Marina Ratner

How to use this course

Marina Ratner proved that certain flows on spaces built from Lie groups cannot behave wildly. Every orbit of a unipotent flow is dense in a nice algebraic subspace, and every invariant measure is the natural measure on such a subspace. These theorems, completed around 1990, gave Grigory Margulis’s proof of the Oppenheim conjecture its general setting and became the basic tool of homogeneous dynamics. Fields Medals to Elon Lindenstrauss (2010) and Maryam Mirzakhani (2014) went to work that grew from them.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons follow her work in order: geodesic flows (her thesis), Bernoulli theory (the 1970s), rigidity of horocycle flows (1982–83), Raghunathan’s conjectures (1990–91), and their p-adic and S-arithmetic extensions (1993–95). The Alex Eskin course on this site shows how her theorems are used for counting and how their spirit reached the moduli space of flat surfaces.

Timeline

Moscow, 1938–1965

Marina Ratner was born in Moscow on 30 October 1938 into a Jewish family of scientists. Her father, Evsei Ratner, was a plant physiologist and a professor at the K. A. Timiryazev Institute of Plant Physiology in Moscow; her mother was a chemist. In the late 1940s her mother lost her job for corresponding with her own mother in Israel, then regarded by the Soviet state as an enemy.

Ratner became interested in mathematics in the fifth grade, encouraged by a school teacher. In 1956, after Khrushchev’s denunciation of Stalin, Moscow State University briefly admitted Jewish applicants on more equal terms, and she was admitted after an oral entrance examination. She studied mathematics and physics and graduated in 1961. Through Andrei Kolmogorov and his circle she became interested in probability theory.

After graduating she spent four years as an assistant in Kolmogorov’s applied statistics group. She also taught in the boarding school for mathematically gifted pupils that Kolmogorov had founded in Moscow. During this period, before her doctorate, she had a daughter, Anna, from a brief marriage.

Sinai’s student and the visa, 1965–1971

In 1965 Ratner returned to Moscow State University to do research under Yakov Sinai, then building the Moscow school of smooth ergodic theory. Sinai and Dmitri Anosov had recently proved that geodesic flows on compact manifolds of negative curvature are ergodic and mixing in a strong sense, and that the uniformly hyperbolic flows now called Anosov flows share these properties. Grigory Margulis was also a student of Sinai, a fact that Hee Oh later recalled when she called Ratner her mathematical aunt.

Ratner’s thesis, Geodesic flows on unit tangent bundles of compact surfaces of negative curvature, earned the degree of Candidate of Sciences, the Soviet equivalent of a doctorate, in 1969. It treated the fine statistical properties of these flows, a theme she continued for several years.

She then took a post as an assistant at the High Technical Engineering School in Moscow. In 1970 she applied for an exit visa to emigrate to Israel and was dismissed. The visa was granted, and in 1971 she left for Jerusalem. Her parents also moved to Israel in the early 1970s. Her sister Judith, who stayed, lost her job because of the family’s emigration and later spent ten years as a refusenik before receiving permission to leave in 1987.

Jerusalem and Berkeley, 1971–1982

From 1971 Ratner lectured at the Hebrew University of Jerusalem, and in 1974–75 she taught at its Pre-academic School. In these years she proved that Anosov flows with their natural invariant measures are Bernoulli, the strongest form of randomness in ergodic theory (Lesson 05). She corresponded with Rufus Bowen at Berkeley, a leader in the symbolic dynamics of hyperbolic systems.

Bowen invited her to Berkeley, and in 1975 she became an acting assistant professor there. Her appointment caused some controversy in the department. She became a regular faculty member in 1976, associate professor in 1979, and full professor in 1982. She held a Sloan Research Fellowship from 1977 to 1979.

At Berkeley she moved from hyperbolic flows, which have positive entropy and behave like coin-tossing, to horocycle flows, which have zero entropy and are much more orderly. In 1978 and 1979 she settled their place in the Ornstein–Feldman–Katok classification. From 1982 she discovered the rigidity of horocycle flows, the work that would lead, eight years later, to her proof of Raghunathan’s conjectures. She was in her forties; the theorems for which she is best known came in her early fifties.

Geodesic and horocycle flows

For the general reader. On a surface of constant negative curvature there are two natural ways to move a unit vector. The geodesic flow pushes it straight ahead, and nearby paths separate exponentially fast. The horocycle flow slides it sideways along a curve of constant curvature, and nearby paths separate only polynomially. Ratner’s career moved from the first flow to the second, and the slow divergence of the second is the source of all its rigidity.

The homogeneous model

Let \(G=\mathrm{SL}(2,\mathbb R)\) act on the upper half-plane \(\mathbb H\) by Möbius maps. \(\mathrm{PSL}(2,\mathbb R)\) acts simply transitively on the unit tangent bundle \(T^1\mathbb H\), so \(T^1\mathbb H\cong\mathrm{PSL}(2,\mathbb R)\). For a lattice \(\Gamma\), a discrete subgroup with \(\Gamma\backslash G\) of finite volume, the space \(X=\Gamma\backslash G\) is the unit tangent bundle of the hyperbolic orbifold \(\Gamma\backslash\mathbb H\). Define

Right multiplication \(x\mapsto xa_t\) is the geodesic flow at unit speed, and \(x\mapsto xu_s\) is the (stable) horocycle flow. Both preserve the Haar probability measure \(\mu\) on \(X\). The key identity is

So the geodesic flow renormalizes the horocycle flow: \(x\,u_s\,a_t = x\,a_t\,u_{e^{-t}s}\), and horocycles are contracted by the geodesic flow.

These are the \(\mathrm{SL}(2,\mathbb R)\) cases of Ratner’s theorems. The point of the general theorems is that nothing comparable to the hyperbolic dynamics of \(a_t\) survives for unipotent flows: geodesic orbits can have fractal closures, horocycle orbits cannot.

Bernoulli and loosely Bernoulli

For the general reader. In the 1970s Donald Ornstein showed that a large class of chaotic systems are all equivalent to coin-tossing, once you know their entropy. Ratner proved that the hyperbolic flows she had studied in Moscow belong to this class. She then turned to the orderly horocycle flows and showed that they sit in a related but finer classification, with a surprising twist for products.

A measure-preserving flow is Bernoulli if its time-one map is isomorphic to a Bernoulli shift. Ornstein proved (1970) that Bernoulli shifts with equal entropy are isomorphic, and Ornstein and Benjamin Weiss proved (1973) that geodesic flows on surfaces of negative curvature are Bernoulli.

The proof uses Markov partitions to code the flow as a suspension of a subshift of finite type, then verifies Ornstein’s “very weak Bernoulli” criterion using the exponential contraction along stable leaves.

Kakutani equivalence

Two ergodic flows are Kakutani equivalent if one is isomorphic to a measurable time change of the other; equivalently, they have isomorphic cross-sections. Kakutani equivalence preserves whether entropy is zero, positive finite or infinite. Jacob Feldman and, independently, Anatole Katok introduced the loosely Bernoulli systems. A zero-entropy ergodic flow is loosely Bernoulli if it is Kakutani equivalent to an irrational linear flow on the 2-torus; a positive-entropy flow is loosely Bernoulli if it is Kakutani equivalent to a Bernoulli flow. Feldman showed that non-loosely-Bernoulli systems exist, and Ornstein, Daniel Rudolph and Weiss developed the general equivalence theory.

The second result showed for the first time that a loosely Bernoulli flow can have a non-loosely-Bernoulli square. The proof of non-equivalence studies how pairs of orbits in the product drift apart, and isolates the polynomial divergence that later became the Ratner property (Lesson 06).

Rigidity of horocycle flows

For the general reader. Ergodic theory usually forgets geometry: two very different systems can be isomorphic as measure spaces. Ratner showed that horocycle flows remember everything. If two such flows are isomorphic in the loosest measurable sense, the surfaces they live on are isometric, and the isomorphism is a geometric map. This was the first glimpse of the rigidity that she later proved in general.

In particular, the horocycle flows of two compact hyperbolic surfaces are isomorphic only if the surfaces are isometric. Compare Lesson 05: all the geodesic flows of the same entropy are isomorphic Bernoulli flows.

Polynomial divergence

Take two nearby points \(x\) and \(xg\) with \(g\) close to the identity. After time \(t\), \(xgu_t=(xu_t)\,(u_{-t}gu_t)\), so the displacement is \(u_{-t}gu_t\). For \(g=\begin{pmatrix}a&b\\c&d\end{pmatrix}\),

The entries are polynomials in \(t\), and the fastest-growing one, the upper-right entry, is the direction of the flow itself. So nearby orbits drift apart slowly, and mainly along the flow. This is the Ratner property: for most nearby pairs, when the orbits finally separate they do so by a shift along the orbit, and they stay at that relative shift for a long, controlled time. Using it, Ratner showed that an isomorphism must map orbits in a way that respects the transverse geometry, which forces it to be algebraic.

Measure classification

For the general reader. In the 1970s M. S. Raghunathan proposed that every orbit of a unipotent flow on a space \(\Gamma\backslash G\) has a closure that is itself a nice algebraic object. S. G. Dani recorded the conjecture in 1981 and proved important cases. Ratner proved it in complete generality by first classifying the invariant measures, which is harder and more powerful. Her proof is long; its core idea is the slow divergence from Lesson 06, used now in any Lie group.

An element \(u\in G\) is Ad-unipotent if \(\mathrm{Ad}(u)-I\) is nilpotent on the Lie algebra. A measure \(\nu\) on \(\Gamma\backslash G\) is algebraic (or homogeneous) if there is a closed subgroup \(L\) and a point \(x\) such that \(xL\) is closed and \(\nu\) is the \(L\)-invariant probability measure on \(xL\).

The 1990 papers treat solvable \(G\) (in Inventiones) and semisimple \(G\) (in Acta); the 1991 paper completes the general case. Margulis and George Tomanov gave a different proof (1994), and Ratner extended the theorem to p-adic and S-arithmetic groups (1993, 1995).

Strategy

Let \(\nu\) be ergodic for a one-parameter unipotent \(u_t\), and let \(L\) be the largest subgroup that preserves \(\nu\). One shows that \(\nu\) is concentrated on a single \(L\)-orbit.

• Polynomial divergence. If \(x\) and \(xg\) are both \(\nu\)-generic, the displacement \(u_{-t}gu_t\) is polynomial in \(t\). Its fastest-growing part points, after renormalization, along the centralizer of \(u_t\), and in particular in the normalizer direction.
• Shearing. Ratner’s argument shows that if generic points can be found arbitrarily close in a direction transverse to \(L\), the drift of the two orbits produces a new element of \(G\) preserving \(\nu\), contradicting the maximality of \(L\).
• Entropy. Ratner used an entropy computation for an element \(a\) normalizing \(U\): the entropy of \(a\) with respect to \(\nu\) is bounded by the expansion rate of \(a\) on \(L\)-leaves, with equality only for measures that are Haar along those leaves. Combined with the shearing step this forces \(\nu\) to be the Haar measure of a closed \(L\)-orbit.

Orbit closures and equidistribution

For the general reader. Once the invariant measures are known, two further statements follow. Every unipotent orbit is dense in a closed algebraic piece of the space, and it spends time in each part of that piece in proportion to its size. These were Raghunathan’s original conjecture and its quantitative strengthening, and they are what applications use.

From measures to orbits

Take any limit \(\nu\) of the averages \(\frac1T\int_0^T\delta_{xu_t}\,dt\). It is \(u_t\)-invariant. Two facts are needed to identify it.

• Non-divergence (Margulis 1975, Dani 1979–86): unipotent orbits do not escape to infinity; outside a compact set they spend only a small fraction of time. So \(\nu\) is a probability measure.
• Linearization (Dani–Margulis; Shah): the ergodic components of \(\nu\) are algebraic by measure classification. The set of points lying in closed orbits of smaller subgroups is a countable union of “tubes”, and a polynomial estimate shows that a generic orbit spends little time near any tube unless it lies in it. So \(\nu\) is the measure on the smallest closed orbit containing \(x\).

Shah extended equidistribution to polynomial trajectories, and Mozes and Shah showed that limits of algebraic measures for unipotent flows are again algebraic.

Applications: Oppenheim and beyond

For the general reader. In 1929 Alexander Oppenheim conjectured that an irrational quadratic form in three or more variables takes values arbitrarily close to zero at integer points. Margulis proved it in 1986 by studying one orbit of an orthogonal group acting on the space of lattices. Ratner’s theorems contain his result as a special case and made such arguments routine across number theory.

Proof via Ratner, for \(n=3\)

Write \(Q=Q_0\circ g\) with \(Q_0=2x_1x_3-x_2^2\) and \(g\in\mathrm{SL}(3,\mathbb R)\), after scaling. Let \(H=\mathrm{SO}(Q_0)^\circ\cong\mathrm{PSL}(2,\mathbb R)\), which is generated by unipotent subgroups. Consider the orbit of the lattice \(g\mathbb Z^3\) under \(H\) in \(X=\mathrm{SL}(3,\mathbb R)/\mathrm{SL}(3,\mathbb Z)\). By Ratner, \(\overline{Hg\mathbb Z^3}=Fg\mathbb Z^3\) for a closed connected subgroup \(F\supseteq H\). Since \(\mathfrak{so}(Q_0)\) is a maximal subalgebra of \(\mathfrak{sl}_3\) (Exercise 9.1), \(F=H\) or \(F=\mathrm{SL}(3,\mathbb R)\).

• If \(F=H\), the orbit \(Hg\mathbb Z^3\) is closed with finite volume, so \(g^{-1}Hg\cap\mathrm{SL}(3,\mathbb Z)\) is a lattice in \(\mathrm{SO}(Q)\). By Borel density this group is Zariski dense in \(\mathrm{SO}(Q)\), and a form preserved by a Zariski-dense group of integer matrices is a multiple of a rational form. This contradicts the hypothesis.
• So \(F=\mathrm{SL}(3,\mathbb R)\) and the orbit is dense. Given \(c\in\mathbb R\), choose a vector \(w\) with \(Q_0(w)=c\); density lets one find \(h\in H\) and \(\gamma\) with \(hg\gamma e_1\) close to \(w\). Then \(Q(\gamma e_1)=Q_0(hg\gamma e_1)\approx c\).

Margulis’s original proof avoided the full measure classification by a direct topological argument. Dani and Margulis then proved that the values at primitive vectors are dense, and Eskin, Margulis and Shahar Mozes made the result quantitative (see the Eskin course, Lesson 06).

Further applications

• Counting integer points on homogeneous varieties (Eskin–Mozes–Shah, 1996).
• Equidistribution of Hecke points and of closed geodesics, and the arithmetic quantum unique ergodicity of Lindenstrauss, which uses an entropy argument for diagonal actions descended from Ratner’s ideas.
• The Eskin–Mirzakhani classification of \(\mathrm{SL}(2,\mathbb R)\)-invariant measures on moduli spaces of translation surfaces, an analogue of Ratner’s theorem in a non-homogeneous space.

p-adic and S-arithmetic versions

For the general reader. Number theorists often work with several kinds of distance at once: the ordinary one, and for each prime \(p\) a “p-adic” distance in which numbers divisible by high powers of \(p\) are small. After her real theorems, Ratner proved that the same rigidity holds for groups built from p-adic numbers and for products mixing real and p-adic factors. These versions are what arithmetic applications most often need.

S-arithmetic lattices

Let \(S\) be a finite set of places of \(\mathbb Q\) containing \(\infty\), and \(\mathbb Z_S\) the ring of rationals whose denominators involve only primes in \(S\). Diagonal embedding makes \(\mathbb Z_S\) a discrete cocompact subring of \(\mathbb Q_S=\prod_{v\in S}\mathbb Q_v\) (Exercise 10.1). By the theorem of Armand Borel and Harish-Chandra, extended to the S-arithmetic case by Borel, for a semisimple \(\mathbb Q\)-group \(\mathbf G\) the group \(\mathbf G(\mathbb Z_S)\) is a lattice in \(G_S=\prod_{v\in S}\mathbf G(\mathbb Q_v)\). For example, \(\mathrm{SL}(2,\mathbb Z[1/p])\) is a lattice in \(\mathrm{SL}(2,\mathbb R)\times\mathrm{SL}(2,\mathbb Q_p)\).

Ratner announced the purely p-adic case in 1993 and treated products of real and p-adic groups in a long paper in 1995. Margulis and Tomanov proved the measure classification over local fields of characteristic zero by a different route in 1994, using entropy and the algebraic structure of unipotent groups. The S-arithmetic theorems underlie, for example, Oppenheim-type results for values of quadratic forms at S-integral points and equidistribution of Hecke orbits.

What changes p-adically

Polynomial divergence still holds, since the displacement \(u_{-t}gu_t\) is a polynomial in \(t\) whatever the field. What changes is the topology: \(\mathbb Q_p\) is totally disconnected, so there are no connected subgroups to work with, and a single unipotent element generates a relatively compact group (Exercise 10.2). The theorem therefore has to be stated for one-parameter groups indexed by \(t\in\mathbb Q_p\), and connectedness arguments in the real proof are replaced by arguments with open compact subgroups.

Recognition, teaching, convictions

Recognition came quickly once the Raghunathan conjectures fell. Ratner was elected to the American Academy of Arts and Sciences in 1992 and to the National Academy of Sciences in 1993. She received the Ostrowski Prize for 1993, presented in Basel in July 1994, and in April 1994 the NAS John J. Carty Award for the Advancement of Science, for her work on unipotent flows. At the International Congress of Mathematicians in Zürich in 1994 she gave a plenary lecture, Interactions between ergodic theory, Lie groups and number theory. She was the third woman to give a plenary address at the Congress, after Emmy Noether in 1932 and Karen Uhlenbeck in 1990.

She was too old for the Fields Medal when her main theorems appeared. Several mathematicians have said she would have been a strong candidate had her work been completed before forty.

Ratner held firm views. In 1986 she boycotted the Congress held in her own city, Berkeley, protesting its de facto exclusion of women from the plenary programme. Yet she rejected the idea that women mathematicians needed special encouragement. In an email to Hee Oh she wrote that a woman who is good at mathematics needs neither encouragement nor a role model. She described mathematics as part of the beauty of nature.

She was a demanding and much admired teacher. She won Berkeley’s Noyce Prize for Excellence in Undergraduate Teaching in 2004. After her formal retirement in 2011 she continued to teach undergraduate courses until the spring of 2017.

Death and legacy

In June 2013 the Hebrew University of Jerusalem, where she had begun her career in exile, awarded her an honorary doctorate and held a conference in her honour, Homogeneous Dynamics, Unipotent Flows, and Applications. Marina Ratner died at her home in El Cerrito, California, on 7 July 2017, aged 78. Her death came a week before that of Maryam Mirzakhani, whose work had extended Ratner’s rigidity to moduli space.

Legacy

• Homogeneous dynamics: her measure classification, orbit closure and equidistribution theorems are the foundation of the subject, presented in textbooks such as Dave Witte Morris’s Ratner’s Theorems on Unipotent Flows (2005).
• Number theory: values of quadratic forms, counting integer points on varieties, equidistribution of Hecke points, and Diophantine approximation on manifolds all rely on her theorems.
• Beyond Lie groups: rigidity theorems for diagonal actions (Einsiedler, Katok, Lindenstrauss), for stationary measures (Benoist and Quint), and for moduli spaces of translation surfaces (Eskin, Mirzakhani, Mohammadi) all adapt her methods.
• Ergodic theory: the Ratner property of slow divergence is a standard tool for proving disjointness and rigidity of zero-entropy flows.

Sources and further reading

Biography

1. J. J. O’Connor, E. F. Robertson, Marina Ratner, MacTutor History of Mathematics.
2. A. Weinstein, Marina Ratner, In Memoriam, University of California, 2019.
3. S. G. Dani, Marina Ratner (1938–2017), Bhāvanā 1(4), 2017.
4. H. Oh, Remembering Marina Ratner, Notices of the AMS, March 2019.
5. Marina Ratner, Wikipedia.
6. Agnes Scott College, Biographies of Women Mathematicians: Marina Ratner.
7. IMU Committee for Women in Mathematics, Marina Ratner passes away, 2017.

Mathematics

1. M. Ratner, Anosov flows with Gibbs measures are also Bernoullian, Israel J. Math. 17 (1974).
2. M. Ratner, Horocycle flows are loosely Bernoulli, Israel J. Math. 31 (1978).
3. M. Ratner, The Cartesian square of the horocycle flow is not loosely Bernoulli, Israel J. Math. 34 (1979).
4. M. Ratner, Rigidity of horocycle flows, Ann. of Math. 115 (1982).
5. M. Ratner, Factors of horocycle flows, Ergodic Theory Dynam. Systems 2 (1982).
6. M. Ratner, Horocycle flows, joinings and rigidity of products, Ann. of Math. 118 (1983).
7. M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990).
8. M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990).
9. M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. 134 (1991).
10. M. Ratner, Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991).
11. M. Ratner, Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups, Duke Math. J. 77 (1995).
12. M. Ratner, Interactions between ergodic theory, Lie groups, and number theory, Proc. ICM Zürich 1994, Birkhäuser, 1995.
13. G. A. Margulis, G. M. Tomanov, Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116 (1994).
14. D. W. Morris, Ratner’s Theorems on Unipotent Flows, University of Chicago Press, 2005; arXiv:math/0310402.
15. Ratner’s theorems, Wikipedia.
16. M. Einsiedler, What is… measure rigidity?, Notices of the AMS 56 (2009).