UDC 519.217

MATHEMATICS

M. E. RATNER

ON AN INVARIANT MEASURE FOR A \(Y\)-FLOW ON A THREE-DIMENSIONAL MANIFOLD

(Presented by Academician A. N. Kolmogorov, 28 X 1968)

In [1] it is proved that if \(T\) is a \(Y\)-diffeomorphism of class \(C^2\) of the two-dimensional torus \(M\), having everywhere dense leaves of the transversal foliations, then in \(M\) there exists a measure \(\mu\), positive on every open set, invariant with respect to \(T\), and \(T\), as an automorphism of the space with measure \((M,\mu)\), is a \(K\)-automorphism. Moreover, for every measurable partition \(\xi\) of the space \(M\), whose elements \(C_\xi\) are intervals of the contracting foliation, the conditional measure \(\mu(\cdot\mid C_\xi)\), induced by \(\mu\), is equivalent on almost every \(C_\xi\) to the normalized Riemannian volume on \(C_\xi\). The listed properties determine the measure \(\mu\) uniquely. The proof in [1] is based on the existence of a Markov partition. In the present paper an analogous assertion is proved for a \(Y\)-flow on a three-dimensional manifold.

1. Markov partition. Let \(W\) be a three-dimensional compact Riemannian manifold of class \(C^\infty\); \(\{T^t\}\) a \(Y\)-flow of class \(C^2\) on \(W\) (see [2]); \(\Gamma_c\) (\(\Gamma_p\)) its contracting (expanding) foliation; \(G_c\) (\(G_p\)) the foliation into leaves
\[ G_c=\bigcup_{t=-\infty}^{\infty} T^t\Gamma_c \quad \left(G_p=\bigcup_{t=-\infty}^{\infty} T^t\Gamma_p\right). \]
It is assumed that all leaves of the foliations \(\Gamma_c\) and \(\Gamma_p\) are everywhere dense in \(W\).

Let the points \(w_0\) and \(w_1\) lie on one small interval \(\gamma_c(w_0)\) of the leaf \(\Gamma_c(w_0)\); let \(M_{w_0}\) and \(M_{w_1}\) be neighborhoods of the points \(w_0\) and \(w_1\) on \(G_p(w_0)\) and \(G_p(w_1)\) such that for \(w\in M_{w_0}\) there is a \(w'\in M_{w_1}\), lying on a small interval \(\gamma_c(w)\), and the mapping \(\pi_{w_0,w_1}: w\to w'\) is a homeomorphism of \(M_{w_0}\) onto \(M_{w_1}\). Let \(\gamma_p\in M_{w_0}\) be an interval of the leaf \(\Gamma_p(w_0)\); let \(\tau(w)\) be a continuous nonnegative function on \(\gamma_p\), not identically equal to zero.

Every set
\[ \Pi=\bigcup_{w\in\gamma_p}\gamma_c(w) \quad \left( V=\bigcup_{w\in\gamma_p}\bigcup_{\tau=0}^{\tau(w)} T^\tau\gamma_c(w) \right) \]
will be called a \(\Gamma_c\)-regular parallelogram (parallelepiped), and the lengths of the intervals \(\gamma_p\) and \(\gamma_c(w_0)\) the dimensions of the parallelogram \(\Pi\). The contracting \(\Delta_c\) (expanding \(\Delta_p\)) boundary of \(V\) will be called
\[ \Delta_c(V)= \bigcup_{y\in\partial\gamma_p}\bigcup_{\tau=0}^{\tau(y)}T^\tau\gamma_c(y) \quad \left( \Delta_p(V)= \bigcup_{w\in\gamma_p}\bigcup_{\tau=0}^{\tau(w)} T^\tau\left(w\cup\pi_{w_0,w_1}(w)\right) \right). \]
We shall call \(\Pi\) the upper face of \(V\). If \(\beta\) is a collection of parallelepipeds \(\{V_i\}\), then \(\Delta_c(\beta)=\bigcup_i\Delta_c(V_i)\) and \(\Delta_p(\beta)=\bigcup_i\Delta_p(V_i)\).

Analogously to [1], we introduce the following definition.

Definition. A finite partition \(\alpha\) into nonintersecting \(\Gamma_c\)-regular parallelepipeds \(\{V_i\}\) is called Markov if for all \(t\ge 0\), \(T^t\Delta_c(\alpha)\subset\Delta_c(\alpha)\), and for all \(t\le 0\), \(T^t\Delta_p(\alpha)\subset\Delta_p(\alpha)\).

Theorem 1. For every \(\varepsilon>0\) there exists a Markov partition into \(\Gamma_c\)-regular parallelepipeds whose upper-face dimensions do not exceed \(\varepsilon\).

2. Special representation.

Let \(a\) be a Markov partition and let \(M\) be the set-theoretic union of the upper faces \(\{\Pi_i\}\) of the parallelepipeds \(\{V_i\}\). Let \(\xi_c\) \((\xi_p)\) be the partition of \(M\) into intervals \(C_{\xi_c}\) \((C_{\xi_p})\), belonging to \(\{\Pi_i\cap \Gamma_c\}\) \((\{\Pi_i\cap C_p\})\). The intervals \(C_{\xi_p}\) are not, generally speaking, intervals of leaves \(\Gamma_p\). However, through each point \(y\in C_{\xi}\) one can draw an interval \(\gamma C_{\xi_p}(y)\) of the leaf \(\Gamma_p(y)\) such that there exists a continuous function \(q(z)\), for \(z\in \gamma C_{\xi_p}(y)\), having the properties: 1) \(q(\gamma C_{\xi_p}(y)\cap C'_{\xi_p})=0\); (2) \(T^{q(z)}z=C_{\xi_p}\); 3) the mapping \(\psi:z\to T^{q(z)}z\) is a homeomorphism of \(\gamma C_{\xi_p}(y)\) onto \(C_{\xi_p}\).

For \(x_0\in \Pi_i\), consider \(C_{\xi_c}(x_0)\) and \(C_{\xi_p}(x_0)\). Let \(u_1\) \((u_2)\) be a smooth coordinate on the leaf \(\Gamma_c(x_0)\) \((\Gamma_p(x_0))\) such that \(u_1(x_0)=0\) \((u_2(x_0)=0)\). Consider on \(C_{\xi_c}(x_0)\) the coordinate \(u_1\), and on \(C_{\xi_p}(x_0)\) the coordinate \(u_2\), transferred from \(\gamma C_{\xi_p}(x_0)\) by means of the mapping \(\psi\). For every point \(w\in \Pi\) there are points \(z_1\in C_{\xi_c}(x_0)\) with coordinate \(u_1\) and \(z_2\in C_{\xi_p}(x_0)\) with coordinate \(u_2\) such that
\(w=C_{\xi_c}(z_2)\cap C_{\xi_p}(z_1)\). Assign to \(w\) the coordinates \((u_1,u_2)\). Then \(\Pi_i\) becomes a smooth manifold with boundary, and \(M\) will consist of a finite number of such manifolds, generally not connected with one another. Consider on \(\Pi_i\) the mapping \(\chi:w\to (u_1,u_2)\). It is a homeomorphism of \(\Pi_i\) onto a rectangle \(P_i\) of the Euclidean plane \((u_1,u_2)\). We introduce on \(\Pi_i\) Riemannian lengths transferred from \(P_i\) by means of the mapping \(\chi^{-1}\).

From the absolute continuity of the foliations \(G_p\) and \(G_c\) (see (2)) it follows that the Riemannian length on \(C_{\xi_c}\) in \(\Pi_i\) is equivalent to the Riemannian length on \(C_{\xi_c}\) induced by the Riemannian metric in \(W\), while the Riemannian length on \(C_{\xi_p}\) is equivalent to the Riemannian length on \(C_{\xi_p}\), transferred from \(\gamma C_{\xi_p}(x)\), \(x\in C_{\xi_p}\), by means of the mapping \(\psi\).

Through \(x\in M\) draw an interval of length \(l(x)\) of the trajectory of the flow \(\{T^t\}\) in the positive direction up to the first intersection with \(M\) at the point \(x'\). The mapping \(U:x\to x'\) is a one-to-one mapping of \(M\) onto itself, and the flow \(\{T^t\}\) is represented as a special flow (see (3)) over \((M,U)\) with function \(l(x)\). The mapping \(U\) on \(M\) is similar in its properties to a \(Y\)-diffeomorphism: it uniformly exponentially contracts (expands) the intervals \(C_{\xi_c}(C_{\xi_p})\). The mapping \(U\) cannot be called a \(Y\)-diffeomorphism, since \(M\) is not a connected closed manifold and \(U\) has, generally speaking, discontinuity points.

The partition \(\{\Pi_i\}\) is Markov for \(U\), and the partition \(\xi_c\) is increasing. The function \(l(x)\) is constant on \(C_{\xi_c}\). Using the Hölder condition for the foliation \(\Gamma_c\) (see (4)), one can prove that \(l(y)\), \(y\in C_{\xi_p}\), satisfies a Hölder condition of positive order on each component of its continuity.

Using (4) and the absolute continuity of the foliation \(G_p\), by the methods of the work [1] one can prove the following theorem.

Theorem 2. In the space \(M\) there exists a measure \(m\), positive on every open set, invariant with respect to \(U\), and \(U\), as an automorphism of the space \(M\) with measure \(m\), is a \(K\)-automorphism with \(K\)-partition \(\xi_c\). The conditional measure \(m(\cdot\mid C_{\xi_c})\), induced by \(m\) on almost every \(C_{\xi_c}\), is equivalent to the normalized Riemannian volume on \(C_{\xi_c}\). These properties determine the measure \(m\) uniquely.

3. Invariant measure.

Using the special representation of the flow \(\{T^t\}\), consider in \(W\) the normalized measure

\[ d\mu=\frac{1}{\bar l}(dm\times dt). \]

\[ \bar l=\int_M l(x)\,dm \]
and the partition \(\eta_c\), consisting of the segments
\[ \{T^u C_{\xi_c}(x),\ 0\le u<l(x),\ x\in M\}. \]
From the invariance of the measure \(m\) with respect to \(U\), the invariance of the measure \(\mu\) with respect to the flow \(\{T^t\}\) follows easily. The partition \(\eta_c\) has the following properties: 1) \(T^{t_1}\eta_c<T^{t_2}\eta_c\) for \(t_1<t_2\);

2)
\[ \prod_{t=-\infty}^{\infty} T^t\eta_c=\varepsilon, \]
where \(\varepsilon\) is the partition into individual points; 3)
\[ \bigcap_{t=-\infty}^{\infty} T^t\eta_c=\nu_Z, \]
where \(\nu_Z\) is the measurable envelope of the partition \(Z\) into full layers of the foliation \(\Gamma_c\).

It follows from (5) that if the foliations \(\Gamma_c\) and \(\Gamma_p\) form a nonintegrable pair, then \(\nu_Z=\nu\), where \(\nu\) is the trivial partition whose only element is the whole space.

Thus the following theorem has been proved:

Theorem 3. In the space \(W\) there exists a measure \(\mu\), positive on every open set, invariant with respect to the flow \(\{T^t\}\), and such that the conditional measure \(\mu(\cdot/C_{\eta_c})\), induced by \(\mu\) on almost every \(C_{\eta_c}\), is equivalent to the normalized Riemannian volume on \(C_{\eta_c}\). If the foliations \(\Gamma_c\) and \(\Gamma_p\) form a nonintegrable pair in the sense of (5), then the flow \(\{T^t\}\) in \((W,\mu)\) is a \(K\)-flow with \(K\)-partition \(\eta_c\). The measure \(\mu\) is uniquely determined by these properties.

For the geodesic flow on a compact manifold of negative curvature, the measure \(\mu\) coincides with the invariant Riemannian volume.

In conclusion, the author expresses gratitude to Ya. G. Sinai, under whose supervision the present work was carried out.

Moscow State University
named after M. V. Lomonosov

Received
20 X 1968

REFERENCES

1. Ya. G. Sinai, Functional Analysis and Its Applications, 2, no. 1 (1968).
2. D. V. Anosov, Proceedings of the V. A. Steklov Mathematical Institute, Academy of Sciences of the USSR, 40 (1967).
3. V. A. Rokhlin, Uspekhi Mat. Nauk, 4, no. 2 (1949).
4. D. V. Anosov, Mathematical Notes, 2, no. 5 (1967).
5. Ya. G. Sinai, Izv. Academy of Sciences of the USSR, Ser. Math., 30, no. 1 (1966).