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• A GEOMETRIC APPROACH FOR CONSTRUCTING SRB

AND EQUILIBRIUM MEASURES IN HYPERBOLIC

DYNAMICS

YAKOV PESIN

1. Lecture I: Definition and ergodic properties of SRBmeasures

1.1. The setting.

(1) f : M M a C1+ diffeomorphism of a compact smooth Riemann-ian manifold M ;

(2) U M an open subset with the property that f(U) U ; such a setU is called a trapping region;

(3) =n0 f

n(U) a topological attractor for f ; we allow the case = M .

is compact, f -invariant, and maximal (i.e., if U is invariant, then ).

1.2. Physical measures. an arbitrary probability measure on . Thebasin of attraction of is

B =

{x U : 1

n

n1k=0

h(fk(x))

h d for any h C1(M)

} is a physical measure if m(B) > 0. An attractor with a physical measureis often referred to as a Milnor attractor.

1.3. Hyperbolic measures. Given x , v TxM , the Lyapunov expo-nent of v at x is

(x, v) = lim supn

1

nlog dfnv.

The function (x, ) takes on finitely many values, 1(x) p(x),where p = dimM which are invariant functions, i.e., i(f(x)) = i(x) forevery i.

A Borel invariant measure on is hyperbolic if i(x) 6= 0 and 1(x) 0 on

that satisfies condition (5) below.(2) There are local stable V s(x) and local unstable V u(x) manifolds at

x; these are C1+ submanifolds given as graphs

V s,u(x) = exp{

(v, s,u(v)) : v Bs,u(0, r(x)) Es,u(x)}

of C1+ functions s,u defined in the ball Bs,u(0, r(x)) centered atzero of radius r(x) > 0 and mapping it into Eu,s(x); we have thats,u(0) = 0 and Ds,u(0) = 0; these functions are constructed viathe Stable Manifold Theorem.

(3) The local stable and unstable manifolds at x satisfy

d(fn(x), fn(y)) C(x)n(x)d(x, y), y V s(x), n 0,d(fn(x), fn(y)) C(x)n(x)d(x, y), y V u(x), n 0

for some Borel function C(x) > 0 on that satisfies Condition (5)below, and some Borel f -invariant function 0 < (x) < 1.

(4) There are the global stable W s(x) and global unstable W u(x) mani-folds at x (tangent to Es(x) and Eu(x), respectively) so that

W s(x) =n0

fn(V s(fnx)), W u(x) =n0

fn(V u(fnx));

these manifolds are invariant under f , i.e., f(W s(x)) = W s(f(x))and f(W u(x)) = W u(f(x)).

(5) The functions C(x) and K(x) can be chosen to satisfy

C(f1(x)) C(x)e(x), K(f1(x)) K(x)e(x),where (x) > 0 is an f -invariant Borel function.

(6) The size r(x) of local manifolds satisfies r(f1(x)) r(x)e(x).One can show that W u(x) for every x (for which the global unstablemanifold is defined).

Since (x) is invariant, it is constant -a.e. when is ergodic, so fromnow on we assume that (x) = is constant on .

Given ` > 1, define regular set of level ` by

` ={x : C(x) `, K(x) 1

`

}.

These sets satisfy:

• THE GEOMETRIC APPROACH FOR CONSTRUCTING SRB MEASURES 3

` `+1,`1 ` = ;

the subspaces Es,u(x) depend continuously on x `; in fact, thedependence is Holder continuous:

dG(Es,u(x), Es,u(y)) M`d(x, y),

where dG is the Grasmannian distance in TM ; the local manifolds V s,u(x) depend continuously on x `; in fact,

the dependence is Holder continuous:

dC1(Vs,u(x), V s,u(y)) L`d(x, y);

r(x) r` > 0 for all x `.1.4. SRB measures. We can choose ` such that (`) > 0. For x `and a small ` > 0 set

Q`(x) =

yB(x,`)`

V u(y).

Let ` be the partition of Q`(x) by Vu(y), and let V s(x) be a local stable

manifold that contains exactly one point from each V u(y) in Q`(x). Thenthere are conditional measures u(y) on each V u(y), and a transverse mea-sure s(x) on V s(x), such that for any h L1() supported on Q`(x), wehave

(1.1)

h d =

V s(x)

V u(y)

h du(y) ds(x).

Let mV u(y) denote the leaf volume on Vu(y).

Definition 1.1. A measure on is called an SRB measure if is hyper-bolic and for every ` with (`) > 0, almost every x ` and almost everyy B(x, `) `, we have the measure u(y) is absolutely continuous withrespect to the measure mV u(y).

For y `, z V u(y) and n > 0 set

un(y, z) =n1k=0

Jac(df |Eu(fk(z)))Jac(df |Eu(fk(y))) .

One can show that for every y ` and z V u(y) the following limit exists

(1.2) u(y, z) = limn

un(y, z) =

k=0

Jac(df |Eu(fk(z)))Jac(df |Eu(fk(y)))

and that u(y, z) depends continuously on y ` and z V u(y).Theorem 1.2. If is an SRB measure on , then the density du(x, ) of theconditional measure u(x) with respect to the leaf-volume mV u(x) on V

u(x)

is given by du(x, y) = u(x)1u(x, y) where

u(x) =

V u(x)

u(x, y) dmu(x)(y)

• 4 YAKOV PESIN

is the normalizing factor.

In particular, we conclude that the measures u(x) and mV u(y) must beequivalent.

The idea of describing an invariant measure by its conditional probabilitieson the elements of a continuous partition goes back to the classical work ofKolmogorov and especially later work of Dobrushin on random fields andrelation (1.2) can be viewed as an analog of the famous Dobrushin-Lanford-Ruelle equation in statistical physics.

1.5. Ergodic properties of SRB measures. Using results of nonuniformhyperbolicity theory one can obtain a sufficiently complete description ofergodic properties of SRB measures.

Theorem 1.3. Let f be a C1+ diffeomorphism of a compact smooth man-ifold M with an attractor and let be an SRB measure on . Then thereis a finite or countable collection of subsets 0,1,2, such that

(1) =i0 i, i j = ;

(2) (0) = 0 and (i) > 0 for i > 0;(3) f |i is ergodic for i > 0;(4) for each i > 0 there is ni > 0 such that i =

nij=1 i,j where the

union is disjoint (modulo -null sets), f(i,j) = i,j+1, f(ni,1) =i,1 and f

ni |i,1 is Bernoulli;(5) if is ergodic, then the basin of attraction B has positive Lebesgue

measure in U .

For smooth measures this theorem was proved by Pesin and an extensionto the general case was given by Ledrappier.

Statement 5 of Theorem 1.3 can be paraphrased as follows: ergodic SRBmeasures are physical. The converse is not true; physical measures need notbe SRB. Easy examples are given by (1) the Dirac measure on an attractingfixed point, which has all exponents negative and hence is not hyperbolic,and by (2) Lebesgue measure for an irrational circle rotation, which has zeroexponent but is still physical.

A more subtle example is given by the time-1 map of the flow illustratedin Figure 1, where the Dirac measure at the hyperbolic fixed point p is ahyperbolic physical measure whose basin of attraction includes all pointsexcept q1 and q2. (In fact, by slowing down the flow near p, one can adaptthis example so that p is an indifferent fixed point and hence, the physicalmeasure is not even hyperbolic.)

1.6. Characterization of SRB measures. The following result providesa complete characterization of SRB measures via their entropies.

• THE GEOMETRIC APPROACH FOR CONSTRUCTING SRB MEASURES 5

pq1

q2

Figure 1. A physical measure that is not SRB.

Theorem 1.4. A measure is an SRB measure if and only if the entropyh(f) of is given by the entropy formula:

h(f) =

i(x)>0

i(x) d(x).

For smooth measures (which are a particular case of SRB measures) theentropy formula was proved by Pesin and its extension to SRB measureswas given by Ledrappier and Strelcyn. The fact that a hyperbolic measuresatisfying the entropy formula is an SRB measure was shown by Ledrappier.

Remark 1.5. Our definition of SRB measure includes the requirement thatthe measure is hyperbolic. In fact, one can extend the notion of SRB mea-sures to those that have some or even all Lyapunov exponents zero (in thelatter case we take W u(x) = {x}). It was proved by Ledrappier and Youngthat a measure satisfies the entropy formula if and only if it is an SRB mea-sure in this more general sense, but we stress that such an SRB measuremay not be physical, i.e., its basin may be of zero Lebesgue measure.

It follows from Theorem 1.3 that f admits at most countably many ergodicSRB measures. One can show that a topologically transitive C1+ surfacediffeomorphism can have at most one SRB measure but the result is nottrue in dimension higher than two.

1.7. An overview of the geometric approach. The idea of the geo-metric approach is to follow the classical Bogolyubov-Krylov procedure forconstructing invariant measures by pushing forward a given reference mea-sure. In our case the natural choice of a reference measure is the Riemannianvolume m restricted to the neighborhood U , which we denote by mU . Wethen consider the sequence of probability measures

(1.3) n =1

n

n1k=0

fkmU .

Any weak* limit point of this sequence of measures is called a natural mea-sure and while in general, it may be a trivial measure, under some additionalhyperbolicity requirements on the attractor one obtains an SRB measure.

• 6 YAKOV PESIN

For attractors with some hyperbolicity one can use a somewhat differentapproach which exploits the fact that SRB measures are absolutely con-tinuous along unstable manifolds. To this end consider a point x , itslocal unstable manifold V u(x), and take the leaf-volume mV u(x) as the ref-erence measure for the above construction. Thus one studies the sequenceof probability measures given by

(1.4) n =1

n

n1k=0

fkmV u(x).

The measures n are spread out over increasingly long pieces of the globalunstable manifold of the point fn(x) and, in some situations, control overthe geometry of the unstable manifold makes it possible to draw conclusionsabout the measures n by keeping track of their densities along unstablemanifolds, and ultim