrm s}_{\mathcal A}$ are satisfied.

To prove that there exists a two-dimensional subspace $E^{\mathrm u}_{\mathcal A}$ we use the cone criterion ([4], Corollary 6.4.8). We show that for the family $H$ of cones of the form $y^2+z^2 \geqslant \gamma^2 x^2$ there exists $\mu>1$ such that

$\begin{equation*} D f_{\mathrm{DA}} (H) \subset \operatorname{Int}H, \qquad \|D f_{\mathrm{DA}} (\mathbf{v})\| \geqslant \mu \|\mathbf{v}\|, \quad \mathbf{v} \in H. \end{equation*} \notag$

We verify the first condition. Let

$\begin{equation*} \begin{pmatrix} \overline x\\ \overline y\\ \overline z \end{pmatrix} =Df_{\mathrm{DA}}\begin{pmatrix} x\\ y\\ z \end{pmatrix}. \end{equation*} \notag$

We show that if

$t\geqslant\gamma |x|$ ,

$\gamma>0$ , then

$\sqrt{\overline{y}^2+\overline{z}^2}=\overline{t} \geqslant \gamma |\overline{x}|$ . To do this we find estimates for the left- and right-hand sides taking the bounds in Lemma 1 into account and bearing in mind that we only need to consider

$t\neq 0$ :

$\begin{equation*} \begin{gathered} \, \begin{aligned} \, \gamma|\overline x| &=\gamma|\lambda^{\mathrm s}\delta_x x+\delta_y y+\delta_z z|\leqslant\gamma\lambda^{\mathrm s}\delta_x |x|+\gamma(|\delta_y y|+|\delta_z z|) \\ &\leqslant \lambda^{\mathrm s} t+2(2-\lambda^{\mathrm s})\gamma t\biggl(\frac{|y|}t+\frac{|z|}t\biggr)\leqslant(\lambda^{\mathrm s}+2(2-\lambda^{\mathrm s})\gamma)t, \end{aligned} \\ \overline t =\sqrt{(\lambda^{\mathrm u}_1 y)^2+(\lambda^{\mathrm u}_2 z)^2}\geqslant \lambda^{\mathrm u}_1 t. \end{gathered} \end{equation*} \notag$

The inequality $(\lambda^{\mathrm s}+2(2-\lambda^{\mathrm s})\gamma)t\leqslant\lambda^{\mathrm u}_1t$ holds for

$\begin{equation*} \gamma\leqslant\frac{\lambda^{\mathrm u}_1-\lambda^{\mathrm s}}{2(2-\lambda^{\mathrm s})} \approx 0.7. \end{equation*} \notag$

We verify that there exists $\mu>1$ such that $\|D f_{\mathrm{DA}} (v)\| \geqslant \mu \|\mathbf{v}\|$ for $\mathbf{v} \in H$ , that is, we verify the inequality $\overline x^2+\overline y^2+\overline z^2\geqslant \mu(x^2+y^2+z^2)$ :

$\begin{equation*} \begin{gathered} \, \overline x^2+\overline y^2+\overline z^2\geqslant\overline y^2+\overline z^2=(\lambda^{\mathrm u}_1 y)^2+(\lambda^{\mathrm u}_2 z)^2\geqslant (\lambda^{\mathrm u}_1)^2t^2, \\ \mu(x^2+y^2+z^2)\leqslant\biggl(\frac{\mu}{\gamma^2}\biggr) t^2+\mu t^2=\biggl(1+\frac{1}{\gamma^2}\biggr)\mu t^2. \end{gathered} \end{equation*} \notag$

Let $(1+1/\gamma^2)\mu t^2\leqslant (\lambda^{\mathrm u}_1)^2t^2$ , that is,

$\begin{equation*} \mu\leqslant\frac{(\lambda^{\mathrm u}_1)^2}{1+1/\gamma^2}. \end{equation*} \notag$

Then

$\mu>1$ exists if

$(\lambda^{\mathrm u}_1)^2>1+1/\gamma^2$ , that is,

$\begin{equation*} \gamma>\frac{1}{\sqrt{(\lambda^{\mathrm u}_1)^2-1}}\approx 0.382. \end{equation*} \notag$

We see that the family of cones $y^2+z^2\geqslant (0.5x)^2$ and $\mu=1.5$ satisfy all the assumptions of the cone criterion. In combination with the calculations for $E^{\mathrm s}_{\mathcal A}$ , this shows that the set $\mathcal A$ is hyperbolic, and therefore $f_{\mathrm{DA}}$ is $\Omega$ -stable.

Lemma 2 is proved.

The nonwandering set of an $\Omega$ -stable diffeomorphism coincides with its chain-recurrent set ([15], Theorem 4.1), so

$\begin{equation*} \mathcal A=\mathrm{NW}(f_{\mathrm{DA}})\setminus\bigcup_{i=1}^8\{v_i\}. \end{equation*} \notag$

2.3. The existence and uniqueness of a two-dimensional expanding attractor

The diffeomorphism $f_{\mathrm{DA}}$ is $\Omega$ -stable, and $\mathrm{NW}(f_{\mathrm{DA}})=\mathcal A\cup\bigcup_{i=1}^8 \{v_i\}$ . Hence $\mathcal A$ is the union of a nonempty finite system of basic sets. In this subsection we find saddle points whose two-dimensional unstable manifolds consist of nonwandering points and are dense in $\mathcal A$ . Then we can prove that $\mathcal A$ contains a two-dimensional expanding attractor and no other basic sets.

Lemma 3. Each domain $D_i$ , $i=1,\dots,8$ , contains two fixed saddle points $\beta^+_i$ and $\beta^-_i$ .

Proof. In the local charts

$(D_i,\psi_i)$ the diffeomorphism

$f_{\mathrm{DA}}$ has the form

$\begin{equation*} (x,y,z)\mapsto (\lambda^{\mathrm s}\delta(x,y,z),\lambda^{\mathrm u}_1y,\lambda^{\mathrm u}_2z), \end{equation*} \notag$

so that fixed points can only lie on the

$Ox$ -axis. We consider the restriction

${f_x=\lambda^{\mathrm s}\delta_0(x)}$ of

$f_{\mathrm{DA}}$ to the

$Ox$ -axis and find its fixed points from the equation

$\lambda^{\mathrm s}\delta_0(x)=x$ (Figure 5). Since

$\varepsilon$ is small, for

$x>0$ the solution lies in the interval

$(r_1-\varepsilon;r_1+\varepsilon)$ , so it is sufficient to solve the equation

$\begin{equation*} \lambda^{\mathrm s}\biggl(\frac{2}{4-\lambda^{\mathrm s}}x+\frac{2 -\lambda^{\mathrm s}}{4-\lambda^{\mathrm s}}r_2\biggr)=x. \end{equation*} \notag$

We obtain

$\begin{equation*} x^*=\frac{4-3\lambda^{\mathrm s}}{\lambda^{\mathrm s}(2-\lambda^{\mathrm s})}. \end{equation*} \notag$

Then the points

$\beta^+_i=\phi_i^{-1}(x^*,0,0)$ and

$\beta^-_i=\phi_i^{-1}(-x^*,0,0)$ are fixed and thus lie in

$\mathcal A$ . As the set

$\mathcal A$ has a hyperbolic structure, the dimensions of the sets

$E^{\mathrm s}_x$ and

$E^{\mathrm u}_x$ ,

$x\in\mathcal A$ , show that

$\beta^+_i$ and

$\beta^-_i$ are hyperbolic saddle points with one-dimensional stable separatrices and two-dimensional unstable ones.

Figure 5.The Lamerey diagram for the restriction of $f_{\mathrm{DA}}$ to the $Ox$ -axis in the coordinate charts under consideration.

The proof is complete.

Lemma 4. The unstable manifolds of the saddle points $\beta^+_i$ and $\beta^-_i$ , $i=1,\dots,8$ , consist of nonwandering points.

Proof. The diffeomorphism

$f_{\mathrm{DA}}$ has a one-dimensional invariant foliation

$W^{\mathrm s}$ , the same as the one for the Anosov diffeomorphism

$\widehat L_A$ . Each leaf

$l^{\mathrm s}$ of

$W^{\mathrm s}$ is dense on the torus; moreover, if

$w\in l^{\mathrm s}$ , then each connected component of the set

$l^{\mathrm s}\setminus\{w\}$ is dense. Consider the one-dimensional leaf

$l^{\mathrm s}$ through

$\beta^+_i$ . The lift of

$l^{\mathrm s}$ to

$\mathbb R^3$ that passes through

$\overline v_i$ contains no other points all of whose coordinates are rational. Hence the connected component

$l^+$ of the set

$l^{\mathrm s}\setminus\{\beta^+_i\}$ that does not contain the fixed source does not pass through any source

$v_j$ ,

$j\neq i$ , and therefore it coincides with the separatrix of

$\beta^+_i$ .

It follows from the construction of $f_{\mathrm{DA}}$ that the disc

$\begin{equation*} \overline D^{\mathrm u}=\biggl\{(x,y,z)\in\mathbb R^3\biggm| x=x^*,\,y^2+z^2<\biggl(\frac{r_2}2\biggr)^2\biggr\} \end{equation*} \notag$

is a subset of the image

$\psi_i(W^{\mathrm u}_{\beta^+_i})$ of the unstable separatrix of

$\beta^+_i$ . Since

$l^+$ is dense on the torus and in the local charts under consideration the image of

$l^+\cap D_i$ is a union of line intervals parallel to the

$Ox$ -axis which is dense in

$\overline D_i$ , the homoclinic points are dense in the set

$\psi^{-1}_i(\overline D^{\mathrm u})$ and therefore also in

$W^{\mathrm u}_{\beta^+_i}$ (Figure 6). Then the fact that points in the nonstable separatrix of

$\beta^+_i$ are nonwandering is obvious because so are homoclinic points. For the unstable separatrices of the points

$\beta^-_i$ the proof is similar.

Figure 6.The intersection of the separatrices of the saddle point $\beta^+_-$ .

The lemma is proved.

Theorem 5. The nonwandering set of $f_{\mathrm{DA}}$ contains a two-dimensional expanding attractor $\Lambda_{\mathrm{DA}}$ .

Proof. It follows from Lemma 4 that the unstable two-dimensional manifolds of fixed saddle points lie in

$\mathcal A$ . By the spectral decomposition theorem (see [1]) there exists a basic set

$\Lambda_{\mathrm{DA}}$ of type⁹^[x]⁹The type of a basic set

$\Lambda$ is the pair of numbers

$(\dim W^{\mathrm u}_x,\dim W^{\mathrm s}_x)$ ,

$x\in\Lambda$ .

$(2,1)$ that contains the two-dimensional unstable manifold of the saddle. Then

$\Lambda_{\mathrm{DA}}$ has topological dimension at least 2. Let

${\Lambda_{\mathrm{DA}}=3}$ ; however, then by Lemma 8.1 in [16]

$\Lambda_{\mathrm{DA}}$ coincides with the supporting manifold

$\mathbb T^3$ , which is impossible since the nonwandering set of

$f_{\mathrm{DA}}$ contains sources. Then the basic set

$\Lambda_{\mathrm{DA}}$ has dimension 2, so it is an attractor (see [17], Theorem 3), which is two-dimensional and expanding.

The proof is complete.

Theorem 6. The set $\mathcal A$ coincides with the attractor $\Lambda_{\mathrm{DA}}$ .

Proof. Since basic sets are compact, it is sufficient to show that the unstable manifold of the saddle point

$\beta^+_1$ is dense in

$\mathcal A$ . Let

$a\in\mathcal A$ , and let

$U_a$ be some neighbourhood of

$a$ . Since periodic points are dense in the nonwandering sets of

$\Omega$ -stable diffeomorphisms, there exists a point

$p\in U_a$ that is periodic with period

$m_p$ . At least one of its one-dimensional separatrices, we denote it by

$l^{\mathrm s}_p$ , is dense on the torus and intersects the unstable manifold of

$\beta^+_1$ . Let

$q$ be one of their points of intersection. Since the stable separatrix

$l^{\mathrm s}_p$ is

$f^{m_p}$ -invariant, it follows that

$\begin{equation*} \lim_{k\to+\infty} d(f^{km_p}(q),p)=0. \end{equation*} \notag$

However,

$q$ also belongs to the invariant manifold of

$\beta^+_1$ , so there exists an index

$k$ such that

$W^{\mathrm u}_{\beta^+_1}\cap U_a\neq\varnothing$ . Thus,

$\mathcal A$ contains no basic sets other than

$\Lambda_{\mathrm{DA}}$ .

The proof is complete.

Summarizing the results of Theorems 4–6 we see that $f_{\mathrm{DA}}$ is an $\Omega$ -stable diffeomorphism and its nonwandering set consists of eight fixed hyperbolic sources and a two-dimensional expanding attractor, that is, it is a generalized DA-diffeomorphism. Note that, as the attractor $\Lambda_{\mathrm{DA}}$ is the complement to the basins of sources, which are homeomorphic to 3-balls, it is connected. One separatrix of each saddle point $\beta_i^+$ and each point $\beta_i^-$ , $i=1,\dots,8$ , lies fully in the basin of the corresponding source, so these saddle points are boundary periodic points of the expanding attractor $\Lambda_{\mathrm{DA}}$ and their unstable manifolds combine into 2-connectives. Hence the attractor obtained is orientable.

§ 3. Constructing a 3-diffeomorphism with generalized Plykin attractor

In this section we present the construction of a 3-diffeomorphism with generalized Plykin attractor and 12 fixed points, that is, we prove Theorem 1.

Consider the quotient space $M_s=\mathbb{T}^3/\Gamma=\mathbb T^3/J$ . This is a three-dimensional orbifold with eight singular (orbifold) points. Then the quotient space $\widetilde M=(\mathbb{T}^3\setminus \bigcup^8_{i=1}\{v_i\})/{\Gamma}$ is a manifold, and the natural projection $\widetilde p\colon \mathbb{T}^3\setminus\bigcup^8_{i=1}\{v_i\}\to \widetilde M$ is a two-sheeted cover. Since $f_{\mathrm{DA}}J=Jf_{\mathrm{DA}}$ , $f_{\mathrm{DA}}$ induces a diffeomorphism $\widetilde f=\widetilde pf_{\mathrm{DA}}\widetilde p^{-1}\colon \widetilde M\to\widetilde M$ with a generalized Plykin attractor $\Lambda=\widetilde p(\Lambda_{\mathrm{DA}})$ , whose stable manifold coincides with $\widetilde M$ . The set $\widetilde M\setminus \Lambda$ is the wandering set of $\widetilde f$ ; it consists of eight connected components $\widetilde B_i$ , $i=1,\dots,8$ , each obtained as a quotient of the basin $W^{\mathrm u}_{v_i}\setminus \{v_i\}$ of a source $v_i$ . It follows from [18] that $\widetilde B_i$ , $i=1,\dots,8$ , is diffeomorphic to $\mathbb RP^2\times \mathbb{R}$ , where $\mathbb RP^2$ is the real projective plane.

To understand the topological structure of the orbit space $\widetilde B_i/\widetilde f$ of the restriction of $\widetilde f$ to each connected component of the wandering set $\widetilde B_i$ , we find a fundamental domain $F_i$ of the restriction of $f_{\mathrm{DA}}$ to the sets $W^{\mathrm u}_{v_i}\setminus \{v_i\}$ , $i=1,\dots,8$ , and project it onto $\widetilde M$ . To do this we consider the local charts $(\psi_i, D_i)$ introduced above. For $x^2+y^2+z^2\leqslant r_1^2$ the diffeomorphism $f_{\mathrm{DA}}$ looks like $f_{\mathrm{DA}}(x,y,z)=(2x,\lambda^{\mathrm u}_1 y,\lambda^{\mathrm u}_2 z)$ in these charts. Then

$\begin{equation*} \begin{aligned} \, F_i& =\psi^{-1}\biggl(\biggl\{(x,y,z)\in\mathbb{R}^3\mid x^2+y^2+z^2\geqslant \biggl(\frac{r_1}2\biggr)^2, \\ &\qquad\qquad \biggl(\frac{x}{2}\biggr)^2+\biggl(\frac{y}{\lambda^{\mathrm u}_1}\biggr)^2+\biggl(\frac{z}{\lambda^{\mathrm u}_2}\biggr)^2\leqslant \biggl(\frac{r_1}{2}\biggr)^2\biggr\}\biggr). \end{aligned} \end{equation*} \notag$

Consider the foliation of

$F_i$ by the ellipsoids with

$\psi_i$ -images

$\begin{equation*} \begin{gathered} \, \biggl(\frac{x}{t}\biggr)^2+\biggl(\frac{y}{(\lambda^{\mathrm u}_1-1)(t-1)+1}\biggr)^2 +\biggl(\frac{z}{(\lambda^{\mathrm u}_2-1)(t-1)+1}\biggr)^2=\biggl(\frac{\lambda^{\mathrm s} r_0}{8}\biggr)^2, \\ t\in[1,2]. \end{gathered} \end{equation*} \notag$

Note that each leaf of this foliation is invariant under the involution

$J$ on

$\mathbb{T}^3$ , so the fundamental domain

$\widetilde F_i$ of the restriction of

$\widetilde f$ to

$\widetilde B_i$ is the projection of the fundamental domain

$F_i$ , that is,

$\widetilde F_i=\widetilde p(F_i)$ ; moreover,

$\widetilde F_i$ is diffeomorphic to

${\mathbb RP^2\times [0,1]}$ because after taking the quotient each ellipse becomes a set diffeomorphic to

$\mathbb RP^2$ . Then the required orbit space

${\widetilde B}_i/{\widetilde f}$ has the form

$\mathbb RP^2\times [0,1]|_{(x,0)\sim (\widetilde{f}(x),1)}$ and is diffeomorphic to

$\mathbb{R}P^2\times S^1$ , since any diffeomorphism on

$\mathbb RP^2$ is isotopic to the identity map [19].

Let $k=1,\dots,4$ , and let

$\begin{equation*} \begin{gathered} \, C_k=\mathbb{R}P^2\times\mathbb{R}, \qquad C_k^0=\mathbb{R}P^2\times \{0\}, \\ C_k^-=\mathbb{R}P^2\times (-\infty,0)\quad\text{and} \quad C_k^+=\mathbb{R}P^2\times (0,+\infty). \end{gathered} \end{equation*} \notag$

Set

$\begin{equation*} C=\bigsqcup_{k=1}^4 C_k, \qquad C^0=\bigsqcup_{k=1}^4 C_k^0\quad\text{and} \quad C^{\pm}=C\setminus C^0=\bigsqcup_{k=1}^4(C_k^-\sqcup C_k^+). \end{equation*} \notag$

Consider a structurally stable diffeomorphism

$g_{RP^2}\colon \mathbb RP^2\to\mathbb RP^2$ with three fixed points: a source

$\alpha$ , a sink

$\omega$ and a saddle point

$\sigma$ (Figure 7).

Figure 7.The diffeomorphism $g_{RP^2}$ on $\mathbb RP^2$ .

Also let $g_R\colon \mathbb R\to\mathbb R$ be the diffeomorphism defined by $g_{R}(x)=2x$ , let $g_k={g_{RP^2}\times g_R}\colon C_k\to C_k$ for $k=1,\dots,4$ , and let $g\colon C\to C$ be the diffeomorphism made up of the $g_k$ . It is obvious that the orbit spaces $C_k^-/g_k$ and $C_k^+/g_k$ are diffeomorphic to $\mathbb RP^2\times\mathbb S^1$ .

Thus, for all $i=1,\dots,8$ and $k=1, \dots,4$ the orbit spaces $C_k^-/g_k$ and $C_k^+/g_k$ are diffeomorphic to ${\widetilde B_i}/\widetilde f$ , and therefore for each permutation $P=\{i_1,j_1,i_2,j_2,i_3,j_3,i_4,j_4\}$ of the string $\{1,\dots,8\}$ there exists a diffeomorphism $h\colon C^{\pm}\to \widetilde M\setminus\Lambda=\bigcup_{i=1}^8 \widetilde B_i$ that conjugates $g|_{C^{\pm}}$ to $\widetilde f|_{\widetilde M\setminus\Lambda}$ and such that

$\begin{equation*} h(C_k^-)=\widetilde B_{i_k}\quad\text{and} \quad h(C_k^+)=\widetilde B_{j_k}, \qquad k=1,\dots,4. \end{equation*} \notag$

Let

$\begin{equation*} M_P=C\bigcup_h \widetilde M, \end{equation*} \notag$

and let

$q\colon C\sqcup \widetilde M\to M_P$ be the natural projection. Let

$f_{P}\colon M_P \to M_P$ be a diffeomorphism coinciding with

$q \widetilde f (q| _{\widetilde M})^{- 1}$ on

$q (\widetilde M)$ and with

$q g (q | _{C})^{- 1}$ on

$q(C)$ . Then, as a diffeomorphism required in Theorem 1, we can choose, for example, the diffeomorphism

$f_P$ corresponding to the permutation

$P=\{1,\dots,8\}$ ; all diffeomorphisms corresponding to permutations form a class

$\mathcal P$ .

§ 4. Homology of supporting manifolds

In this section we calculate the integer homology of the supporting manifolds $M_P$ for the diffeomorphisms $f_{P}$ constructed in § 3. This will prove Theorem 2.

4.1. Auxiliary constructions and isomorphisms

We introduce some notation:

$\begin{equation*} \begin{gathered} \, C_k^1=\mathbb{R}P^2\times[-1,1], \qquad C^1=\bigsqcup_{k=1}^4C_k^1, \\ N_P=q(C^1), \qquad M_1=M_P\setminus\operatorname{Int}{N}_P\quad\text{and} \quad N_1=\partial{M}_1. \end{gathered} \end{equation*} \notag$

Since the set

$q(C^0)$ is closed in

$M_P$ and lies in

$\operatorname{Int}{N}_P$ , the inclusion

$(M_P\setminus q(C^0),N_P\setminus q(C^0))\to(M_P,N_P)$ induces an isomorphism

$H_n(M_P\setminus q(C^0),N_P\setminus q(C^0))\cong H_n(M_P,N_P)$ . Since

$N_P\setminus q(C^0)=q(C^1\setminus C^0)\approx N_1\times[0,1)$ , the inclusion

$(M_1,N_1)\to(M_P\setminus q(C^0),N_P\setminus q(C^0))$ is a homotopy equivalence of pairs, and so induces an isomorphism

$H_n(M_1,N_1)\cong H_n(M_P\setminus q(C^0),N_P\setminus q(C^0))$ . As a result, we obtain isomorphisms

$H_n(M_1,N_1)\cong H_n(M_P,N_P)$ induced by the inclusion

$(M_1,N_1)\to(M_P,N_P)$ .

Set

$\begin{equation*} \widetilde{M}_1=(q|_{\widetilde{M}})^{-1}(M_1)\quad\text{and} \quad \widetilde{N}_1=\partial\widetilde{M}_1=(q|_{\widetilde{M}})^{-1}(N_1). \end{equation*} \notag$

Then

$q\colon \widetilde{M}_1\to M_1$ is a homeomorphism and

$q(\widetilde{N}_1)=N_1$ . Hence

$q$ induces an isomorphism

$H_n(\widetilde{M}_1,\widetilde{N}_1)\cong H_n(M_1,N_1)$ .

Let $M_s=\mathbb{T}^3/\Gamma$ , let $p\colon \mathbb{R}^3\to\mathbb{T}^3$ and $p_J\colon \mathbb{T}^3\to M_s$ be the natural projections, and let $p_s=p_J\circ p$ . Set

$\begin{equation*} \widetilde{N}=(q|_{\widetilde{M}})^{-1}(q(C^1\setminus{C}^0)), \qquad V=\{p_J(v_1),\dots,p_J(v_8)\}\quad\text{and}\quad N_s=\widetilde{N}\cup V. \end{equation*} \notag$

Then

$\widetilde{M}=M_s\setminus{V}$ , and therefore

$\widetilde{M}\subset M_s$ and

$\widetilde{N}\subset M_s$ ; moreover,

$N_s$ is the closure of

$\widetilde{N}$ in the orbifold

$M_s$ . For the same reasons as above we have isomorphisms

$H_n(\widetilde{M}_1,\widetilde{N}_1)\cong H_n(M_{s}\setminus{V},N_{s}\setminus{V})\cong H_n(M_{s},N_{s})$ , which are induced by the inclusions of pairs

$(\widetilde{M}_1,\widetilde{N}_1)\to(M_{s}\setminus{V},N_{s}\setminus{V})$ and

$(M_{s}\setminus{V},N_{s}\setminus{V})\to(M_s,N_s)$ .

Combining these results, we arrive at the following statement.

Lemma 5. Let $\imath_s\colon \widetilde{M}\to M_s$ be the inclusion and $q\colon C\sqcup \widetilde M\to M_P$ be the natural projection. Then the maps of pairs $\imath_s\colon (\widetilde{M}_1,\widetilde{N}_1)\to(M_s,N_s)$ and $q\colon (\widetilde{M}_1,\widetilde{N}_1)\to(M_P,N_P)$ induce isomorphisms

$\begin{equation*} H_n(M_s,N_s)\cong H_n(\widetilde{M}_1,\widetilde{N}_1)\cong H_n(M_P,N_P). \end{equation*} \notag$

The action of the group $\Gamma$ on $\mathbb{T}^3$ has a fundamental domain equal to the image $Q=p(\overline{Q})$ of the parallelepiped $\overline{Q}=[0,1]^2\times[0,1/2]$ . Furthermore, $\partial{Q}=p([0,1]^2\times 0)\cup{p}([0,1]^2\times(1/2))$ . The involution $J\in\Gamma$ acts on $\partial{Q}$ by the formula $J(p(x,y,\epsilon))=p(1-x,1-y,\epsilon)$ , where $\epsilon\in\{0,1/2\}$ . This produces a natural cell decomposition of $M_s$ (Figure 8). It consists of the zero-dimensional cells $e^0_i=p_J(v_i)=p_s(\overline{v}_i)$ , $i=1,\dots,8$ , the three-dimensional cell $e^3=p_s(\operatorname{Int}\overline{Q})$ , the one-dimensional cells

$\begin{equation*} \begin{alignedat}{2} e^1_1&=p_s\biggl(\biggl(0,\frac12\biggr)\times0\times0\biggr), &\qquad e^1_2 &=p_s\biggl(\biggl(0,\frac12\biggr)\times0\times\biggl(\frac12\biggr)\biggr), \\ e^1_3 &=p_s\biggl(0\times\biggl(0,\frac12\biggr)\times0\biggr), &\qquad e^1_4 &=p_s\biggl(0\times\biggl(0,\frac12\biggr)\times\biggl(\frac12\biggr)\biggr), \\ e^1_5 &=p_s\biggl(\biggl(0,\frac12\biggr)\times\biggl(\frac12\biggr)\times0\biggr), &\qquad e^1_6 &=p_s\biggl(\biggl(0,\frac12\biggr)\times\biggl(\frac12\biggr)\times\biggl(\frac12\biggr)\biggr) \end{alignedat} \end{equation*} \notag$

and

$\begin{equation*} e^1_7 =p_s\biggl(0\times0\times\biggl(0,\frac12\biggr)\biggr), \end{equation*} \notag$

and the two-dimensional cells

$\begin{equation*} \begin{gathered} \, e^2_1 =p_s\biggl((0,1)\times0\times\biggl(0,\frac12\biggr)\biggr), \qquad e^2_2=p_s\biggl(0\times(0,1)\times\biggl(0,\frac12\biggr)\biggr), \\ e^2_3=p_s\biggl((0,1)\times\biggl(0,\frac12\biggr)\times0\biggr) \quad\text{and}\quad e^2_4=p_s\biggl((0,1)\times\biggl(0,\frac12\biggr)\times\biggl(\frac12\biggr)\biggr). \end{gathered} \end{equation*} \notag$

Figure 8.A cell decomposition of $M_s$ .

We denote the closure of the 1-cell $e^1_i$ by $a_i$ , $i=1,\dots,7$ , and the closure of $e^2_j$ by $V_j$ , $j=1,\dots,4$ . Then $a_1=[e^0_1e^0_3]$ , $a_2=[e^0_2e^0_4]$ , $a_3=[e^0_1e^0_5]$ , $a_4=[e^0_2e^0_6]$ , $a_5=[e^0_5e^0_7]$ , $a_6=[e^0_6e^0_8]$ and $a_7=[e^0_1e^0_2]$ . Each closed 2-cell $V_j$ is homeomorphic to $S^2$ and is a (‘pillow’) orbifold with four singular points. More precisely,

$\begin{equation} e^0_1,e^0_2,e^0_3,e^0_4\in V_1, \quad e^0_1,e^0_2,e^0_5,e^0_6\in V_2, \quad e^0_1,e^0_3,e^0_5,e^0_7\in V_3\quad\text{and}\quad e^0_2,e^0_4,e^0_6,e^0_8\in V_4. \end{equation} \tag{4.1}$

4.2. The relative homology of the pair $(M_P,N_P)$

For indices $\zeta\ \in \{s,P\}$ the homology sequence

$\begin{equation} \dotsb \to H_n(N_{\zeta})\xrightarrow{\imath_*^n} H_n(M_{\zeta})\xrightarrow{\jmath_*^n} H_n(M_{\zeta},N_{\zeta}) \xrightarrow{\partial_*^n} H_{n-1}(N_{\zeta}) \to \dotsb \end{equation} \tag{4.2}$

of the pair

$(M_{\zeta},N_{\zeta})$ is exact.

The above cell decomposition of $M_s$ shows that $\partial{e}^2_j=\partial{V}_j=0$ for $j=1,\dots,4$ and, if the orientations of 2-cells are as indicated in Figure 1, $\partial{e}^3=2V_3-2V_4$ . Direct calculations show that $Z_1(M_s)=0$ . Therefore,

$\begin{equation} H_3(M_s)=H_1(M_s)=0, \qquad H_0(M_s)\cong\mathbb{Z} \end{equation} \tag{4.3}$

and

$\begin{equation} H_2(M_s)=\bigl\langle[V_1],[V_2],[V_3],[V_3-V_4]\parallel 2[V_3-V_4]\bigr\rangle \cong\mathbb{Z}^3\times\mathbb{Z}_2. \end{equation} \tag{4.4}$

By construction $\widetilde{N}=(q|_{\widetilde{M}})^{-1}(q(C^1\setminus{C}^0))\subset\bigsqcup_{i=1}^8\widetilde{B}_i$ . Let $\widetilde{N}_i=\widetilde{N}\cap\widetilde{B}_i$ , and let $N_{si}=\widetilde{N}_i\cup\{e^0_i\}$ be the closure of $\widetilde{N}_i$ in $M_s$ . Then $N_{si}$ is a closed neighbourhood of the singular point $e^0_i$ and $N_s=\bigsqcup_{i=1}^8N_{si}$ .

The sets $\widetilde{N}_i$ are homeomorphic to the cylinder $\mathbb{R}P^2\times(0,1]$ or $\mathbb{R}P^2\times[-1,0)$ , and the neighbourhoods $N_{si}$ are homeomorphic to a cone over $\mathbb{R}P^2$ . Hence

$\begin{equation} H_3(N_s)=H_2(N_s)=H_1(M_s)=0\quad\text{and} \quad H_0(N_s)\cong\mathbb{Z}^8. \end{equation} \tag{4.5}$

For all $i=1,\dots,4$ set $\overline{V}_i=V_i+C_2(N_s)$ . Plugging (4.3)–(4.5) into (4.2) for ${\zeta=s}$ we obtain the short exact sequences

$\begin{equation} 0 \to H_3(M_s,N_s)\to 0,\qquad 0 \to H_1(M_s,N_s)\to \mathbb{Z}^8\to \mathbb{Z} \to 0 \end{equation} \tag{4.6}$

and

$\begin{equation} 0\to\mathbb{Z}^3\times\mathbb{Z}_2 \xrightarrow{\jmath_*^2}H_3(M_s,N_s)\to 0. \end{equation} \tag{4.7}$

By (4.6)

$\begin{equation} H_3(M_s,N_s)=0\quad\text{and} \quad H_1(M_s,N_s)\cong\mathbb{Z}^7. \end{equation} \tag{4.8}$

Since

$\jmath_*^2([V_i])=[\overline{V}_i]$ , from (4.7) and (4.4) we obtain

$\begin{equation} H_2(M_s,N_s)= \langle[\overline{V}_{1}],[\overline{V}_{2}],[\overline{V}_{3}],[\overline{V}_{3}-\overline{V}_{4}]\parallel 2[\overline{V}_{3}-\overline{V}_{4}]\rangle \cong\mathbb{Z}^3\times\mathbb{Z}_2. \end{equation} \tag{4.9}$

Let $V_{j1}=V_j\cap M_{s1}$ , $V_{jP}=q(V_{j1})$ and $\overline{V}_{jP}=V_{jP}+C_2(N_P)$ for $j=1,\dots,4$ . Then from (4.8), (4.9) and Lemma 5 we deduce the following result.

Lemma 6. The following formulae hold:

$\begin{equation*} H_3(M_P,N_P)=0, \qquad H_1(M_P,N_P)\cong\mathbb{Z}^7 \end{equation*} \notag$

and

$\begin{equation*} H_2(M_P,N_P)= \langle[\overline{V}_{1P}],[\overline{V}_{2P}],[\overline{V}_{3P}],[\overline{V}_{3P}-\overline{V}_{4P}]\parallel 2[\overline{V}_{3P}-\overline{V}_{4P}]\rangle \cong\mathbb{Z}^3\times\mathbb{Z}_2. \end{equation*} \notag$

4.3. The homology of the manifold $M_P$

Lemma 7. For each permutation $P=\{i_1,j_1,i_2,j_2,i_3,j_3,i_4,j_4\}$ there exists $m\in\{1,2,3\}$ such that

$\begin{equation*} H_3(M_P)=0, \qquad H_2(M_P)\cong\mathbb{Z}^3\times\mathbb{Z}_2\quad\textit{and} \quad H_1(M_P)\cong\mathbb{Z}^4\times\mathbb{Z}_2^m. \end{equation*} \notag$

Proof. Since

$q\colon C^1\to N_P$ is a homeomorphism, it follows that

$\begin{equation} H_3(N_P)=H_2(N_P)=0\quad\text{and} \quad H_1(N_P)\cong H_0(N_P)\cong\mathbb{Z}_2^4. \end{equation} \tag{4.10}$

By (4.10) and Lemma 6, from (4.2) for

$\zeta=P$ we obtain the exact sequence

$\begin{equation} 0 \to H_3(M_P)\to 0 \to H_2(M_P)\xrightarrow{\jmath_*^2} \mathbb{Z}^3\times\mathbb{Z}_2 \xrightarrow{\partial_*^2} \mathbb{Z}_2^4\xrightarrow{\imath_*^1} H_1(M_P)\to \mathbb{Z}^4\to 0. \end{equation} \tag{4.11}$

By (4.11)

$H_3(M_P)=0$ and

$H_2(M_P)\cong\mathbb{Z}^3\times\mathbb{Z}_2^t$ , where

$t\in\{0,1\}$ . In addition, by the universal coefficients theorem

$\begin{equation*} H_3(M_P,\mathbb{Z}_2)\cong \operatorname{Tor}(\mathbb{Z}^3\times\mathbb{Z}_2^t,\mathbb{Z}_2)\cong\mathbb{Z}_2^t. \end{equation*} \notag$

On the other hand, as

$M_P$ is a closed 3-manifold, we have

$\begin{equation*} H_3(M_P,\mathbb{Z}_2)\cong\mathbb{Z}_2. \end{equation*} \notag$

Comparing these formulae, we conclude that

$t=1$ and

$H_2(M_P)\cong\mathbb{Z}^3\times\mathbb{Z}_2$ .

By construction $\widetilde{N}_1=\bigsqcup_{i=1}^8\widetilde{N}_{1i}$ , where $\widetilde{N}_{1i}\approx\mathbb{R}P^2$ and $\widetilde{N}_{1i}\subset\widetilde{B}_i$ ; we also have $N_1=q(\widetilde{N}_1)$ . Let $\widetilde{u}_i$ be the 1-cycle in $\widetilde{N}_{1i}$ with homology class equal to the generator of the group $H_1(\widetilde{N}_{1i})\cong\mathbb{Z}_2$ . Then $H_1(\widetilde{N}_1)=\langle[\widetilde{u}_1],\dots,[\widetilde{u}_8]\rangle\cong\mathbb{Z}^8$ . Set $u_i=q(\widetilde{u}_i)$ for $i=1,\dots,8$ . Since the cycles $u_{i_k}$ and $u_{j_k}$ are homologous in $N_P$ , it follows that

$\begin{equation} H_1(N_P)=\bigl\langle[u_1],\dots,[u_8]\parallel [u_{i_k}]=[u_{j_k}], \, k=1,\dots,4\bigr\rangle \cong\mathbb{Z}_2^4. \end{equation} \tag{4.12}$

For each $j=1,\dots,4$ the boundary $\partial{V}_{j1}$ is a sum of the four cycles in the set $\widetilde{u}_1,\dots,\widetilde{u}_8$ that correspond to the singular vertices in $V_j$ . By definition the homomorphism $\partial_*^2$ in (4.11) satisfies the equalities $\partial_*^2([\overline{V}_{jP}])=[\partial{V}_{jP}]=[q(\partial{V}_{j1})]$ . Hence by (4.1) and Lemma 6 the subgroup $\operatorname{im}\partial_*^2$ of $H_1(N_P)$ is generated by the elements

$\begin{equation} \partial_*^2([\overline{V}_{1P}]) =w_1=[u_1]+[u_2]+[u_3]+[u_4], \end{equation} \tag{4.13}$

$\begin{equation} \partial_*^2([\overline{V}_{2P}]) =w_2=[u_1]+[u_2]+[u_5]+[u_6], \end{equation} \tag{4.14}$

$\begin{equation} \partial_*^2([\overline{V}_{3P}]) =w_3=[u_1]+[u_3]+[u_5]+[u_7] \end{equation} \tag{4.15}$

and

$\begin{equation} \partial_*^2([\overline{V}_{3P}]-[\overline{V}_{4P}]) =w_4=[u_1]+\dots+[u_8]. \end{equation} \tag{4.16}$

Since the permutation $P$ involves all elements of $\{1,\dots,8\}$ , it follows from (4.16) and (4.12) that $w_4=0$ . Therefore, $\operatorname{im}\partial_*^2\cong\mathbb{Z}_2^{\mathrm s}$ , where $0\leqslant{s}\leqslant3$ .

If in the permutation $P=\{i_1,j_1,i_2,j_2,i_3,j_3,i_4,j_4\}$ we replace a pair $\{i_k,j_k\}$ by $\{j_k,i_k\}$ , then $h(C_k^-)=\widetilde{B}_{j_k}$ and $h(C_k^+)=\widetilde{B}_{i_k}$ . By interchanging two pairs $\{i_k,j_k\}$ and $\{i_l,j_l\}$ in $P$ , $k\ne l$ , we simply renumber the cylinders in the list $C_1,\dots,C_4$ . These transformations do not change the topological type of $M_P$ , so in what follows we assume without loss of generality that

$\begin{equation} i_k<j_k \quad \text{for all } k=1,\dots,4 \quad \text{and} \quad 1=i_1<i_2<i_3<i_4. \end{equation} \tag{4.17}$

By (4.13), (4.12) and (4.17) the equality $w_1\!=\!0$ is only possible when $\{\mkern-1mu i_1,j_1,i_2,j_2\mkern-1mu\}$ is a permutation of the elements in the string $\{1,2,3,4\}$ such that $i_1=1$ , and $\{i_3,j_3,i_4,j_4\}$ is a permutation of elements in the string $\{5,6,7,8\}$ . If we have $w_2=0$ in addition, then $j_1=2$ by (4.14). However, by (4.15), for $i_1=1$ and $j_1=2$ we have $w_3\ne0$ . Thus, $\operatorname{im}\partial_*^2\ne0$ , and therefore $s>0$ .

We see that $\operatorname{im}\partial_*^2\cong\mathbb{Z}_2^{\mathrm s}$ , where $0\!\leqslant\! s\!\leqslant\!3$ . Moreover, by (4.11) we have ${\operatorname{im}\imath_*^1\!\cong\!\mathbb{Z}_2^m}$ , where $m=4-s$ . But then by the same formula (4.11) we have $H_1(M_P)\cong\mathbb{Z}^4\times\mathbb{Z}_2^m$ and $0\leqslant{m}\leqslant3$ .

Lemma 7 is proved.

Lemma 8. Each case $m=1,2,3$ in Lemma 7 can be realized. In other words, for each $m\in\{1,2,3\}$ there exists a permutation $P$ such that

$\begin{equation*} H_1(M_P)\cong\mathbb{Z}^4\times\mathbb{Z}_2^m. \end{equation*} \notag$

Proof. Let

$P=\{1,2,3,4,5,6,7,8\}$ . Then

$[u_1],[u_3],[u_5],[u_7]$ is a basis of

$H_1(N_P)$ , and the relations

$[u_1]=[u_2]$ ,

$[u_3]=[u_4]$ ,

$[u_5]=[u_6]$ and

$[u_7]=[u_8]$ hold in

$H_1(N_P)$ . These relations and formulae (4.13) and (4.14) show that

$w_1=w_2=0$ . Therefore,

$\operatorname{im}\partial_*^2=\langle w_3\rangle\cong\mathbb{Z}_2$ . Thus, in our case

$s=1$ and

$m=4-s=3$ .

For $P=\{1,2,3,5,4,6,7,8\}$ the basis of $H_1(N_P)$ is formed by $[u_1],[u_3],[u_4]$ and $[u_7]$ , and the relations in (4.12) are as follows:

$\begin{equation*} [u_1]=[u_2], \qquad [u_3]=[u_5], \qquad [u_4]=[u_6]\qquad\text{and} \qquad [u_7]=[u_8]. \end{equation*} \notag$

Furthermore, by (4.13)–(4.15) we have

$w_1=w_2=[u_3]+[u_4]$ and

$w_3=[u_1]+[u_7]$ . The coordinates of

$w_1$ and

$w_2$ with respect to the basis

$[u_1],[u_3],[u_4],[u_7]$ form the matrix

$\begin{equation*} A= \begin{pmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1\\ \end{pmatrix}. \end{equation*} \notag$

Since

$\operatorname{rank}{A}=2$ , it follows that

$\operatorname{im}\partial_*^2=\langle w_1,w_3\rangle\cong\mathbb{Z}_2^2$ . Hence in this case

$s=2$ and

$m=2$ .

Finally, let $P=\{1,3,2,5,4,6,7,8\}$ . Then $[u_1]$ , $[u_2]$ , $[u_4]$ , $[u_7]$ is a basis of $H_1(N)$ and we have $[u_1]=[u_3]$ , $[u_2]=[u_5]$ , $[u_4]=[u_6]$ and $[u_7]=[u_8]$ . These relations show that

$\begin{equation*} w_1=[u_2]+[u_4], \qquad w_2=[u_1]+[u_4]\qquad\text{and} \qquad w_3=[u_2]+[u_7]. \end{equation*} \notag$

Hence the coordinates of

$w_1,w_2$ and

$w_3$ with respect to this basis form the matrix

$\begin{equation*} B= \begin{pmatrix} 0 & 1 & 1 & 0\\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1 \end{pmatrix}. \end{equation*} \notag$

Since

$\operatorname{rank}B=3$ , we have

$\operatorname{im}\partial_*^2=\langle w_1,w_2,w_3\rangle\cong\mathbb{Z}_2^3$ . Thus, for

$P$ in question we have

$s=3$ and

$m=1$ .

The lemma is proved.

Lemmas 7 and 8 yield Theorem 2.

§ 5. Existence of an energy function

Now we show that diffeomorphisms in $\mathcal P$ have energy functions, that is, we prove Theorem 3.

Proof of Theorem 3. Let

$f_P\colon M_P\to M_P$ be a diffeomorphism in

$\mathcal P$ with generalized Plykin attractor

$\Lambda$ . By construction

$M_P=C\bigcup_h \widetilde M$ , where the set

$C$ consists of four cylinders over the real projective plane, that is,

$C=\bigsqcup_{k=1}^4 C_k$ , where

$C_k=\mathbb{R}P^2\times\mathbb{R}$ , and the image of

$\widetilde M$ under the natural projection

$q\colon C\sqcup\widetilde M\to M_P$ is the stable manifold of the attractor

$\Lambda$ .

Recall that the restriction of the projection $q$ to $C$ is a diffeomorphism onto its image $q(C)=M_P\setminus\Lambda$ , which conjugates $f_P|_{M_P\setminus\Lambda}$ smoothly to the diffeomorphism $g$ made up of maps $g_k\colon C_k\to C_k$ of the form $g_k=g_{RP^2}\times g_R$ , where $g_{RP^2}$ is a structurally stable diffeomorphism of the real projective plane whose nonwandering set consists of three fixed points: a source $\alpha$ , a sink $\omega$ and a saddle point $\sigma$ (see Figure 7), and $g_R(x)=2x$ for $x\in\mathbb{R}$ .

Pixton [9] proved in 1977 that Morse–Smale diffeomorphisms of closed surfaces have energy functions, so $g_{RP^2}$ has one. We present a geometric version of the construction of such a function in the lemma below.

Lemma 9. Let $p_{RP^2}\colon \mathbb S^2\to \mathbb {R}P^2$ be a two-sheeted cover and $p_z\colon \mathbb R^3\to\mathbb R$ be the canonical projection onto the $Oz$ -axis. Then there exists an immersion $H\colon \mathbb S^2\to\mathbb R^3$ such that $\varphi_{RP^2}=p_z\circ H\circ p^{-1}_{RP^2}\colon \mathbb RP^2\to\mathbb R$ is a well-defined function which is an energy function for $g_{RP^2}$ .

Proof. Let

$\overline{g}_{RP^2}\colon \mathbb S^2\to\mathbb S^2$ be a lift of

$g_{RP^2}$ such that the nonwandering set of

$\overline g_{RP^2}$ consists of six fixed points (Figure 9). Consider a smooth map

$H\colon \mathbb S^2\to\mathbb{R}^3$ of the 2-sphere to three-dimensional Euclidean space such that the height function

$p_z\circ H$ commutes with

$p_{RP^2}$ , so that for all

$a\in\mathbb RP^2$ the set

$p_zHp_{RP^2}^{-1}(a)$ consists of a single point and the image of the fixed-point set of

$\overline g_{RP^2}$ is the critical set of the function

$p_z\circ H$ .

Figure 9.The lift of $g_{RP^2}$ to the sphere $\mathbb S^2$ .

We define the required image $H(\mathbb S^2)$ of the sphere parametrically. For $r\in[0,8]$ and $s\in [0, 2\pi]$ we consider the map $\Theta=(\Theta_x, \Theta_y, \Theta_z)\colon [0,8]\times [0,2\pi]\to\mathbb R^3$ such that the first two components have the form

$\begin{equation*} \Theta_x(r,s)=f(r)\cos s\quad\text{and} \quad \Theta_y(r,s)=f(r)\sin s, \end{equation*} \notag$

where

$f$ is the function

$\begin{equation*} f(r)= \begin{cases} r & \text{for } r\in[0,3], \\ -r+6 & \text{for } r\in(3,5], \\ r-4 & \text{for } r\in(5,6], \\ -r+8 & \text{for } r\in(6,8], \end{cases} \end{equation*} \notag$

and the third component

$\Theta_z (r,s)$ has the form

$\begin{equation*} \Theta_z(r,s)= \begin{cases} -\sqrt{4-r^2} & \text{for } r\in[0,2], \\ 0.5 (\cos(2s)+3) \biggl(\dfrac{\arcsin(2r-5)} {\pi}+0.5\biggr)+2 & \text{for } r\in(2,3], \\ \sqrt{1-((-r+6)-2)^2}+0.5 \cos(2s)+3.5 & \text{for } r\in(3,5], \\ \sqrt{1-((-r+6)-2)^2}+0.5 \cos(2s)+3.5 & \text{for } r\in(5,6], \\ - \sqrt{4-(- r+8)^2}+2 & \text{for } r\in(6,8]. \end{cases} \end{equation*} \notag$

The set $H(\mathbb S^2)$ thus obtained is shown in Figure 10.

Figure 10.The image of the sphere under the immersion $H$ .

It turn, we can parametrize the sphere by the map $\Upsilon=(\Upsilon_x,\Upsilon_y,\Upsilon_z)\colon [0,2\pi]\times[-\pi/2,\pi/2]\to \mathbb S^2\subset\mathbb{R}^3$ , where

$\begin{equation*} \Upsilon_x=\cos(\phi)\cos(\psi),\qquad \Upsilon_y=\sin(\phi)\cos(\psi)\qquad\text{and}\qquad \Upsilon_z=\sin(\psi) \end{equation*} \notag$

for

$\phi\in[0,2\pi]$ and

$\psi\in[-\pi/2,\pi/2]$ .

Let $H_{sq}\colon [0,2\pi]\times[-\pi/2,\pi/2]\to[0,8]\times[0,2\pi]$ be the map of the form

$\begin{equation*} H_{sq}(\phi,\psi)=\biggl(\frac{8(\psi+{\pi}/{2})}{\pi},\phi\biggr). \end{equation*} \notag$

Then

$H=\Theta\,{\circ}\, H_{sq}\,{\circ}\, \Upsilon^{-1}$ is the required immersion. The function

$p_z\circ H$ is a Lyapunov function for the diffeomorphism¹⁰^[x]¹⁰Note that although we have not formally defined the diffeomorphisms

$g_{RP^2}$ and

$\overline g_{RP^2}$ , Figure 7 specifies a conjugacy class of diffeomorphisms of the real projective plane, in which we can select a diffeomorphism

$g_{RP^2}$ such that

$p_z\circ H$ is a Lyapunov function for its lift.

$\overline g_{RP^2}$ , which is the lift of

$g_{RP^2}$ . The critical points of

$p_z\circ H$ coincide with the nonwandering points of

$\overline g_{RP^2}$ . Furthermore, we can verify directly that

$p_z\circ H$ commutes with the covering map

$p_{RP^2}$ , so that

$\varphi_{RP^2}= p_zHp_{RP^2}^{-1}$ is an energy function for our system on

$\mathbb{R}P^2$ .

Lemma 9 is proved.

We continue the proof of Theorem 3. The cascade $g_R$ has the energy function $\varphi_R(x)=-x^2$ . Then the fact that the direct product of $g_{RP^2}$ and $g_R$ has an energy function $\varphi_k\colon C_k\to\mathbb R$ , $k=1,\dots,4$ , is a consequence of [20]. Moreover, we can take it in the form

$\begin{equation*} \varphi_k(w,x)=\tan^{-1}(\varphi_{RP^2}(w)+\varphi_R(x))+\frac{\pi}{2}, \end{equation*} \notag$

so that

$\begin{equation*} \lim_{x \to \infty} \varphi_k(w,x)=0, \quad \text{where } w \in \mathbb RP^2\quad\text{and} \quad x\in\mathbb{R}. \end{equation*} \notag$

Let

$\varphi\colon C \to \mathbb R$ be the function made up of the

$\varphi_k(w,x)$ . Consider the function

$\Phi\colon M_P\to \mathbb{R}$ defined by

$\begin{equation*} \Phi(a)= \begin{cases} 0 &\text{for } a\in\Lambda, \\ \varphi((q|_{C})^{-1}(a)) & \text{otherwise}. \end{cases} \end{equation*} \notag$

By the definition of the $\varphi_k$ , $k=1,\dots,4$ , the function $\Phi$ is continuous.

Although $\Phi$ is an energy function for $f_{P}$ on the set $M_P\setminus \Lambda$ , it is not smooth on the whole of $M_P$ in general. By the smoothing lemma in [14] there exists a function $\xi\colon \mathbb R \to \mathbb R$ such that $\xi \circ \Phi$ is the required energy function for $f_{P}$ .

Theorem 3 is proved.



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Citation: M. K. Barinova, O. A. Kolchurina, E. I. Yakovlev, “On 3-diffeomorphisms with generalized Plykin attractors”, Sb. Math., 215:9 (2024), 1135–1158

Citation in format AMSBIB

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\by M.~K.~Barinova, O.~A.~Kolchurina, E.~I.~Yakovlev

\paper On 3-diffeomorphisms with generalized Plykin attractors

\jour Sb. Math.

\yr 2024

\vol 215

\issue 9

\pages 1135--1158

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