MATHEMATICS

V. I. ARNOL'D and Ya. G. SINAI

ON SMALL PERTURBATIONS OF AUTOMORPHISMS OF THE TORUS

(Presented by Academician A. N. Kolmogorov, January 20, 1962)

§ 1. Let the two-dimensional torus \(T^2\) be realized as the unit square of the plane \((x_1, x_2)\) with pairwise identified sides. An automorphism of the torus \(T^2\) is a transformation \(x \to Ax = \bar{x}\), defined by an integral matrix \(A=\|a_{ij}\|\) with determinant \(\pm 1\), and acting on the torus by the formula
\[ \bar{x}_i=\sum_j a_{ij}x_j \pmod 1,\quad i=1,2. \]

Suppose that the matrix \(\|a_{ij}\|\) has two real eigenvalues, not equal in modulus to 1. Then, if \((\alpha_1,1)\) is an eigenvector of the matrix \(A^*\) with eigenvalue \(\lambda_1\), \(|\lambda_1|<1\), the system of straight lines on the torus
\[ dx_2+\alpha_1dx_1=0 \tag{1} \]
has, with respect to the automorphism \(A\), the following properties:

I. Every straight line \(\Gamma\) from the family (1) is carried under the action of \(A\) into a straight line \(A\Gamma\) from the family (1), i.e. the family (1) is invariant with respect to \(A\).

II. There exists \(\mu_1\), \(\mu_1>1\), such that the lengths \(s(l)\), \(s(Al)\) of segments \(l\), \(Al\), lying on the straight lines \(\Gamma\) and \(A\Gamma\), satisfy the inequality
\[ s(Al)\geq \mu_1 s(l). \tag{2} \]

Similarly, if one takes the system of straight lines
\[ dx_2+\alpha_2dx_1=0, \tag{1'} \]
where \((\alpha_2,1)\) is an eigenvector of the transformation \(A^*\) with eigenvalue \(\lambda_2\), \(|\lambda_2|>1\), then for it properties I and II′ are fulfilled, where II′ is formulated in the same way as II, only instead of inequality (2) there appears the inequality
\[ s(Al)\leq \mu_2s(l),\quad 0<\mu_2<1. \tag{2'} \]

§ 2. It turns out that the property of the transformation \(A\) of possessing a family of curves with properties I, II (and I, II′) is coarse, i.e. is preserved under small perturbations of the automorphism \(A\) by nonlinear terms. Namely, let \(A_\varepsilon=A+\varepsilon B(x)\), i.e. \(x\to A_\varepsilon x=Ax+\varepsilon B(x)\), where \(B(x)=(b_1(x_1,x_2), b_2(x_1,x_2))\), \(b_i(x)\) are functions periodic in each argument with period 1 and three times continuously differentiable.

Theorem 1. For sufficiently small \(\varepsilon>0\) there exists a system of curves
\[ dx_2+\widetilde{\alpha}_1(x,\varepsilon)\,dx_1=0, \tag{3} \]
possessing, with respect to the automorphism \(A_\varepsilon\), properties I and II; the function \(\widetilde{\alpha}_1(x,\varepsilon)\) has continuous derivatives of bounded variation with respect to \(x\), and is continuous with respect to \(\varepsilon\). Among the solutions of (3) there are no closed curves.

Proof. We shall use the method of successive approximations. Suppose that the curves
\[ dx_2+\alpha_1^n(x,\varepsilon)\,dx_1=0 \]
have already been constructed. Apply to them the transformation \(A_\varepsilon\). If the matrix
\[ \|a_{ij}+\varepsilon\,\partial b_i/\partial x_j\|^{-1} \]
has the form
\[ \|\bar{a}_{ij}\|+\varepsilon\|g_{ij}(x,\varepsilon)\|, \]
where \(\bar{a}_{ij}\) are the elements of the matrix \(A^{-1}\), and \(\|g_{ij}(x,\varepsilon)\|\) is a matrix depending continuously on \(x\) and on \(\varepsilon\) and bounded, then the result-

the system of curves will be written in the form \(dx_2+\alpha_1^{n+1}(x,\varepsilon)\,dx_1=0\), where

\[ \alpha_1^{n+1}(x,\varepsilon)= \frac{(\bar a_{11}+\varepsilon g_{11}(x,\varepsilon))\alpha_1^n(A_\varepsilon^{-1}x)+(\bar a_{21}+\varepsilon g_{21}(x,\varepsilon))} {(\bar a_{12}+\varepsilon g_{12}(x,\varepsilon))\alpha_1^n(A_\varepsilon^{-1}x)+(\bar a_{22}+\varepsilon g_{22}(x,\varepsilon))}. \tag{4} \]

Lemma 1. If \(\max\limits_x |\alpha_1^n(x,\varepsilon)-\alpha_1|<\delta\), then there exist \(\mu\), \(0<\mu<1\), and \(C<\infty\), depending only on \(A\) and \(B\), such that
\[ \max_x |\alpha_1^{n+1}(x,\varepsilon)-\alpha_1|\leq \mu\delta+C\varepsilon. \]

Proof. We rewrite equality (4) as follows:

\[ \alpha_1^{n+1}(x,\varepsilon)= \frac{\bar a_{11}\alpha_1+\bar a_{21}+(\alpha_1^n(A_\varepsilon^{-1}x)-\alpha_1)\bar a_{11} +\varepsilon(g_{11}\alpha_1^n+g_{21})} {\bar a_{12}\alpha_1+\bar a_{22}+(\alpha_1^n(A_\varepsilon^{-1}x)-\alpha_1)\bar a_{12} +\varepsilon(g_{12}\alpha_1^n+g_{22})}. \]

From the fact that \((\alpha_1,1)\) is an eigenvector of the matrix \(A^*\), it is not hard to derive that
\(\bar a_{11}\alpha_1+\bar a_{21}=\alpha_1/\lambda_1\),
\(\bar a_{12}\alpha_1+\bar a_{22}=1/\lambda_1\). Therefore

\[ \alpha_1^{n+1}(x,\varepsilon)= \frac{\alpha_1+\lambda_1[(\alpha_1^n(A_\varepsilon^{-1}x)-\alpha_1)\bar a_{11} +\varepsilon(g_{11}\alpha_1^n+g_{21})]} {1+\lambda_1[(\alpha_1^n(A_\varepsilon^{-1}x)-\alpha_1)\bar a_{12} +\varepsilon(g_{12}\alpha_1^n+g_{22})]}, \]

\[ |\alpha_1^{n+1}(x,\varepsilon)-\alpha_1|= \left| \lambda_1 \frac{(\alpha_1-\alpha_1^n)(-\bar a_{12}\alpha_1+\bar a_{11}) +\varepsilon[g_{11}\alpha_1^n+g_{21}-\alpha_1(g_{12}\alpha_1^n+g_{22})]} {1+\lambda_1[(\alpha_1^n(A_\varepsilon^{-1}x)-\alpha_1)\bar a_{12} +\varepsilon(g_{12}\alpha_1^n+g_{22})]} \right|. \]

To complete the proof of the lemma it is enough to take into account that, analogously to the preceding, \(-\alpha_1\bar a_{12}+\bar a_{11}=\lambda_1\).

It follows from Lemma 1 that for any \(\delta>0\) one can find \(\varepsilon_1(\delta)\) such that
\(\max\limits_x |\alpha_1^n(x,\varepsilon)-\alpha_1|<\delta\) for all \(n\), if \(\varepsilon<\varepsilon_1(\delta)\).

We now estimate the difference \(|\alpha_1^{n+1}(x,\varepsilon)-\alpha_1^n(x,\varepsilon)|\). Using (4), it is easy to find that

\[ \alpha_1^{n+1}(x,\varepsilon)-\alpha_1^n(x,\varepsilon) = D_n(x,\varepsilon) [\alpha_1^n(A_\varepsilon^{-1}x,\varepsilon)-\alpha_1^{n-1}(A_\varepsilon^{-1}x,\varepsilon)], \]

where

\[ D_n(x,\varepsilon)= \frac{\bar a_{11}\bar a_{22}-\bar a_{12}\bar a_{21} +\varepsilon^2(g_{11}g_{22}-g_{12}g_{21}) +\varepsilon(g_{11}\bar a_{22}+g_{22}\bar a_{11}-g_{21}\bar a_{12}-g_{12}\bar a_{21})} {[(\bar a_{12}\alpha_1^n+\bar a_{22})+\varepsilon(g_{12}\alpha_1^n+g_{22})] [(\bar a_{12}\alpha_1^{n-1}+\bar a_{22})+\varepsilon(g_{12}\alpha_1^{n-1}+g_{22})]}. \]

But, by Lemma 1, for any \(\delta>0\) and \(\varepsilon<\varepsilon_2(\delta)\) for all \(n\)
\[ |\bar a_{12}\alpha_1^n+\bar a_{22}-1/\lambda_1|\leq \delta. \]
Since \(\bar a_{11}\bar a_{22}-\bar a_{12}\bar a_{21}=\pm1\) and \(|\lambda_1|<1\), there exists \(\rho\), \(0<\rho<1\), such that for all \(n\)

\[ \max_x |\alpha_1^{n+1}(x,\varepsilon)-\alpha_1^n(x,\varepsilon)| \leq \rho \max_x |\alpha_1^n(x,\varepsilon)-\alpha_1^{n-1}(x,\varepsilon)|. \]

Consequently, the sequence \(\alpha_1^n(x,\varepsilon)\) converges uniformly on \(T^2\). Put
\[ \tilde\alpha(x,\varepsilon)=\lim_{n\to\infty}\alpha_1^n(x,\varepsilon). \]
It is easy to see that the system of curves \(\Gamma_1\)

\[ dx_2+\bar\alpha_1(x,\varepsilon)\,dx_1=0 \tag{5} \]

has properties I and II with respect to the automorphism \(A_\varepsilon\). The continuous dependence of \(\tilde\alpha_1(x,\varepsilon)\) on \(\varepsilon\) is obvious. The existence of derivatives of \(\tilde\alpha_1(x,\varepsilon)\) with respect to \(x\) of bounded variation was established by V. I. Oseledets by methods analogous to the preceding ones.

We now prove that among the curves (5) there are no closed curves. Suppose that such a curve \(\Gamma\) exists. Then, by property II, the length of the curve \(A_\varepsilon^{-n}\Gamma\) satisfies the inequality \(s(A_\varepsilon^{-n}\Gamma)\leq\mu_1^{-n}s(\Gamma)\). But the curve \(A_\varepsilon^{-n}\Gamma\) cannot then be a curve of (5), since it cannot satisfy

to the inequality \(|\tilde a_1-a_1|<\delta\), which holds for the curves (5). The theorem is proved.

Remark. Similarly one can prove the existence of curves \(\Gamma_2\),
\(dx_2+\tilde a_2(x,\varepsilon)\,dx_1=0\), possessing properties I and II′.

§ 3. Consider two (incompatible) equations on the torus

\[ dx_2=f_i(x_1,x_2)\,dx_1 \quad (i=1,2), \tag{6} \]

where the first derivatives of the functions \(f_i(x_1,x_2)\) of period 1 in \(x_1\) and \(x_2\) have bounded variation. Poincaré (\(^1\)) defined rotation numbers \(\omega_i\) for equations of the form (6).

Theorem 2. Let \(\omega_1,\omega_2\) be irrational and

\[ -\infty<c_1<f_1<C_1<c_2<f_2<C_2<\infty, \tag{7} \]

where \(c_1,\ldots,C_2\) are constants. Then there exists a homeomorphism of the torus \(x\leftrightarrow y\), straightening the integral curves \(\Gamma_i\) of both equations (6), i.e. transforming them into straight lines \(\Gamma_i'\): \(dy_2=\omega_i dy_1\) \((i=1,2)\).

Proof. \(1^\circ\). Denote by \(\Gamma_i(x_0)\) the integral curves of equations (6) passing through the point \(x_0=(x_1^0,x_2^0)\), and by \(\Gamma_i'(y_0)\) the straight line \(y_2-y_2^0=\omega_i(y_1-y_1^0)\). Let \(p_1,p_2\) be two integer points. Consider on the \(x\)-plane the point \(q(p_1,p_2)=\Gamma_1(p_1)\cap\Gamma_2(p_2)\), and on the \(y\)-plane the point \(q'(p_1,p_2)=\Gamma_1'(p_1)\cap\Gamma_2'(p_2)\). Define the mapping \(y\to x\) by assigning to the point \(q'(p_1,p_2)\) the point \(q(p_1,p_2)\).

\(2^\circ\). Lemma 2. The mapping \(q'\to q\) is uniformly continuous.

Proof. By Denjoy’s theorem (\(^2\)), there exists a continuous transformation of the torus taking the lines \(\Gamma_1\) into the straight lines \(\Gamma_1'\). Therefore, for every \(\varepsilon>0\) there exists \(\delta_1(\varepsilon)>0\) such that if the distance between two straight lines \(\Gamma_1'(p_1),\Gamma_1'(p_3)\) is less than \(\delta_1\), then the distance between the curves \(\Gamma_1(p_1),\Gamma_1(p_3)\) is everywhere less than \(\varepsilon\). Analogous arguments applied to \(\Gamma_2'\) and \(\Gamma_2\) give \(\delta_2(\varepsilon)\). In view of condition (7), if the distance between the points \(q'(p_1,p_2)\), \(q'(p_3,p_4)\) is less than \(\delta_1(\varepsilon)\) and \(\delta_2(\varepsilon)\), then the distance between the points \(q(p_1,p_2)\), \(q(p_3,p_4)\) is less than \(K\varepsilon\) (where \(K\) depends only on \(C_1\) and \(c_2\)). The lemma is proved.

\(3^\circ\). Since the sets of points \(q(p_1,p_2)\) and \(q'(p_1,p_2)\) are everywhere dense, in view of (\(^2\)) and (7), the mapping \(q'\to q\) can be extended by continuity to the whole \(y\)-plane. The resulting homeomorphism of the plane determines the required homeomorphism of the torus, since the straight lines \(\Gamma_i'\) are mapped into the curves \(\Gamma_i\). Theorem 2 is proved.

§ 4. Theorem 3. An ergodic automorphism of the two-dimensional torus is structurally stable \(^*\). This means that, under the conditions of Theorem 1 and for sufficiently small \(\varepsilon\), there exists a homeomorphism of the torus \(x\leftrightarrow y\), transforming the perturbed automorphism into the unperturbed one:

\[ y(A_\varepsilon x)=Ay(x). \tag{8} \]

Proof. For small \(\varepsilon\), \(A_\varepsilon\) has one fixed point \(O_\varepsilon\) on the \(x\)-plane. We take it as the origin on the \(x\)-plane. Construct the homeomorphism of Theorem 2 from the curves \(\Gamma_i\) obtained in § 2 (5). Since these curves on the torus are not closed, the rotation numbers \(\omega_i\) are irrational. On the other hand, \(\omega_i(\varepsilon)\) depend continuously on \(\varepsilon\). Consequently, they are constant. Therefore the straight lines \(\Gamma_i'\) have the directions of the eigenvectors of \(A\).

The curves \(\Gamma_i(O_\varepsilon)\) are mapped into themselves under the action of the transformation \(A_\varepsilon\), as are the straight lines \(\Gamma_i'(O)\) under the action of \(A\). Moreover, \(A\) takes \(q'(p_1,p_2)\) to \(q'(Ap_1,Ap_2)\), while \(A_\varepsilon\) takes \(q(O_\varepsilon+p_1,O_\varepsilon+p_2)\) to \(q(O_\varepsilon+Ap_1,O_\varepsilon+Ap_2)\). Therefore (8) is fulfilled for \(x=q(p_1,p_2)\), and hence, by continuity, for all \(x\).

Remark. If \(A_\varepsilon\) is analytic, then, according to (\(^3\)), the curves \(\Gamma_1\) or

\(^*\) Or “rough,” in the terminology of Andronov–Pontryagin.

\(\Gamma_2\) can be straightened locally by an analytic transformation. However, the homeomorphism constructed in § 3 may fail to be differentiable. Indeed, the eigenvalues of \(A_\varepsilon\) in \(O_\varepsilon\) may differ from the eigenvalues of \(A\). If the homeomorphism \(x \leftrightarrow y\) is absolutely continuous, then, for sufficiently small \(\varepsilon\), the transformation \(A_\varepsilon\) has an invariant measure absolutely continuous with respect to Lebesgue measure and is metrically isomorphic to \(A\). However, we do not know whether the condition of absolute continuity is fulfilled even in the case of analytic measure-preserving perturbations.

§ 5. In the \(n\)-dimensional case we have been able to prove the following.

Theorem 4. If the matrix \(A=\|a_{ij}\|\) has \(n\) real eigenvalues, \(k\) of which are greater than 1 and the remaining ones less than 1, and \(A_\varepsilon=A+\varepsilon B\), then on the torus \(T^n\) there exists a system of \((n-k)\)-dimensional unclosed smooth surfaces, invariant with respect to \(A_\varepsilon\), and such that for any piece \(l\) of a surface the volume \(V(l)\leq \mu^l V(A_\varepsilon l)\), where \(\mu';\ 0<\mu'<1\).

Theorem 5. Let the \(n\)-dimensional torus \(T^n\) be decomposed into \((n-1)\)-dimensional smooth surfaces \(\Gamma: x_n=g(x_1,\ldots,x_{n-1})\), and let the functions \(g\) be twice continuously differentiable. If none of these surfaces is closed, then there exists a homeomorphism of the torus \(x \leftrightarrow y\) straightening the surfaces \(\Gamma\), i.e. taking them into the planes \(\Gamma': y_n=\omega_1 y_1+\cdots+\omega_{n-1}y_{n-1}+C\).

Proof. \(1^\circ\). Let \(\Gamma(x_n)\) be the surface \(\Gamma\) passing through the point \((0,\ldots,0,x_n)\). Denote by \(Q_p(x_n)\) (where \(p=(p_1,\ldots,p_{n-1})\)) the point \((p,x'_n)\in\Gamma(x_n)\). Let \(p_1,\ldots,p_{n-1}\) be integers. Then \(Q_p\) may be regarded as a mapping of the circle \(p=0\) onto itself. It is easy to see that \(Q_{p+q}=Q_pQ_q\), so that all the mappings \(Q_p\) commute.*

\(2^\circ\). Lemma 3. Suppose a finite number of commuting twice continuously differentiable homeomorphisms of the circle \(Q_1,\ldots,Q_r\) is given. Then either there exists a homeomorphism of the circle transforming them into rotations, or there exists an \(N\) such that \(Q_1^N,\ldots,Q_r^N\) have a common fixed point.

Indeed, if at least one of the transformations, for example \(Q_1\), has an irrational rotation number \((^1)\), then, for a certain choice of parameter on the circle, it is a rotation through an irrational angle. Since all the transformations \(Q\) commute with \(Q_1^k\) \((k=1,2,\ldots)\), they commute with all rotations and, consequently, themselves are rotations. If all rotation numbers are rational, then for some \(N\) each of the transformations \(R_1=Q_1^N,\ldots,R_r=Q_r^N\) has fixed points. Therefore there exists

\[ \lim_{n_1\to\infty} R_1^{n_1}\lim_{n_2\to\infty} R_2^{n_2}\cdots \lim_{n_r\to\infty} R_r^{n_r}x_0=z. \]

It is easy to see that \(z\) is a fixed point of all \(R_1,\ldots,R_r\).

\(3^\circ\). Apply Lemma 3 to the transformations \(Q_i=Q_{1,0,\ldots,0},\ldots,Q_{0,\ldots,0,1}\). If all \(Q^N\) have a common fixed point \(z\), then the surface \(\Gamma(z)\) is closed, contrary to the hypothesis of the theorem. Consequently, there exists a parameter \(\varphi(x_n)\) such that \(\varphi(Q_i x_n)=\varphi(x_n)+\omega_i\). Now define \(x_0(x)\) by the condition \(x\in\Gamma(x_0)\) and set \(y_i=x_i\) \((1\leq i<n)\), \(y_n=\varphi(x_0(x))+\omega_1x_1+\cdots+\omega_{n-1}x_{n-1}\). It is easy to see that \(x\leftrightarrow y\) is the required homeomorphism of the torus.

Theorem 3 was communicated to the authors in the form of a conjecture by S. Smale. The authors express their gratitude to him, and also to D. V. Anosov and E. G. Belaga, for useful discussions.

Moscow State University
named after M. V. Lomonosov

Received
17 I 1962

REFERENCES

\(^{1}\) A. Poincaré, On curves defined by differential equations, Moscow–Leningrad, 1947.
\(^{2}\) A. Denjoy, J. de Math., 11, Fasc. IV, 333 (1932).

* The same device makes it possible to study decompositions of skew products with circle fiber. The question reduces to finding representations of the fundamental group of the base into the group of mappings of the circle onto itself.