Yakov Sinai

How to use this course

Yakov Sinai made deterministic chaos a subject of precise mathematics. His first major result, the entropy of a dynamical system, gave mathematicians a number that measures how unpredictable a system is. His billiards showed that the simplest mechanical models of a gas behave randomly. Later he carried the same probabilistic intuition into statistical physics, random media and fluid dynamics. The 2014 Abel Prize honoured him for fundamental contributions to dynamical systems, ergodic theory and mathematical physics.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The order follows Sinai’s work: entropy (1959), billiards (1963–1970), Markov partitions and Gibbs measures (1968–1972), lattice phase diagrams (1975–1976), random walks in random environment (1982), and stochastic Burgers turbulence (2000). Sinai is living; the biographical lessons keep to his career and public record.

Timeline

Moscow, 1935–1952

Yakov Grigorevich Sinai was born in Moscow on 21 September 1935. Both his parents, Grigory Sinai and Nadezda Kagan, were microbiologists. His grandfather Veniamin Fedorovich Kagan was a geometer who headed the Department of Differential Geometry at Moscow State University until 1952, and who was known for his work on the foundations of geometry and on tensor analysis. The mathematician and mechanician Grigory Barenblatt, later an authority on similarity methods and turbulence, was his half-brother.

The family spent part of the Second World War in evacuation. Sinai began school in Moscow in 1943, after their return. He grew up in a household of working scientists.

Admission

In 1952 he applied to the Faculty of Mechanics and Mathematics of Moscow State University, known as Mekhmat. It was the last year of Stalin’s rule, and Jewish applicants faced open discrimination. According to the MacTutor biography, Sinai was initially failed in the entrance examination, and the decision was reversed after his grandfather appealed to the rector, Ivan Petrovsky. He entered Mekhmat that autumn.

Mekhmat and Kolmogorov, 1952–1960

Mekhmat in the 1950s was one of the strongest mathematics departments in the world. Sinai heard analysis from Mikhail Lavrentiev, classical mechanics from Nikolai Chetaev and algebra from Eugene Dynkin. He first worked with Chetaev and then with Dynkin, who ran a seminar for students in probability. A problem posed by Dynkin led to his first paper, on the distribution of the first positive sum of a sequence of independent random variables, published in 1957, the year he graduated.

The decisive step was entering the circle of Andrei Kolmogorov. In 1954 Kolmogorov had announced the persistence of quasi-periodic motion in perturbed Hamiltonian systems, a result later completed by Vladimir Arnold and Jürgen Moser (KAM theory). He now turned to the opposite question: how to measure the randomness of systems that are not quasi-periodic. Sinai became his student and, within two years, the co-author of one of the central concepts of the field.

In 1960 he joined the Laboratory of Probabilistic and Statistical Methods at Moscow State University as a researcher, a post he held until 1971. Sources date his doctoral degree under Kolmogorov to 1960 (Wikipedia) or 1963 (the Abel Prize biography and Princeton); the difference probably reflects the two levels of the Soviet degree system.

The entropy years, 1958–1963

In 1958 Kolmogorov introduced entropy as an invariant of measure-preserving transformations, adapting Claude Shannon’s information theory. His definition applied to a restricted class of systems and was hard to compute with. In 1959 Sinai gave the definition that is now standard: the supremum of the entropy rate over all finite partitions. He also proved that a single generating partition computes it. The resulting invariant is called the Kolmogorov–Sinai entropy, or metric entropy. Lesson 04 develops it.

The new invariant answered a question that had been open since the 1930s. Bernoulli shifts with different probability vectors have the same spectral invariants, so the methods of John von Neumann and Paul Halmos could not tell them apart. Entropy could: the shifts with probabilities \((\tfrac12,\tfrac12)\) and \((\tfrac13,\tfrac13,\tfrac13)\) have entropies \(\log 2\) and \(\log 3\), so they are not isomorphic.

Sinai’s interests moved quickly from abstract classification to physics. In 1962 he gave an invited address at the International Congress of Mathematicians in Stockholm. In 1963 he announced a proof that a system of hard spheres in a box is ergodic, a rigorous form of Ludwig Boltzmann’s ergodic hypothesis. The announcement set the programme for the next decade of his work and for half a century of work by others.

Kolmogorov–Sinai entropy

For the general reader. Watch a system through a coarse lens that reports only which of a few boxes it is in. If the system is regular, after a while each new observation tells you almost nothing new. If it is chaotic, each observation carries fresh information. Entropy measures that average amount of new information per step, maximized over all possible lenses.

Definitions

Let \((X,\mathcal B,\mu)\) be a probability space and \(T:X\to X\) measurable with \(\mu(T^{-1}A)=\mu(A)\) for all \(A\in\mathcal B\). For a finite measurable partition \(\xi=\{C_1,\dots,C_k\}\), set

where \(\xi\vee\eta\) is the common refinement \(\{C\cap D\}\). The sequence \(a_n=H(\xi^{(n)})\) is subadditive, \(a_{m+n}\le a_m+a_n\), so the limit below exists and equals the infimum of \(a_n/n\).

Isomorphic systems have equal entropy, because an isomorphism carries partitions to partitions. The supremum is hard to compute directly. Sinai’s theorem reduces it to a single partition.

Proof sketch

For any partition \(\eta\), one has \(h(T,\eta)\le h(T,\zeta)+H(\eta\mid\zeta)\), where \(H(\eta\mid\zeta)\) is conditional entropy. Apply this with \(\zeta=\bigvee_{|k|\le m}T^k\xi\). Its entropy rate equals \(h(T,\xi)\), since shifting finitely many times does not change the rate. Because \(\xi\) generates, the partitions \(\zeta_m\) approximate every set in \(\mathcal B\), and the martingale convergence theorem gives \(H(\eta\mid\zeta_m)\to0\). Hence \(h(T,\eta)\le h(T,\xi)\) for every \(\eta\).

Bernoulli shifts

Let \(X=\{1,\dots,k\}^{\mathbb Z}\) with product measure \(\mu=p^{\otimes\mathbb Z}\) and the left shift \(\sigma\). The partition by the zeroth coordinate generates, and \(\xi^{(n)}\) consists of cylinders of length \(n\) with independent coordinates, so

In 1970 Donald Ornstein proved the converse question for this class: two Bernoulli shifts with equal entropy are isomorphic. Entropy is a complete invariant for Bernoulli systems. In the early 1960s Sinai had proved a weaker statement in the same direction: every ergodic system of positive entropy has Bernoulli factors of all smaller or equal entropies.

Dispersing billiards and hard spheres

For the general reader. A ball rolls without friction on a square table whose opposite edges are glued together, so it never meets a wall. In the middle stands a round post. Two balls launched almost identically separate after hitting the post, and they separate more after every bounce. Sinai proved that this simple system is as random as a coin-tossing sequence, in a precise sense. It became the model for the chaotic motion of gas molecules.

The setting

Let \(Q=\mathbb T^2\setminus\bigcup_{i}B_i\), where the \(B_i\) are finitely many disjoint closed regions with smooth boundaries of strictly positive curvature. A point moves in \(Q\) with unit speed in straight lines and reflects off \(\partial Q\) with angle of incidence equal to angle of reflection. The billiard flow \(\Phi^t\) acts on \(Q\times S^1\) and preserves the Liouville measure \(dq\,d\theta\). The billiard map acts on the collision space of pairs \((r,\varphi)\), where \(r\) is arclength on \(\partial Q\) and \(\varphi\in[-\tfrac\pi2,\tfrac\pi2]\) is the angle with the normal. It preserves \(d\nu=(2|\partial Q|)^{-1}\cos\varphi\,dr\,d\varphi\).

Giovanni Gallavotti and Donald Ornstein showed in 1974 that the flow is in fact Bernoulli. The billiard is the first mechanical system of physical origin proved to be ergodic.

The mechanism

Follow a one-parameter family of nearby trajectories, a wavefront, and let \(B\) be the curvature of the front, positive when the front is spreading out. During free flight of time \(t\) the radius of curvature grows linearly, so

At a reflection off a boundary of curvature \(\kappa>0\), at angle \(\varphi\) to the normal, the mirror formula gives

Every collision makes a front more dispersing, and a dispersing front stays dispersing in flight. The length of a front grows by the factor \(1+\tau B\) over a free flight of length \(\tau\), so infinitesimal separations grow exponentially. This gives hyperbolicity with positive Lyapunov exponent. Unlike an Anosov system, a billiard has singularities: trajectories tangent to a scatterer, where the map is not continuous. The difficult part of Sinai’s proof controls these singularities. He adapted the argument that Eberhard Hopf had used for geodesic flows: local stable and unstable curves exist almost everywhere, are long enough, and cross each other, so time averages are constant on large sets.

Hard spheres

Boltzmann’s ergodic hypothesis states that a gas wanders through its whole energy surface. In 1963 Sinai announced the ergodicity of \(N\) hard balls in a box or torus. The case of two discs on a torus reduces to his billiard (Exercise 5.2). For more balls the configuration space is a dispersing billiard in high dimension with only semi-dispersing walls, cylinders instead of strictly convex bodies, and the proof is far harder. Sinai and Nikolai Chernov treated systems of two discs and three-dimensional balls in 1987. Domokos Szász, Nándor Simányi and others carried the programme to large classes of cases over the following decades. His student Leonid Bunimovich discovered in the 1970s that focusing boundaries can also give chaos, as in the stadium billiard.

Markov partitions and symbolic dynamics

For the general reader. A chaotic map of a surface can be described by a list of symbols that records which region the orbit visits at each step. Sinai showed in 1968 that for an important class of chaotic maps the regions can be chosen so cleverly that the allowed symbol sequences obey a simple rule: which symbol may follow which. Questions about the map become questions about these sequences, which are much easier to handle.

Anosov diffeomorphisms

A diffeomorphism \(f\) of a compact Riemannian manifold \(M\) is Anosov if \(TM=E^s\oplus E^u\) splits continuously and invariantly, with constants \(C>0\), \(0<\lambda<1\) such that \(\|Df^n v\|\le C\lambda^n\|v\|\) for \(v\in E^s\), \(n\ge0\), and \(\|Df^{-n}v\|\le C\lambda^n\|v\|\) for \(v\in E^u\). Dmitri Anosov proved in the 1960s that such maps, and geodesic flows on manifolds of negative curvature, are structurally stable and, with a smooth invariant volume, ergodic. The standard example is the cat map \(f_A(x)=Ax \bmod \mathbb Z^2\) with \(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\), whose eigenvalues are \(\lambda_\pm=(3\pm\sqrt5)/2\).

Rectangles and the Markov property

A rectangle is a closed set \(R\), the closure of its interior, such that for \(x,y\in R\) the local stable manifold of \(x\) and the local unstable manifold of \(y\) meet in a single point of \(R\). A finite cover \(\{R_1,\dots,R_m\}\) by rectangles with disjoint interiors is a Markov partition if, whenever \(x\in\operatorname{int}R_i\) and \(f(x)\in\operatorname{int}R_j\), the image \(f(W^u_{R_i}(x))\) contains \(W^u_{R_j}(f(x))\) and \(f(W^s_{R_i}(x))\subseteq W^s_{R_j}(f(x))\). Unstable sides stretch across, and stable sides fit inside.

Roy Adler and Benjamin Weiss had constructed such partitions for hyperbolic automorphisms of the two-torus in 1967. Rufus Bowen extended Sinai’s construction to the basic sets of Axiom A diffeomorphisms in 1970, with a simpler proof based on shadowing of pseudo-orbits. The coding is one-to-one off the images of the boundaries, a set of zero measure for the measures that matter (Lesson 07).

Why it helps

Through \(\pi\), invariant measures of \(f\) correspond to shift-invariant measures on \(\Sigma_T\), which can be built like Markov chains. Periodic points are counted by traces, \(\#\{\underline x:\sigma^n\underline x=\underline x\}=\operatorname{tr}T^n\), and the topological entropy of \(f\) equals \(\log\rho(T)\), where \(\rho(T)\) is the spectral radius of \(T\).

Gibbs measures and SRB measures

For the general reader. If you start a chaotic system at a random point and record long-run averages, which probability distribution do you see? For a volume-preserving system it is volume. For a dissipative system, whose orbits collapse onto a thin attractor, volume is not invariant. Sinai showed in 1972 that the answer is still a single natural measure, and he found it by borrowing the Gibbs distributions of statistical mechanics. David Ruelle and Rufus Bowen extended the theory, and the measures carry all three names.

Thermodynamic formalism

Let \(\sigma\) be a transitive subshift of finite type and \(\phi\) a Hölder continuous function on it. The pressure is

Sinai’s insight was that a Markov partition turns a hyperbolic map into a one-dimensional lattice spin system, with the symbol sequence as a configuration and \(-\phi\) as an energy. Gibbs states of one-dimensional systems with rapidly decaying interactions are unique, with no phase transitions, and this uniqueness is what makes the natural measure unique.

SRB measures

Ruelle proved the analogous statement for Axiom A attractors, and Bowen and Ruelle treated flows. The entropy identity is a case of the Pesin formula, which Yakov Pesin proved for smooth invariant measures of non-uniformly hyperbolic systems. Ledrappier and Young later showed that the formula characterizes SRB measures in general. SRB measures are the reason physicists can speak of the statistics of a dissipative chaotic system, such as the Lorenz attractor, at all.

Phase transitions: Pirogov–Sinai theory

For the general reader. Water can be ice, liquid or vapour, and at special values of temperature and pressure two or three phases coexist. Mathematical physicists study such phase diagrams in lattice models, where each site of a grid carries a spin or a particle. In 1975 and 1976 Sergei Pirogov and Sinai gave a general method to compute the phase diagram at low temperature, even when no symmetry fixes it, and it became a standard tool.

Gibbs states and the Peierls argument

In the Ising model on \(\mathbb Z^2\), each site carries \(\sigma_x=\pm1\) and the energy in a finite box \(\Lambda\) with boundary condition \(\bar\sigma\) is \(H_\Lambda(\sigma)=-\sum_{\langle xy\rangle}\sigma_x\sigma_y-h\sum_x\sigma_x\), the first sum over nearest-neighbour pairs meeting \(\Lambda\). The finite-volume Gibbs measure is proportional to \(e^{-\beta H_\Lambda}\). A phase transition at \(h=0\) means that the limits with all-plus and all-minus boundary conditions differ.

With plus boundary conditions, draw the dual edges separating unlike neighbours. They form closed curves, the contours. A contour of length \(\ell\) costs energy \(2\ell\). Erasing it, by flipping the spins inside, gives an injective map on configurations, so the probability that a given contour is present is at most \(e^{-2\beta\ell}\). This is Rudolf Peierls’s argument of 1936, made rigorous by Robert Griffiths and Roland Dobrushin in 1964 and 1965.

The key idea is a contour expansion for each phase, in which contours whose interiors favour a different phase are given a modified weight. This makes the expansion converge in every phase at once. Earlier, Robert Minlos and Sinai had analysed phase separation at low temperature (1967), and Pavel Bleher and Sinai had given one of the first rigorous treatments of the renormalization group at a critical point, for Dyson’s hierarchical model (1973).

Sinai’s random walk

For the general reader. An ordinary random walk spreads out like the square root of time. Now let the coin used at each site be biased, with biases chosen at random once and for all, fair on average. Sinai proved in 1982 that the walker then becomes trapped in deep valleys of a random landscape and spreads only like the square of the logarithm of time: after a million steps it has moved about two hundred sites rather than a thousand.

The model

Let \(\omega=(p_x)_{x\in\mathbb Z}\) be i.i.d. random variables in \((0,1)\). Given \(\omega\), the walk \(X_n\) starts at 0 and moves from \(x\) to \(x+1\) with probability \(p_x\) and to \(x-1\) with probability \(1-p_x\). Set \(\rho_x=(1-p_x)/p_x\) and define the potential

The walk is a reversible Markov chain with invariant measure proportional to \(e^{-V(x)}(1+\rho_x)\), so it lives near the minima of \(V\). Fred Solomon showed in 1975 that the walk is recurrent for almost every \(\omega\) if and only if \(\mathbb E\log\rho_0=0\).

Harry Kesten and Andrei Golosov independently computed the law of \(b_\infty\) in 1986. It has density

The valley heuristic

To climb a potential barrier of height \(H\) the walk needs time of order \(e^{H}\), as in Arrhenius’s law. By time \(n\) it can cross barriers of height up to about \(\log n\), but not more. By Donsker’s theorem, \(V\) on scale \(L\) looks like \(\sigma\) times a Brownian motion, so barriers of height \(\log n\) occur at distances of order \((\log n)^2/\sigma^2\). At time \(n\) the walk sits at the bottom of the smallest valley around the origin that has depth at least \(\log n\); \(b_n\) is the rescaled position of that bottom. The proof makes this precise. Golosov later showed a sharper localization: the walk stays within a bounded distance of a point determined by the environment, with high probability.

Burgers turbulence

For the general reader. The Burgers equation is the simplest model of a fluid that forms shocks, sudden jumps in velocity. Driven by random forces, it gives a one-dimensional caricature of turbulence. In 2000 Weinan E, Konstantin Khanin, Alexander Mazel and Sinai showed that the long-run statistics are unique: whatever the starting state, the solution forgets it, and at each moment it is determined by the past history of the forcing alone.

The equation

On the circle \(\mathbb T=\mathbb R/\mathbb Z\) consider

where the \(F_k=G_k'\) are smooth with mean zero and the \(B_k\) are independent standard Brownian motions. The mean \(\int_{\mathbb T}u\,dx=c\) is conserved. In the inviscid limit \(\varepsilon\to0\) the solutions are the entropy solutions, and for \(u=\partial_x\psi\) they satisfy the Lax–Oleinik variational formula

with the stochastic integral understood after integration by parts. Fluid particles follow minimizing curves; shocks are points reached by two different minimizers.

The proof brings Aubry–Mather theory of minimizing orbits into a random setting. In earlier work (1997) the same authors used minimizers and shocks to derive the tails of the velocity-gradient distribution. Khanin, his former student, has continued the programme in higher dimensions with Rufus Iturriaga and others.

Landau Institute and Princeton

In 1968 Sinai was among the mathematicians who signed a letter protesting the forced psychiatric confinement of the logician Alexander Esenin-Volpin. Signatories faced professional penalties. Sinai was invited to speak at the 1970 International Congress in Nice and was not allowed to travel. The MacTutor biography links the long delay in his promotion at Moscow State University to the letter.

In 1971 he became a senior researcher at the L. D. Landau Institute for Theoretical Physics of the USSR Academy of Sciences, where he worked alongside theoretical physicists. The setting suited him. His work of the 1970s and 1980s is mathematical physics in the strict sense: Gibbs states, phase diagrams, the renormalization group, random Schrödinger operators. In the early 1980s, with Isaak Khalatnikov, Evgeny Lifshitz, Konstantin Khanin and Lev Shchur, he studied the chaotic oscillations of the early universe near the cosmological singularity, where the dynamics reduces to a continued-fraction map. He also held a professorship at Moscow State University, and his seminar there trained a large school.

The 1990 Congress in Kyoto heard his plenary lecture on hyperbolic billiards. He was elected to the USSR Academy of Sciences in 1991. In 1993 he became professor of mathematics at Princeton University, while keeping his position at the Landau Institute; he was Thomas Jones Professor in 1997–98 and Moore Distinguished Scholar at Caltech in 2005. At Princeton he turned increasingly to fluid dynamics, including, with Dong Li, a renormalization-group construction of finite-time blow-up for complex-valued solutions of the three-dimensional Navier–Stokes equations (2008). His wife, the mathematical physicist Elena Vul, has been a co-author on several papers. He chaired the Fields Medal Committee in 2002 and became professor emeritus in 2023.

Honours, school and influence

On 26 March 2014 the Norwegian Academy of Science and Letters announced that Sinai would receive the Abel Prize for his fundamental contributions to dynamical systems, ergodic theory and mathematical physics. Crown Prince Haakon presented it in Oslo on 20 May. The citation named entropy, billiards, Markov partitions, SRB measures, Sinai’s walk, Pirogov–Sinai theory, the Bleher–Sinai renormalization group, Burgers turbulence and the spectral theory of discrete Schrödinger operators. He was the first Princeton faculty member to receive the prize.

Students

Sinai has supervised more than fifty doctoral students. They include Grigory Margulis, Marina Ratner, Anatole Katok, Valery Oseledets, Leonid Bunimovich, Nikolai Chernov, Konstantin Khanin, Dmitry Dolgopyat, Svetlana Jitomirskaya, Leonid Polterovich and Corinna Ulcigrai. Among them are a Fields Medallist and Abel laureate (Margulis), the author of the rigidity theorems for unipotent flows (Ratner), the author of the multiplicative ergodic theorem (Oseledets) and the discoverer of the stadium billiard (Bunimovich). Few twentieth-century schools have shaped a field so completely.

Honours

• Prizes: Boltzmann Medal (1986); Dannie Heineman Prize for Mathematical Physics (1990); Markov Prize of the USSR Academy (1990); Dirac Medal of the ICTP (1992); Wolf Prize in Mathematics (1997); Moser Prize of SIAM (2001); Nemmers Prize (2002); Lagrange Prize (2008); Henri Poincaré Prize and Dobrushin Prize (2009); Leroy P. Steele Prize for Lifetime Achievement (2013); Abel Prize (2014); Marcel Grossmann Award (2015).
• Academies: American Academy of Arts and Sciences (1983); USSR, later Russian, Academy of Sciences (1991); Hungarian Academy of Sciences (1993); US National Academy of Sciences (1999); Brazilian Academy of Sciences (2000); Academia Europaea (2008); Foreign Member of the Royal Society (2009); Norwegian Academy of Science and Letters (2014).
• Honorary degrees: Warsaw (1993), Budapest University of Technology and Economics (2002), Hebrew University of Jerusalem (2005), Warwick (2010).

Influence

• Ergodic theory: entropy is the central invariant of the field, from Ornstein’s isomorphism theory to the measure rigidity results of Margulis, Ratner and their successors.
• Hyperbolic dynamics: Markov partitions and SRB measures are the standard toolkit for chaotic systems, and dispersing billiards are the laboratory for rigorous statistical mechanics, including the Lorentz gas and diffusion.
• Physics: Pirogov–Sinai theory underlies the rigorous study of low-temperature phases; Sinai’s walk is the basic model of anomalous diffusion in disordered media.

Connections

People linked to Yakov Sinai across this collection. A name without a link has no page here yet. See Yakov Sinai in the network of all 337 lives →

Teachers and advisors

• Andrey Kolmogorov 1957-1963

Students

• Grigory Margulis
• Marina Ratner
• Anatole Katok
• Corinna Ulcigrai
• Dmitry Dolgopyat
• Konstantin Khanin
• Leonid Bunimovich
• Leonid Polterovich
• Nikolai Chernov
• Svetlana Jitomirskaya
• Valery Oseledets

Influenced by

• Ludwig Boltzmann
• Eugene Dynkin

Influenced

• Domokos Szász
• Donald Ornstein

Colleagues and collaborators

• Alexander Mazel
• David Ruelle
• Dmitri Anosov
• Isaak Khalatnikov
• Pavel Bleher
• Robert Minlos
• Rufus Bowen
• Sergei Pirogov 1975-1976
• Weinan E 1997-2000

Family

• Elena Vul
• Grigory Barenblatt
• Veniamin Kagan

Also mentioned in this life

• Svante Arrhenius
• Paul Dirac
• Fibonacci
• Josiah Willard Gibbs
• Joseph-Louis Lagrange
• Henri Lebesgue
• Joseph Liouville
• Hendrik Antoon Lorentz
• Aleksandr Lyapunov
• Andrey Markov
• Claude-Louis Navier
• Henri Poincaré
• Erwin Schrödinger
• Claude Shannon
• George Gabriel Stokes
• John von Neumann

Fields

• ergodic theory 7 lives
• dynamical systems 7 lives
• mathematical physics 45 lives
• probability theory 23 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Yakov Grigorevich Sinai, MacTutor History of Mathematics.
2. Yakov Sinai, Wikipedia.
3. Norwegian Academy of Science and Letters, Abel Prize 2014 citation.
4. Princeton University, Sinai receives Abel Prize for lifelong influence on mathematics, 26 March 2014.
5. H. Holden and R. Piene (eds.), The Abel Prize 2013–2017, Springer, 2019 (includes an autobiographical essay by Sinai).

Mathematics

1. Ya. G. Sinai, On the concept of entropy for a dynamical system, Dokl. Akad. Nauk SSSR 124 (1959).
2. Ya. G. Sinai, Dynamical systems with elastic reflections, Russian Math. Surveys 25 (1970).
3. Ya. G. Sinai, Markov partitions and C-diffeomorphisms, Funct. Anal. Appl. 2 (1968).
4. Ya. G. Sinai, Gibbs measures in ergodic theory, Russian Math. Surveys 27 (1972).
5. S. A. Pirogov and Ya. G. Sinai, Phase diagrams of classical lattice systems, Theor. Math. Phys. 25 (1975) and 26 (1976).
6. Ya. G. Sinai, The limiting behavior of a one-dimensional random walk in a random medium, Theory Probab. Appl. 27 (1982).
7. W. E, K. Khanin, A. Mazel and Ya. Sinai, Invariant measures for Burgers equation with stochastic forcing, Ann. of Math. 151 (2000).
8. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975.
9. N. Chernov and R. Markarian, Chaotic Billiards, AMS Mathematical Surveys and Monographs 127, 2006.
10. Ya. G. Sinai, Introduction to Ergodic Theory, Princeton University Press, 1976; and Topics in Ergodic Theory, Princeton, 1994.
11. O. Zeitouni, Random walks in random environment, Lectures on Probability Theory and Statistics, Lecture Notes in Math. 1837, Springer, 2004.
12. S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems, Cambridge University Press, 2017 (Chapter 7, Pirogov–Sinai theory).