Reports of the Academy of Sciences of the USSR

1. Volume 136, No. 3

MATHEMATICS

Ya. G. Sinai

GEODESIC FLOWS ON COMPACT SURFACES OF NEGATIVE CURVATURE

(Presented by Academician A. N. Kolmogorov on 30 VIII 1960)

In the work of E. Hopf \((^6)\) the ergodicity of the dynamical system generated by the geodesic flow in the space of line elements of a compact two-dimensional surface of negative curvature was proved. In the present work, by methods analogous to those applied in \((^4)\), the spectrum and the character of mixing of this dynamical system are investigated.

§ 1. Let \(\widetilde L\) be the interior of the unit circle of the plane \(z = x_1 + i x_2\), endowed with the Riemannian metric

\[ ds^2 = \lambda^2(x_1, x_2)\, \frac{dx_1^2 + dx_2^2}{(1 - x_1^2 - x_2^2)^2}. \tag{1} \]

Assume that: 1) the function \(\lambda(x_1, x_2)\) is differentiable 10 times; 2) there exist positive constants \(a, b\), \(a \leq b\), such that \(a \leq \lambda(x_1, x_2) \leq b\); 3) the Gaussian curvature \(K(x_1, x_2)\), determined by the metric (1), is negative, and there exist positive constants \(c, d\), \(c \geq d\), such that \(-c \leq K(x_1, x_2) \leq -d\). The geodesic lines of the surface \(\widetilde L\) are close in their properties to the straight lines of the Lobachevsky plane \((^{6,7})\). Thus, the notion of a geodesic that is positively asymptotic to a given one and passes through a chosen point has meaning (see, for example, \((^6)\)).

Let \(\Gamma\) be a discrete subgroup of the group of fractional-linear transformations of the plane \(z\) taking the unit circle onto itself. We shall be interested only in the case when \(\Gamma\) has a compact fundamental domain (in the sense of distances in the Lobachevsky plane). If \(\lambda(x_1, x_2)\) is invariant with respect to all \(\gamma \in \Gamma\), then the identification of points passing into one another under the action of transformations from \(\Gamma\) leads to a certain compact two-dimensional Riemannian manifold \(L\) of negative curvature.

Consider the spaces \(\widetilde M, M\), formed from the line elements of the surfaces \(\widetilde L, L\), respectively. In \(\widetilde M\) and \(M\) one can define the geodesic flow \(\{T^t\}\), i.e. a one-parameter group of transformations, where the individual transformation \(T^t\) consists in the fact that the line element \(l \in \widetilde M\) (or \(l \in M\)) is shifted along the geodesic determined by it in a given direction by a distance \(t\). In the spaces \(\widetilde M\) and \(M\) there exist measures \(d\widetilde\mu\) and \(d\mu\), invariant with respect to the flow \(\{T^t\}\) \((^6)\).

§ 2. Horocycles on the manifolds \(\widetilde L, L\). By horocycles on \(\widetilde L\) we shall mean, by analogy with the Lobachevsky plane, orthogonal trajectories to a family of positively asymptotic geodesics. They are constructed in the following way.* Let \(l \in \widetilde M\), and let \(z(t)\) be the support of the line element \(T^t l\).

* For the first time, as far as we know, horocycles on surfaces of variable negative curvature were constructed in \((^7)\). Some properties of horocycles discussed below are also indicated there.

Draw the geodesic circle \(R_t\) with center at \(z(t)\) and radius \(t\). The boundary of the closure of the aggregate of the points \(R_t\), as \(t\) runs through all values from \(0\) to \(\infty\), will be called the horocycle generated by the line element \(l\), and will be denoted by \(H(l)\). If \(G_1\) is a geodesic positively asymptotic to the geodesic \(G(l)\) determined by the line element \(l\), then the set \(H(l)\cap G_1\) consists of exactly one point. If \(F\) is a subset of \(H(l)\), then the aggregate of line elements whose base points belong to \(F\), and whose directions are orthogonal to \(H(l)\) and determine geodesics positively asymptotic to \(G(l)\), will be called belonging to \(F\).

Let us note the following properties of horocycles:

1. The horocycle \(H(l')\), generated by a line element \(l'\) belonging to the horocycle \(H(l)\), coincides with \(H(l)\). From this property it follows that a horocycle is an orthogonal trajectory to the family of positively asymptotic geodesics.

2. On any geodesic \(G_1\), positively asymptotic to \(G(l)\), the horocycles \(H(l)\) and \(H(T^t l)\) cut out an arc of length \(t\).

3. The geodesic curvature of a horocycle. To compute the geodesic curvature \(\chi\) of the horocycle \(H(l)\) at the point \(z\), one must draw through \(z\) the geodesic \(G_1\) positively asymptotic to \(G(l)\), and consider on \(G_1\) the equation \(\chi' + \chi^2 + K = 0\). The solution of this equation bounded on the entire geodesic \(G_1\) exists, is unique, and \(\chi(z)\) is the geodesic curvature of the horocycle \(H(l)\) at the point \(z\). The function \(\chi(z)\) satisfies the inequalities \(\sqrt{d}\leq \chi(z)\leq \sqrt{c}\).

4. Horocycles make it possible to introduce the following coordinate system in the space \(\widetilde M\). Let \(o\) be an arbitrary point of \(\widetilde L\) and let \(l\in \widetilde M\). Then there exists only one geodesic \(G\) passing through \(o\) and positively asymptotic to \(G(l)\) \({}^{(6)}\). By \(\varphi\) denote the angle which the geodesic \(G\) makes with a fixed direction at \(o\). Next introduce the quantity \(s\), equal to the length of the segment of the geodesic \(G\) between the point \(o\) and the point of intersection of \(G\) with \(H(l)\), taken with a plus sign if \(o\) lies inside \(H(l)\), and with a minus sign otherwise, and the quantity \(\rho\), equal to the length of the arc of the horocycle \(H(l)\) from the base point of \(l\) to the point \(G\cap H(l)\), taken with a plus sign if the rotation of the geodesic \(G\) about the point \(G\cap H(l)\) to make it coincide with the base of \(l\) occurs counterclockwise, and with a minus sign in the opposite case. The coordinates \(\varphi, s, \rho\) determine a line element of the surface \(\widetilde L\) uniquely. The invariant measure \(d\tilde\mu\) is written in these coordinates in the following way: \(d\tilde\mu=f(\varphi,s,\rho)\,d\varphi\,ds\,d\rho\), where \(f(\varphi,s,\rho)\) is a continuous positive function. Moreover, for any \(d>0\) there exists a constant \(\alpha(d)\) such that for all line elements whose base points are at a distance from the point \(o\) not exceeding \(d\), \(f(\varphi,\rho,s)\geq \alpha(d)\).

5. It is easy to see that horocycles constructed from two line elements \(l\) and \(l'\), congruent to one another with respect to the subgroup \(\Gamma\), are congruent with respect to \(\Gamma\). Therefore horocycles may be regarded as lines on the surface \(L\).

§ 3. The investigation carried out below is based on the notion of a Kolmogorov dynamical system or, what is the same thing, a \(K\)-system \({}^{(1,2,4)}\).

Theorem 1. The geodesic flow in the space \(M\) of line elements of the surface \(L\) is a \(K\)-system.

Proof. Let \(\pi=(l_1,l_2)\) be the set of line elements belonging to the arc \((z_1,z_2)\) of the horocycle \(H(l_1)\) (or \(H(l_2)\)), \(z_i\) being the base point of \(l_i\), \(i=1,2\). Under the action of the transformation \(T^t\) the set \(\pi\) is transformed into the set \(\pi^t\) of line elements belonging to the arc \((z_1^t,z_2^t)\) of the horocycle \(T^tH(l_1)=H(T^t l)\).

Lemma 1. The limit

\[ \lim_{z_2\to z_1}\lim_{\tau\to 0}\frac{1}{\tau} \left( \frac{\rho(z_1,z_2)}{\rho(z_1^\tau,z_2^\tau)}-1 \right), \]

where \(\rho(z_1^\tau,z_2^\tau)\) is the length of the horocycle segment \(H(T^\tau l)\), exists and is equal to the geodesic curvature \(\varkappa\) of the horocycle \(H(l_1)\) at the point \(z_1\).

The result of the lemma is obtained by means of the Gauss–Bonnet formula, applied to the quadrilateral bounded by the horocycle segments \((z_1,z_2)\) and \((z_1^\tau,z_2^\tau)\) and by the geodesic arcs joining \(z_1\) with \(z_1^\tau\) and \(z_2\) with \(z_2^\tau\). From the lemma and inequalities (2) it follows that, for any \(\tau \le 0\),

\[ \exp(-\sqrt{\bar d}\,\tau)\rho(x_1,x_2) \le \rho(x_1^\tau,x_2^\tau) \le \exp(-\sqrt{c}\,\tau)\rho(x_1,x_2). \tag{3} \]

Let now \(R\) be a certain fundamental domain of the subgroup \(\Gamma\), chosen in the form of a non-Euclidean polygon with a finite number of sides and containing the point \(o\) in its interior (see § 2, item 4). Consider the set \(D_0\) of line elements of the surface \(L\), whose coordinates satisfy the inequalities

\[ 0\le \varphi_l<2\pi,\qquad -\alpha(1+\varphi_l)\le s_l\le \alpha(1+\varphi_l),\qquad -\beta\le \rho_{T^{-s_l}l}\le \beta. \]

By choosing \(\alpha\) and \(\beta\), one can arrange that, for every \(l\in D_0\) and every nonidentity transformation \(\gamma\in\Gamma\), one has \(\gamma l\bar{\in}D_0\). Then the set \(D_0\) may be regarded as a subset of \(M\).

Consider the measurable partition \(\xi\) of the set \(D_0\), each element of which is specified by fixed values of the coordinates \(\varphi\) and \(s\) and consists of the line elements \(l\in D_0\) for which \(\varphi_l=\varphi\) and \(s_l=s\). Next, let \(\xi'\) be the measurable partition of the space \(M\), coinciding with the partition \(\xi\) on the set \(D_0\) and degenerate on the set \(M-D_0\) (i.e., \(M-D_0\) is an element of the partition \(\xi'\)). Put \(\xi^0=\prod_{t\le \tau}T^t\xi'\), \(\zeta=\prod_{|t|\le \tau}T^t\xi'\), where \(\tau>\beta+2\pi\alpha\).

Lemma 2. There exists a set \(\Omega\subset M\) of measure zero, invariant with respect to the flow \(\{T^t\}\), such that two line elements \(l_1\in M-\Omega\) and \(l_2\in M-\Omega\) can belong to one element of the partition \(\xi^0\) only if they belong to one and the same horocycle.

The proof of Lemma 2 is simple and is omitted.

Lemma 3. The set of elements of the partition \(\xi^0\) consisting of a single point has measure zero. Almost every element of the partition \(\xi^0\) consists of line elements \(l\in M\) belonging to a finite segment of some horocycle.

We indicate the main points of the proof of Lemma 3. Let \(\Xi_k\) be the set of line elements of the partition \(\zeta\) intersecting more than one element of the partition \(T^{-k\tau}\zeta\). On the basis of (2) and (3), for \(\mu(\Xi_k)\) one can obtain the estimate \(\mu(\Xi_k)\le C_1\exp(-C_2K)\), where \(C_1\) and \(C_2\) are positive constants. Since \(\sum_k\mu(\Xi_k)<\infty\), the assertion of the lemma follows from the fact that almost every element of the partition \(\zeta\) belongs to a finite number of the sets \(\Xi_k\).

Lemma 4. For the system of partitions \(\xi^t=T^t\xi^0\) the relation

\[ \prod_{-\infty}^{\infty}\xi^t=\varepsilon \mod 0, \]

holds, where \(\varepsilon\) is the partition of the space \(M\) into individual points.

Proof. It is easy to show that there exists a number \(\sigma>0\) such that the lengths of the horocycle segments making up the elements of the partition \(\xi^0\) do not exceed \(\sigma\). Then, on the basis of Lemma 1, the lengths of the horocycle segments

cycles that constitute elements of the partition \(\zeta^t\), are bounded by the constant \(\sigma \exp(-\sqrt d\,t)\). Therefore, if the linear elements \(l_1\) and \(l_2\) lie, for all \(t\), in one and the same element of the partition \(\zeta^t\), then the distance between their carriers is equal to zero and, consequently, they coincide. The assertion of the lemma now follows from V. A. Rokhlin’s theorem on bases ((\(^{3}\), p. 123)).

Thus, a system of partitions \(\zeta^t\), \(\zeta^t=T^t\zeta^0\), has been obtained with the following properties: 1) \(\zeta^t \geqslant \zeta^{t_1}\mod 0\) for \(t>t_1\), which follows from the construction, and 2)

\[ \prod_{-\infty}^{\infty}\zeta^t=\varepsilon \mod 0 \]

—see Lemma 4. To complete the proof of the theorem it remains to show that

\[ \bigcap_{-\infty}^{\infty}\zeta^t=\nu \mod 0, \]

where \(\nu\) is the trivial partition, whose only element is the whole space.

Corollary. The dynamical system generated by the geodesic flow \(\{T^t\}\) has countably multiple Lebesgue spectrum and is mixing of all degrees (see (\(^{5}\))).

§ 4. With the aid of the theorem proved above one can transfer to the case under consideration the main result of the work (\(^{8}\)).

Theorem 2. Let a bounded real function \(f\), given on \(M\), satisfy the conditions:

1) there exist numbers \(\alpha>0\), \(\varepsilon>0\), \(\varepsilon_1>0\), \(C_1>0\), \(C_2\geqslant0\) such that, for all \(x>0\),

\[ \mu\left\{\,l:\sup_{l_1:\sigma(l_1,l)<x} \left|\int_0^\alpha f(T^\tau l)\,d\tau-\int_0^\alpha f(T^\tau l_1)\,d\tau\right| > \frac{C_1}{\log^{1+\varepsilon}\sigma} \,\right\} \leqslant \frac{C_2}{|\log^{1+\varepsilon_1}\sigma|}, \]

where \(\sigma(l_1,l)\) is the distance in the Riemannian space \(M\);

2)

\[ D_T(f)=\int_M\left[\int_0^T f(T^t l)\,dt\right]^2\,d\mu\sim cT,\quad \text{where } c>0; \]

3) for every \(\varepsilon>0\) there exist \(N(\varepsilon)\) and \(T(\varepsilon)\) such that, for all \(t>T(\varepsilon)\),

\[ \frac{1}{D_T(f)} \int_{\left\{\,l:\left|\int_0^T f(T^\tau l)\,d\tau\right|>N(\varepsilon)\sqrt{D_T(f)}\,\right\}} \left[\int_0^T f(T^\tau l)\,d\tau\right]^2 d\mu \leqslant \varepsilon. \]

Then, for any fixed \(s\), \(-\infty<s<\infty\),

\[ \lim_{T\to\infty} \mu\left\{\,l: \frac{\int_0^T f(T^t l)\,dt-T\bar f}{\sqrt{D_T(f)}}<s \,\right\} = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{s}\exp\left(-\frac{u^2}{2}\right)\,du, \]

where

\[ \bar f=\int_M f(l)\,d\mu. \]

It can be shown that there exist functions satisfying all the requirements of Theorem 2.

Moscow State University
named after M. V. Lomonosov

Received
23 VIII 1960

CITED LITERATURE

\(^{1}\) A. N. Kolmogorov, DAN, 119, No. 5 (1958).
\(^{2}\) V. A. Rokhlin, UMN, 15, No. 4 (1960).
\(^{3}\) V. A. Rokhlin, Matem. sborn., 25, No. 1 (1949).
\(^{4}\) Ya. G. Sinai, DAN, 131, No. 4 (1960).
\(^{5}\) Ya. G. Sinai, Teoriya veroyatn. i ee primenen., 5, No. 3 (1960).
\(^{6}\) E. Hopf, UMN, 4, No. 2 (1949).
\(^{7}\) A. Grant, Duke Math. J., 5, No. 2 (1939).
\(^{8}\) Ya. G. Sinai, DAN, 133, No. 6 (1960).