THE WORK OF FEDERICO RODRIGUEZ HERTZ ON ERGODICITY OF DYNAMICAL SYSTEMS

A BSTRACT . We review recent advances on ergodicity of partially and nonuniformly hyperbolic systems describing, in particular, important contributions of Federico Rodriguez Hertz and his collaborators.


HOPF ARGUMENT
One inspiring source for the development of the modern theory of dynamical systems is the work of Poincaré on the three body problem. Poincaré made a crucial observation that some systems are so complicated that it is impossible to obtain exact formulas for every trajectory and so one has to aim for a less precise description of the trajectories.
Ergodicity is a basic example of such an approach. Namely, if the system is ergodic then almost every trajectory is uniformly distributed in the phase space.
By now there are several methods to establish ergodicity, the most successful of which are harmonic analysis and the Hopf argument. The former method requires the system to have a high degree of symmetry while the later does not have this constraint. The Birkhoff Ergodic Theorem shows that the set R of regular points has full measure with respect to each invariant probability measure.
We say that x and y in R are equivalent if for any continuous function φ we haveφ(x) =φ(y). Then µ is ergodic provided that µ-almost all x and y are FIGURE 1

. A Hopf chain
We say that x and y in R are equivalent if for any continuous function φ we haveφ(x) =φ(y). Then µ is ergodic provided that µ-almost all x and y are equivalent. On the other hand, (1) implies that if x, y ∈ R and y ∈ W u (x) ∪ W s (x), then x and y are equivalent. So the Hopf method is to find for every (or most) pair (x, y) a Hopf chain x = z 0 , z 1 ,..., z N = y so that z j +1 ∈ W u (z j ) ∪ W s (z j ). (2) We say that x and y belong to the same accessibility class if they can be connected by a Hopf chain (2).
The problem is that the existence of a Hopf chain connecting x and y does not imply that x and y are equivalent since the intermediate points may not belong to R. Therefore extra work is needed to establish ergodicity. The first case where the Hopf argument was implemented is uniformly hyperbolic systems.
We need some terminology. Let Λ be a hyperbolic attractor, that is Λ = n≥0 f n U , where U is an open forward invariant set. Λ is called (uniformly) hyperbolic if there is an f -invariant splitting where d f expands E u and contracts E s . Given an invariant measure µ, its basin is the set of points whose forward dynamics is described statistically by µ. That is For uniformly hyperbolic systems the Hadamard-Perron Theorem [2] guarantees existence of unstable manifolds tangent to E u and stable manifolds tangent to E s . Uniform transversality of stable and unstable manifolds makes uniformly hyperbolic systems a perfect ground for applying the Hopf argument. In this case the problem of intermediate points was resolved by Hopf [27] under the assumption that stable and unstable foliations are C 1 by showing that x and y can be joined by many chains, and, by Fubini theorem, most chains are good.
The C 1 assumption was later removed by Anosov and Sinai [2,3] We say that x and y belong to the same accessibility class if they can be connected by a Hopf chain (2).
The problem is that the existence of a Hopf chain connecting x and y does not imply that x and y are equivalent since the intermediate points may not belong to R. Therefore extra work is needed to establish ergodicity. The first case where the Hopf argument was implemented is uniformly hyperbolic systems.
We need some terminology. Let Λ be a hyperbolic attractor, that is Λ = where d f expands E u and contracts E s . Given an invariant measure µ, its basin is the set of points whose forward dynamics is described statistically by µ. That is For uniformly hyperbolic systems the Hadamard-Perron Theorem [2] guarantees existence of unstable manifolds tangent to E u and stable manifolds tangent to E s . Uniform transversality of stable and unstable manifolds makes uniformly hyperbolic systems a perfect ground for applying the Hopf argument. In this case the problem of intermediate points was resolved by Hopf [27] under the assumption that stable and unstable foliations are C 1 by showing that x and y can be joined by many chains, and, by Fubini theorem, most chains are good.
The C 1 assumption was later removed by Anosov and Sinai [2,3] by showing that stable and unstable foliations are absolutely continuous. A foliation is called absolutely continuous if any set of zero Lebesgue measure intersects almost every leaf on a set of zero leafwise measure. Absolute continuity allows to apply Fubini theorem which is needed to show that most chains on Figure 2 are good. JOURNAL OF MODERN DYNAMICS VOLUME 10, 2016, 175-189 where p j is any periodic point in Λ j .
In addition, Λ j can be further decomposed as Λ j = r j k=1 Λ j k so that f Λ j k = Λ j (k+1) mod r j and f r j | Λ j k is mixing.
There are two natural ways to extend this result: partial hyperbolicity and nonuniform hyperbolicity. Federico Rodriguez Hertz made important contributions in both cases. Partially hyperbolic systems are discussed in Section ?? while nonuniformly hyperbolic systems are treated in Section ??.   Combining this idea with the work of Smale on structure of hyperbolic attractors [50], work of Sinai and Bowen on symbolic coding of hyperbolic systems [48,10], and the work of Sinai and Ruelle on invariant measures for symbolic systems [49,47], one obtains the following result.
THEOREM 1 (see [11]). We can decompose Λ = m j =1 Λ j , where Λ j are invariant and there are ergodic measures µ j supported on Λ j such that m j =1 B(µ j ) has full measure in U .
Moreover, B(µ j ) is equal up to a set of Lebesgue measure 0 to the homoclinic class where p j is any periodic point in Λ j .
In addition, Λ j can be further decomposed as Λ j = r j k=1 Λ j k so that f Λ j k = Λ j (k+1) mod r j and f r j | Λ j k is mixing.
There are two natural ways to extend this result: partial hyperbolicity and nonuniform hyperbolicity. Federico Rodriguez Hertz made important contributions in both cases. Partially hyperbolic systems are discussed in Section 2 while nonuniformly hyperbolic systems are treated in Section 3.
JOURNAL OF MODERN DYNAMICS VOLUME 10,2016,[175][176][177][178][179][180][181][182][183][184][185][186][187][188][189] For partially hyperbolic systems the distributions E u and E s are uniquely integrable, and they are tangent to foliations W u and W s ( [26]). This sets the stage for the Hopf argument. In fact, it was shown by Brin and Pesin [13] under a number of technical conditions, including the assumption that W u and W s are Lipshitz, then two regular points in the same accessibility class are equivalent. Unfortunately, the Lipshitz condition is quite rarely satisfied (unless there are additional symmetries). So a lot of non trivial work went into weakening the Lipshitz assumption. This was done by Grayson, Pugh, Shub, Burns, and Wilkinson [24,37,38,16]. They succeeded to weaken the regularity assumptions on the center direction to center bunching. The diffeomorphism is called center bunched if (d f |E c ) is close to conformal. The weakest center bunching assumption is due to [16]. It For our purposes, it suffices to note that if dimE c = 1, then the middle term becomes one and so (4) is always satisfied (for a suitable Riemannian metric) since d f expands E u and contracts E s .
The upshot of the above cited works is that if f is volume preserving and center bunched, then almost all points in the same accessibility class are equivalent. Therefore, it is convenient to make the following definition. We say that f is essentially accessible if any measurable set consisting of accessibility classes has either 0 or full measure. If there is only one accessibility class, then we say that f has accessibility property. THEOREM 2 ([16]). If f is volume preserving, center bunched, and essentially accessible, then it is ergodic.
Let us call diffeomorphism f stably ergodic if there exists r > 0 such that any diffeomorphism g which is C r -close to f is ergodic. Theorem 2 allows to show that many classical examples of partially hyperbolic systems are stably ergodic ( [15,30,39,53]).
Let me give a sample result of this type. Let f : M → M be an Anosov diffeomorphism, G be a compact Lie group and τ : M → G be a smooth function. Consider a compact extension F f ,τ acting on M ×G by the formula
(c) ( [15]) If M is a nilmanifold and F f ,τ has the accessibility property, then it is stably ergodic.
In view of Theorem 5 below it is interesting to note that, while essential accessibility is sufficient for ergodicity, it does not imply stable ergodicity. Indeed if G = T 2 and τ(x) = (t (x)u 1 , t (x)u 2 ), then F f ,τ will be ergodic for a typical function t provided that u 1 /u 2 ∈ Q. However, by small perturbation one can JOURNAL OF MODERN DYNAMICS VOLUME 10, 2016, 175-189 make u 1 /u 2 rational, u 1 /u 2 = k 1 /k 2 , in which case F f ,τ is not ergodic since φ(x, g 1 , g 2 ) = e 2πi (k 1 g 1 +k 2 g 2 ) is an invariant function.
2.2. Difficulties in proving ergodicity. Theorem 3 and similar results provide an evidence for the conjecture of Pugh and Shub saying that stable ergodicity is dense among partially hyperbolic systems. While this conjecture is still open, Federico Rodriguez Hertz obtained several strong results in that direction. Before describing these results in detail let me mention the difficulty: the accessibility classes could be complicated.
To explain the problem let me compare the accessibility classes in dynamics with accessibility classes in control theory. In the setting of partially hyperbolic systems the accessibility class of x denoted by A(x) is the set of points which can be joined to x by a piecewise smooth curve where each piece is tangent to either E u or E s . More generally, fix two plane fields E 1 and E 2 and let A(x) be the set of points which can be joined to x by a piecewise smooth curve where each piece is tangent to either E 1 or E 2 . If E 1 and E 2 are smooth, then the accessibility classes are nice.
Let Lie(E 1 , E 2 ) be the smallest subspace in the space of vector fields on M which contains all vector fields tangent to either E 1 or E 2 and which is closed with respect to taking commutators. Let THEOREM 4 (see e.g. [32,37]).
So in the smooth case accessibility classes are manifolds and their tangent spaces could be obtained by local analysis. Unfortunately, for partially hyperbolic systems, the regularity of E u and E s is much lower than needed for applying Theorem 4. In fact, it was shown in [25] that even in the uniformly hyperbolic case E u and E s are generically not better than Hölder. Moreover, in many cases one can show that sufficient regularity of stable and unstable foliations implies that the system is algebraic (see e.g. [7]).

Work of Rodriguez Hertz.
The first important contribution of Federico Rodriguez Hertz is his thesis [41] dealing with stable ergodicity of linear toral automorphisms with two dimensional center. Before that work there were no tools for proving stable ergodicity of non accessible systems. The precise statement of [41] is the following. Let me go briefly over the main steps of the proof. The key fact is that accessibility classes are reasonable. While they are not homogeneous spaces as in Theorem 3 they enjoy local homogeneity. Namely let T be a submanifold transversal to E u ⊕ E s . Given x, y ∈ T which belong to the same accessibility class, there exists a continuous local map φ mapping a small neighborhood of x in T to a small neighborhood of y in T which preserves accessibility classes. The map φ is obtained by composing the holonomy maps along the Hopf chain joining x and y. In particular, in general, φ has low regularity. If φ were smooth we could conclude that accessibility classes are manifolds ( [40,54]). In the present setting low dimensional topology arguments yield LEMMA 6. If dim(E c ) ≤ 2, then the accessibility classes are topological manifolds.
To present the idea of the argument consider the case dim(E c ) = 1. Take a curve T transversal to E u ⊕ E s (see Figure 3). Take x ∈ T and consider all curves tangent to E u ∪ E s starting at x and ending at T. There are two cases. Either all such curves end at x in which case A(x) is a codimension 1 surface or there is a curve γ which does not end at x. Shortening all legs of γ by a factor t and adding two extra legs to land at T , if necessary, we obtain a family of curves γ t with γ 0 (1) = x, γ 1 (1) = y. Thus A(x) ∩ T contains a segment. Now local homogeneity implies that A(x) ∩ T is open proving the lemma. 180 DMITRY DOLGOPYAT transversal to E u ⊕ E s . Given x, y ∈ T which belong to the same accessibility class, there exists a continuous local map φ mapping a small neighborhood of x in T to a small neighborhood of y in T which preserves accessibility classes. The map φ is obtained by composing the holonomy maps along the Hopf chain joining x and y. In particular, in general, φ has low regularity. If φ were smooth we could conclude that accessibility classes are manifolds ([?, ?]). In the present setting low dimensional topology arguments yield LEMMA 6. If dim(E c ) ≤ 2, then the accessibility classes are topological manifolds.
To present the idea of the argument consider the case dim(E c ) = 1. Take a curve T transversal to E u ⊕E s (see Figure ??). Take x ∈ T and consider all curves tangent to E u ∪ E s starting at x and ending at T. There are two cases. Either all such curves end at x in which case A(x) is a codimension 1 surface or there is a curve γ which does not end at x. Shortening all legs of γ by a factor t and adding two extra legs to land at T , if necessary, we obtain a family of curves γ t with γ 0 (1) = x, γ 1 (1) = y. Thus A(x) ∩ T contains a segment. Now local homogeneity implies that A(x) ∩ T is open proving the lemma. The above argument also shows that dimA(x) is upper semicontinuous. Next, using algebraic topology the author shows that the partition into accessibility classes is minimal, that is, there are no nontrivial open sets consisting of accessibility classes. Now Lemma ?? shows that there are only three possibilities:  The above argument also shows that dimA(x) is upper semicontinuous. Next, using algebraic topology the author shows that the partition into accessibility classes is minimal, that is, there are no nontrivial open sets consisting of accessibility classes. Now Lemma 6 shows that there are only three possibilities: A to a center space is a rotation. For a small perturbation, there is a fixed point close to 0 which posses an invariant manifold where the dynamics is close to a rotation.) (iii) A(x) ∩ T is discrete for each x. A standard argument then shows that the accessibility partition of the perturbed map f is close to the accessibility partition of the linear model. The accessibility classes of the linear map are Diophantine planes. One then invokes a KAM result of Moser to promote the topological conjugacy to the smooth conjugacy which in turn implies that f is essentially accessible.
Apart from playing an important role in the proof of Theorem 5, Lemma 6 has several spectacular applications in case dim(E c ) = 1 which are described below.

THEOREM 7 ([44]
). If f is a partially hyperbolic volume preserving diffeomorphism of a three dimensional nilmanifold, then either f has accessibility property or the manifold is T 3 .
In particular, there are manifolds where all partially hyperbolic diffeomorphisms are ergodic.

THEOREM 8 ([45]). For partially hyperbolic systems with one dimensional center accessibility is open and dense.
(The openness of accessibility was established in [20] while density is due to [45]).
To prove both theorems the authors consider the set Γ( f ) of points whose accessibility class has codimension 1. It is a closed invariant set saturated by stable and unstable manifolds. Thus there are three possibilities. Either In case (iii), the fact that f is area preserving and there only finitely many gaps in Γ of a given size implies that the boundary leaves of Γ( f ) have Anosov dynamics and, hence, they contain many periodic points. Now to prove Theorem 8 it suffices to prove that generically there is at least one open accessibility class (which rules out case (ii)) and all periodic points belong to open classes (which rules out case (iii)). Both of those genericity statements could be achieved by local perturbations.
In the proof of Theorem 7 cases (ii) and (iii) are ruled out by using deep results about two dimensional foliations of three dimensional manifolds.
Pesin theory (see [5]) guarantees that for almost every x there exist an unstable manifold W u (x) tangent to E + and a stable manifold W s tangent to E − . This makes it possible to speak about Hopf chains and accessibility classes. In particular, given a point x its Hopf brush is a set of points which are connected to x by a two-leg chain. It is proven in [34] that Hopf brushes have positive Lebesgue measure.
We say that an invariant measure µ has the SRB property 1 if it has conditional densities on unstable manifolds, that is, µ admits a disintegration where γ(t ) is the unstable manifold passing through t and µ γ has a density on γ. THEOREM 9 (Pesin, Pugh-Shub, see [5,34,36]). If µ has the SRB property, then for µ almost any x, Λ(x) ⊂ B(µ). In particular, there are at most countably many SRB measures.
In general, Λ(x) can be a complicated fractal set. In particular, there indeed may be countably many SRB measures ( [21]) and basins of different measures may be intermingled ( [18,29]).
While in many specific examples (such as Hénon attractors, Lorenz equation, dispersive billiards, etc.) the uniqueness of SRB measures has been established ( [6,17,35,55]), the general results on ergodicity were lacking due to the complicated geometry of Hopf brushes.

Work of Rodriguez Hertz.
Jointly with Jana Rodriguez Hertz, Ali Tahzibi and Raul Ures, Federico Rodriguez Hertz has obtained deep results about geometry of Hopf brushes, especially in the low dimensional setting. Let µ be a measure with nonzero Lyapunov exponents and the SRB property. Given a saddle p let The intuition behind this theorem is that if µ(Λ(p)) > 0, then most points on W u (p) are forward typical for µ and this ensures that W u (p) and hence ∪ x∈W u (p) W s (x) belong to one component (see Figure ??).

COROLLARY 11 ([?]). Let f be a topologically transitive map of a surface. Then every hyperbolic measure with the SRB property is ergodic. Hence such a measure is unique.
To explain the idea of the proof we need the notion of Pesin set. We refer the reader to [?] for the precise definition. Roughly speaking, a Pesin set is a set of points where the functions c(x), λ(x) − 1, and ∠(E + (x), E − (x)) (6) are not too small and they do not deteriorate too quickly along the orbit of x. In particular, the points from a given Pesin set have a uniform lower bounds on sizes of stable and unstable manifolds. On the other hand given a hyperbolic measure µ one can find a Pesin set of measure arbitrary close to 1 (at the expense of allowing a worse bound on the functions in (??)).
The proof of Corollary ?? relies on an elegant analytic trick. Let µ 1 , µ 2 be two hyperbolic ergodic measures with the SRB property.
Recall from Section ?? that for B(µ j ) is saturated by stable manifolds. Since µ j have the SRB property, the absolute continuity of the W u implies that, for µ j almost every point x, Lebesgue almost every point inside W u (x) is µ j typical. If this happens we say that W u (x) is supp(µ j ) saturated by unstable manifolds.
Let R 1 be a rectangle containing many stable manifolds having the following properties (see Figure ??  Theorem 10 provides an extension of Theorem 1 to the nonuniformly hyperbolic setting. The intuition behind this theorem is that if µ(Λ(p)) > 0, then most points on W u (p) are forward typical for µ and this ensures that W u (p) and hence ∪ x∈W u (p) W s (x) belong to one component (see Figure 4). ([42]). Let f be a topologically transitive map of a surface. Then every hyperbolic measure with the SRB property is ergodic. Hence such a measure is unique.

COROLLARY 11
To explain the idea of the proof we need the notion of Pesin set. We refer the reader to [5] for the precise definition. Roughly speaking, a Pesin set is a set of points where the functions are not too small and they do not deteriorate too quickly along the orbit of x. In particular, the points from a given Pesin set have a uniform lower bounds on sizes of stable and unstable manifolds. On the other hand given a hyperbolic measure µ one can find a Pesin set of measure arbitrary close to 1 (at the expense of allowing a worse bound on the functions in (6)).
The proof of Corollary 11 relies on an elegant analytic trick. Let µ 1 , µ 2 be two hyperbolic ergodic measures with the SRB property.
Recall from Section 1 that for B(µ j ) is saturated by stable manifolds. Since µ j have the SRB property, the absolute continuity of the W u implies that, for µ j almost every point x, Lebesgue almost every point inside W u (x) is µ j typical. If this happens we say that W u (x) is supp(µ j ) saturated by unstable manifolds. (b) they belong to a given Pesin set; (c) they fully cross R 1 .
This can be achieved taking a density point of some Pesin set of µ 1 and letting R 1 to be a sufficiently small rectangle around it. We denote by W s the union of the stable manifolds having properties (a)-(c).
Let W be a piece of an unstable manifold which is typical for µ 2 , is contained inside R 1 and fully crosses R 1 . The existence of such manifold follows from the topological transitivity. The authors show, using uniform absolute continuity of W s , that W s can be extended to a C 1 foliation of R 1 . Then Sard Theorem guarantees that almost every leaf of that foliation intersects W transversely. Therefore W contains many points from B(µ 1 ) and since it is typical for µ 2 we can conclude that µ 2 = µ 1 proving the uniqueness.

ERGODICITY AND RIGIDITY
Federico Rodriguez Hertz obtained significant geometric information on the structure of ergodic decomposition in both partially hyperbolic and nonuniformly hyperbolic settings, which constitutes a significant advance in this classical subject. Perhaps equally important is that his work provides a new point of view on ergodicity of hyperbolic systems. Namely, he treated ergodicity as a rigidity problem. If the majority of systems are believed to be ergodic, then it makes sense to classify the nonergodic examples and to see if the system at hand belongs to a small list of exceptions. This new point of view allows one to bring to the spotlight new powerful tools of rigidity theory, in particular topological and geometric methods. Currently, the interplay between hyperbolicity and rigidity is an active research topic 2 . Below I present a selection of works in this area making emphasis on the papers related to the results discussed in the two previous sections. 2 The present survey concentrates on using rigidity methods to study ergodic properties of hyperbolic systems. It is also very fruitful to use ergodic properties of hyperbolic systems to Let R 1 be a rectangle containing many stable manifolds having the following properties (see Figure 5): (a) they belong to B(µ 1 ); (b) they belong to a given Pesin set; (c) they fully cross R 1 .
This can be achieved taking a density point of some Pesin set of µ 1 and letting R 1 to be a sufficiently small rectangle around it. We denote by W s the union of the stable manifolds having properties (a)-(c).
Let W be a piece of an unstable manifold which is typical for µ 2 , is contained inside R 1 and fully crosses R 1 . The existence of such manifold follows from the topological transitivity. The authors show, using uniform absolute continuity of W s , that W s can be extended to a C 1 foliation of R 1 . Then Sard Theorem guarantees that almost every leaf of that foliation intersects W transversely. Therefore W contains many points from B(µ 1 ) and since it is typical for µ 2 we can conclude that µ 2 = µ 1 proving the uniqueness.

ERGODICITY AND RIGIDITY
Federico Rodriguez Hertz obtained significant geometric information on the structure of ergodic decomposition in both partially hyperbolic and nonuniformly hyperbolic settings, which constitutes a significant advance in this classical subject. Perhaps equally important is that his work provides a new point of view on ergodicity of hyperbolic systems. Namely, he treated ergodicity as a rigidity problem. If the majority of systems are believed to be ergodic, then it makes sense to classify the nonergodic examples and to see if the system at hand belongs to a small list of exceptions. This new point of view allows one to bring to the spotlight new powerful tools of rigidity theory, in particular topological and geometric methods. JOURNAL OF MODERN DYNAMICS VOLUME 10, 2016, 175-189 Currently, the interplay between hyperbolicity and rigidity is an active research topic 2 . Below I present a selection of works in this area making emphasis on the papers related to the results discussed in the two previous sections.

Lyapunov exponents.
Since hyperbolicity provides a powerful tool for studying statistical properties of dynamical systems, it is desirable to obtain methods for proving that Lyapunov exponents are non-zero. For products of independent random matrices this was done by Furstenberg [23]. The dependent case remained much less understood even though significant progress was achieved in the work of Ledrappier [31]. The situation changed in the last decade due to the work Bonatti, Gomez-Mont, Viana, Avila and their collaborators ( [9,52,4]). For cocycles over nonuniformly hyperbolic systems it was proven in [52] that the top Lyapunov exponent is positive on a open and dense set of smooth cocycles. To explain the idea of this work let A be a matrix valued cocycle over a diffeomorphism f . Consider an induced action on the projective bundle The key observation of [52] is that if Lyapunov exponents of A are small (compared to the exponents of f ), then F is non-uniformly partially hyperbolic. This allows one to employ the method of [31] to show that if the Lyapunov exponents vanish, then the accessibility classes of F are small, which does not occur generically. Avila and Viana extend the techniques used for hyperbolic systems to partially hyperbolic setting. Here is a sample of their results.

THEOREM 12 ([4]). Let g be a symplectic diffeomorphism of T 4 which is close to
where A has two dimensional center. Then either g has non-zero Lyapunov exponents or g is conjugated to f . In particular, g is Bernoulli.
Note that the last statement strengthens Theorem 5 in the symplectic setting.

Random diffeomorphisms.
The results presented above deal with ergodicity of single diffeomorphism. In this section we discuss ergodicity of several diffeomorphisms. Let ( f 1 , f 2 . . . , f k ) be a finite collection of transformations of a space M preserving the same measure µ. We say that this collection is ergodic with respect to µ if every measurable set invariant under each transformation in our collection has either zero or full measure. Equivalently, one can consider a skew product Σ × M on Σ = {1, 2, . . . , k} Z given by F (ω, x) = (σω, f ω 0 (x)), 2 The present survey concentrates on using rigidity methods to study ergodic properties of hyperbolic systems. It is also very fruitful to use ergodic properties of hyperbolic systems to obtain new results in rigidity theory. The work of Federico Rodriguez Hertz in that direction is described in the companion survey [51]. JOURNAL OF MODERN DYNAMICS VOLUME 10,2016,[175][176][177][178][179][180][181][182][183][184][185][186][187][188][189] where σ is the shift. F preserves the measures which are products of a Bernoulli measure on Σ with µ. (The meaning of F is that the transformations applied at different times are chosen independently from our collection.) According to Kakutani Theorem [28] the ergodicity of the collection ( f 1 , f 2 , . . . , f k ) is equivalent to the ergodicity F with respect to any product measure described above.
Since the dynamics of hyperbolic systems has many common features with the shift σ, one can expect that the problem of ergodicity of several diffeomorphisms is similar to the problem of ergodicity of skew products over hyperbolic systems. The later skew products are often partially hyperbolic and the results of Section 2 apply. However, the geometry of partially hyperbolic systems is richer than the geometry of Bernoulli shift, so the problem of ergodicity of several maps is less understood.
The next result presents an instructive example of a stably ergodic system, where each map in the system has poor ergodic properties. THEOREM 13. [22] Let f 1 and f 2 be volume preserving maps of S 2d such that f j are close to rotations R j . If (R 1 , R 2 ) generate a dense subgroup of SO d +1 , then the collection ( f 1 , f 2 ) is ergodic.
Similarly to [41], a key step is to develop a KAM scheme and to show that a lack of stochasticity (in this case the presence of zero Lyapunov exponents) gives sufficient information to conclude vanishing of the obstructions to conjugacy to the linear model. I would like also to mention that, recently, KAM machinery allowed to obtain several advances in the rigidity theory (see e.g. [19] and references therein) which go beyond the present survey. On the other hand, a recent work of Aaron Brown and Federico Rodriguez Hertz [14] provides a significant generalization of Theorem 13, at least, in the two dimensional case.

Homological equation. The homological equation
where φ is a given function and Φ is an unknown plays a key role in several branches of dynamics, including rigidity theory, limit theorems and time changes. In particular, in many cases one would like to understand the regularity of the transfer function Φ. If f is volume preserving, then integrating both sides of (7) we obtain that, if (7) has a solution, then φ has zero mean. For functions of zero mean the study of (7) is closely related to the ergodicity of the skew product on M × R given by (see e.g. [1,12,33]).
For partially hyperbolic systems the regularity of solutions to (7) is given by the following result. THEOREM 14 ([54]). Let f be a partially hyperbolic volume preserving diffeomorphism with the accessibility property. Let φ ∈ C ∞ (M ). If (7) admits a measurable solution, then it admits a smooth solution.

CONCLUSION
The last two decades saw significant advances in understanding ergodic and other statistical properties of a large number of classical dynamical systems. Federico Rodriguez Hertz has made a significant contributions to this subject. The present survey contains merest sketches of his work and related developments. More detailed reviews are contained in [8,46]. However, I urge the readers who want to get a better appreciation of this subject to go to the original papers for a wealth of beautiful ideas and powerful techniques.