$\mathcal{A}^{u}$. We claim that the following partition is subordinate to
$W^{u}$

$\xi=\bigvee_{j=0}^{\infty}f^{j}\xi_{0}.$

The most complicated part is to prove that $\xi(x)$ contains a neighbourhood
of $x$ inside $W^{u}(x)$ for $\mu$-almost every $x$. Denote by
$\partial\mathcal{P}$ the union of boundaries of atoms of $\mathcal{P}$. We
first use that $\mu(\partial\mathcal{P})=0$ and the $f$-invariance of $\mu$ to
find a Borel set $X$ full for $\mu$ such that
$f^{j}(x)\notin\partial\mathcal{P}$ for every $x\in X$ and $j\in\mathbb{Z}$.
This implies that for every $x\in X$, $\xi_{k}(x)$ contains an open
neighbourhood of $x$ for every $k\in\mathbb{N}$ where

$\xi_{k}=\bigvee_{j=0}^{k}f^{j}\xi_{0}.$

The fact that $\xi(x)$ contains an open neighbourhood of $x$ comes from the
fact that

$\xi(x)=\xi_{k(x)}(x)$

for some $k(x)\in\mathbb{N}$. This $k(x)$ is obtained from an argument à la
Borel–Cantelli. Let us explain it. Using once more the $f$-invariance of $\mu$
as well as the second property characterizing $\mathcal{P}$, we see that

$\sum_{j=0}^{\infty}\mu\left[f^{j}\left(\mathcal{N}_{\lambda^{j}}(\partial\mathcal{P})\right)\right]<\infty.$

By Borel–Cantelli’s theorem there exists a Borel set $X$ of full $\mu$-measure
such that for every $x\in X$ we have
$d(f^{-k}(x),\partial\mathcal{P})>\lambda^{k}$ for every $k$ greater than some
$k(x)$. One easily shows that when $k\geq k(x)$ we have
$\xi_{k}(x)=\xi_{k+1}(x)$, for the contrary would imply that
$f^{k+1}(\partial\mathcal{P})\cap\xi_{k}(x)\neq\emptyset$. Using that plaques
of $\mathcal{A}^{u}$ have uniform diameters (say smaller than $1$) we find
that $d(x,f^{k+1}(\partial\mathcal{P}))\leq 1$. Using the uniform expansion of
$f$ along $W^{u}$, one would find that
$d(f^{-(k+1)}(x),\mathcal{P}))\leq\lambda^{k+1}$, which is absurd by the
definition of $k(x)$.

######  Remark D.5.

In order to treat the general case one has to use Pesin’s theory in order to
get uniform expansion in sets of positive measure. This is done by defining
Pesin’s sets and analysing the first return maps to these Pesin’s sets.
Pesin’s sets $\Lambda$ (we use Katok–Mendoza’s terminology [KH]) are sets of
positive measure (but which are not invariant) enjoying the following
properties

  * •

the size of local unstable manifolds of elements of $\Lambda$ is uniformly
bounded from below; more precisely, there exists $\delta>0$ such that for
every $x\in\Lambda$, the preimage of $W^{u}_{loc}(x)$ under the exponential
map at $x$ contains the graph of a $C^{1}$-map from the $\delta$-neighbourhood
of $0$ in $E^{u}(x)$ to $E^{c}(x)\oplus E^{s}(x)$;

  * •

the dynamics is uniformly expanding along local unstable manifolds inside
$\Lambda$; more precisely there exist constants $0<\mathchar
271\relax<\lambda/100$ and $C>0$ such that for for every $x\in\Lambda$ every
$n\geq 1$ and $m\in\mathbb{Z}$

$\left|\left|D_{x}f^{-n}|_{E^{u}(f^{m}(x))}\right|\right|\leq
Ce^{(\lambda-\mathchar 271\relax)n}e^{-|m|\mathchar 271\relax}.$

where $\lambda>0$ is the smallest positive Lyapunov exponent of $f$ for $\mu$.

Of course, if one wants to increase the measure of Pesin sets, one looses
control on the constants $C$ and $\mathchar 271\relax$. Nevertheless the
argument sketched in the case of uniformly expanded foliations can be adapted
even if one only guarantees the uniformity of the expansion in positive
measure sets. This analysis is essentially an argument given by Mañé in [Ma].
We won’t enter here into the details and suggest the reader to consult the
classical references: [Ma, LS, LYI, Yo2].

#####  D.1.2. Entropy along the unstable direction

The next step of the proof of Ledrappier and Young is to define the entropy
along the unstable direction. Recall (Section 8.1.2) that when $\eta_{1}$ and
$\eta_{2}$ are measurable partitions of $M$, $H_{\mu}(\eta_{1}\mid\eta_{2})$
denotes the conditional entropy of $\eta_{1}$ given $\eta_{2}$ and that when
$\eta$ is an increasing partition we have

$h_{\mu}(f,\eta)=H_{\mu}(\eta\mid f\eta).$

The next proposition allows us to define the entropy along the unstable
direction (cf. Definition 8.3).

######  Proposition D.6.

Let $f:M\to M$ be a $C^{1+\beta}$ diffeomorphism of a compact manifold. Let
$\xi_{1}$ and $\xi_{2}$ be two measurable partitions subordinate to the
unstable foliation $W^{u}$ of $f$. Then

$h_{\mu}(f,\xi_{1})=h_{\mu}(f,\xi_{2}).$

###### Proof.

Let us detail the argument proving this proposition. Let $\xi_{1}$ and
$\xi_{2}$ be two measurable partitions subordinate to $W^{u}$. It is enough to
prove that $h_{\mu}(f,\xi_{1})=h_{\mu}(f,\xi_{1}\vee\xi_{2})$. The great idea
of the proof is to note that since $f\xi_{i}\prec\xi_{i}$ we have for every
$n\geq 0$

$f^{n}\xi_{1}\vee f^{n}\xi_{2}\prec\xi_{1}\vee
f^{n}\xi_{2}\prec\xi_{1}\vee\xi_{2}.$

By $f$-invariance of $\mu$, the entropies of $f$ conditional to the first and
last partitions coincide so we deduce that

$h_{\mu}(f,\xi_{1}\vee\xi_{2})=h_{\mu}(f,\xi_{1}\vee
f^{n}\xi_{2})=H_{\mu}\left(\left.\xi_{1}\vee f^{n}\xi_{2}\right|f\xi_{1}\vee
f^{n+1}\xi_{2}\right),$ (D.1)

the last equality coming from the definition of conditonal entropy. Before
carrying on with the proof, observe that $f$ expands the unstable foliation,
so eventually the atoms of $\xi_{1}$ should be included in atoms of
$f^{n}\xi_{2}$, for $n$ large enough. This gives a good hint that the entropy
conditional to $\xi_{1}\vee f^{n}\xi_{2}$ should tend to the entropy
conditional to $\xi_{1}$. Let us give a formal explanation of this intuition.

We will use a formula of conditional entropy proved in Rokhlin’s classical
paper [Rok, §5.9]. For measurable partitions $\mathcal{A},\mathcal{B}$ and
$\mathcal{C}$ we have that

$H_{\mu}(\mathcal{A}\vee\mathcal{B}|\mathcal{C})=H_{\mu}(\mathcal{A}|\mathcal{C})+H_{\mu}(\mathcal{B}|\mathcal{A}\vee\mathcal{C}).$
(D.2)

Applying (D.2) with $\mathcal{A}=\xi_{1}$, $\mathcal{B}=f^{n}\xi_{2}$ and
$\mathcal{C}=f\xi_{1}\vee f^{n+1}\xi_{2}$ and having in mind that $\xi_{1}\vee
f\xi_{1}=\xi_{1}$ we find

$\displaystyle H_{\mu}\left(\left.\xi_{1}\vee f^{n}\xi_{2}\right|f\xi_{1}\vee
f^{n+1}\xi_{2}\right)$ $\displaystyle=H_{\mu}\left(\xi_{1}\left|f\xi_{1}\vee
f^{n+1}\xi_{2}\right.\right)$
$\displaystyle+H_{\mu}\left(\left.f^{n}\xi_{2}\right|\xi_{1}\vee f\xi_{1}\vee
f^{n+1}\xi_{2}\right)$ $\displaystyle=H_{\mu}\left(\xi_{1}\left|f\xi_{1}\vee
f^{n+1}\xi_{2}\right.\right)+H_{\mu}\left(\xi_{2}\left|f^{-n}\xi_{1}\vee
f\xi_{2}\right.\right).$

Let us recapitulate. We just prove that the following equality holds for every
$n\geq 0$

$h_{\mu}(f,\xi_{1}\vee\xi_{2})=H_{\mu}\left(\xi_{1}\left|f\xi_{1}\vee
f^{n+1}\xi_{2}\right.\right)+H_{\mu}\left(\xi_{2}\left|f^{-n}\xi_{1}\vee
f\xi_{2}\right.\right).$ (D.3)

Observe that $f^{-n}\xi_{1}$ generates (by Item 4. of the definition) so the
second term tends to $0$ as $n\to\infty$. We must now prove that the first
term converges to $H_{\mu}(\xi_{1}|f\xi_{1})=h_{\mu}(f,\xi_{1})$. We clearly
have $H_{\mu}(\xi_{1}|f\xi_{1}\vee f^{n+1}\xi_{2})\leq
H_{\mu}(\xi_{1}|f\xi_{1})$ (see [Rok, §5.10]).

We will now use that $f$ expands the unstable manifold so that for most points
$x\in M$, the atom $f\xi_{1}(x)$ is contained in an atom of $f^{n+1}\xi_{2}$.
Consider the Borel set $D_{n}$ of such $x$. On the one hand
$f\xi_{1}=f\xi_{1}\vee f^{n+1}\xi_{2}$ in restriction to $D_{n}$. On the other
hand, since $f^{-1}$ contracts unstable manifolds we have $\mu(D_{n})\to 1$ as
$n\to\infty$. This yields $\lim_{n\to\infty}H_{\mu}(\xi_{1}|f\xi_{1}\vee
f^{n+1}\xi_{2})\geq H_{\mu}(\xi_{1}|f\xi_{1})$, thus concluding the proof. ∎

This allows us to give sense to the following definition (see Section 8.3).

######  Definition D.7 (Entropy along the unstable direction).

The $\mu$-entropy of $f$ along the unstable direction is the value

$h_{\mu}^{u}(f)=h_{\mu}(f,\xi),$

where $\xi$ is any measurable partition subordinate to the unstable foliation
$W^{u}$.

#####  D.1.3. Local entropy

It is often convenient to work with a local version of entropy which is due to
M. Brin and A. Katok see [BK]. The construction is quite general, but for the
sake of clarity we will state their results in our context.

Let us first define dynamical balls. Given $x\in M$, $n\in\mathbb{N}$ and
$r>0$ we define

$B_{n}(x,r)=\left\\{y\in
M;\,d(f^{i}(x),f^{i}(y))<r,\forall\,i=0,\ldots,n-1\right\\}.$

Let $\mu$ be an ergodic $f$-invariant measure. Given $x\in M$, set

$\underline{h}_{\mu}(f,x)=\lim_{r\to 0}\mathop{\underline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu(B_{n}(x,r)),$

and

$\overline{h}_{\mu}(f,x)=\lim_{r\to 0}\mathop{\overline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu(B_{n}(x,r)).$

######  Theorem D.8 (Brin–Katok).

Let $f:M\to M$ be a $C^{1+\beta}$ diffeomorphism of a compact manifold and
$\mu$ an ergodic $f$-invariant measure. Then for $\mu$-almost every $x\in M$

$h_{\mu}(f)=\underline{h}_{\mu}(f,x)=\overline{h}_{\mu}(f,x)=\lim_{r\to
0}\lim_{n\to\infty}-\frac{1}{n}\log\,\mu(B_{n}(x,r)).$

Following this classical work, F. Ledrappier and L.-S. Young adopted a
pointwise approach for defining the entropy along the unstable direction,
which works well in the ergodic case.

We will let $d^{u}$ denote the Riemannian distance on unstable manifolds
induced by the ambient Riemannian structure. Given $x\in M$, $n\in\mathbb{N}$
and $r>0$ we define

$B^{u}_{n}(x,r)=\left\\{y\in
W^{u}(x);\,d^{u}(f^{i}(x),f^{i}(y))<r,\forall\,i=0,\ldots,n-1\right\\}.$ (D.4)

We will now consider a partition $\xi$ subordinate to $W^{u}$ and a system
$(\mu_{x}^{u})_{x\in M}$ of conditional measures of $\mu$ associated with
$\xi$, uniquely defined up to a $\mu$-negligible set by Rokhlin’s theorem (see
Appendix B). We will define

$\underline{h}^{u}_{\mu}(f,x,\xi)=\lim_{r\to 0}\mathop{\underline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{u}(B^{u}_{n}(x,r)),$

and

$\overline{h}_{\mu}(f,x,\xi)=\lim_{r\to 0}\mathop{\overline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{u}(B^{u}_{n}(x,r)).$

In the second part of their work, F. Ledrappier and L.-S. Young prove the
following theorem (see [LYII, Proposition 7.2.1. and Corollary 7.2.2.]).

######  Theorem D.9.

Let $f:M\to M$ be a $C^{1+\beta}$ diffeomorphism of a compact manifold and let
$\mu$ be an ergodic $f$-invariant measure. Then for $\mu$-a.e. $x\in M$

$\underline{h}^{u}_{\mu}(f,x,\xi)=\overline{h}^{u}_{\mu}(f,x,\xi)=\lim_{r\to
0}\lim_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{u}(B^{u}_{n}(x,r)),$

and the common value is

$H_{\mu}(\xi\mid f\xi)=h^{u}_{\mu}(f).$

####  D.2. Measures satisfying the entropy formula

#####  D.2.1. All the expansion occurs in the unstable direction

The principal accomplishment of Ledrappier–Young’s first paper [LYI] is the
proof of the following key result which says that all the expansion of $f$
occurs in the unstable direction (cf. Section 9.1).

######  Theorem D.10.

Let $f:M\to M$ be a $C^{1+\beta}$ diffeomorphism of a compact manifold and
$\mu$ be an ergodic $f$-invariant probability measure. Then

$h_{\mu}(f)=h_{\mu}^{u}(f).$

In [L], Ledrappier had already proved a similar statement for measures without
zero Lyapunov exponents, and, as we will see later on, knew how to deduce the
conclusion of Theorem D.1 from the equality

$h_{\mu}^{u}(f)=\sum_{\lambda^{j}>0}m^{j}\lambda^{j}.$

#####  D.2.2. The uniformly hyperbolic case

The case of uniformly hyperbolic dynamics is certainly an oversimplification
of the general context. Nevertheless, we may find useful to understand the
skeleton of the Ledrappier–Young’s delicate argument and the difficulties
therein.

Let us assume here that $f$ is an Anosov diffeomorphism. This means that there
is a $Df$-invariant splitting $TM=E^{s}\oplus E^{u}$ where $E^{s}$ and $E^{u}$
are respectively uniformly contracted and expanded by $Df$.

Using the local product structure, every sufficiently small ball is contained
in a foliated chart for $W^{u}$ of the form $D^{s}\times D^{u}$ where $D^{s}$
and $D^{u}$ are small stable and unstable discs respectively. Moreover in such
a chart the Riemannian distance is uniformly equivalent to the Lyapunov
distance which we may define as the $L^{1}$-distance of the product
$D^{s}\times D^{u}$.

Given $x\in M$ and sufficiently small $r>0$, the dynamical ball $B_{n}(x,r)$
(for the Lyapunov distance) is inside $B^{s}_{n}(x,r)\times B^{u}_{n}(x,r)$.
Since $W^{s}$ is uniformly contracted, the dynamical ball $B^{s}_{n}(x,r)$
does not depend on $n\in\mathbb{N}$. Using the transverse continuity of the
restriction of the induced Riemannian structure on leaves of $W^{u}$ we see
that there exist $r_{1}<r_{2}$ converging to $0$ with $r$ such that

$\bigcup_{y\in
B^{s}_{n}(x,r)}B_{n}^{u}(y,r_{1})\mathop{\subset}B_{n}(x,r)\mathop{\subset}\bigcup_{y\in
B^{s}_{n}(x,r)}B_{n}^{u}(y,r_{2}).$ (D.5)

Let $\mu$ be an ergodic $f$-invariant probability measure and $\mu_{y}^{u}$, a
system of conditional measures of $\mu$ along unstable plaques of a chart
$D^{s}\times D^{u}$ containing $B_{n}(x,r)$. Using (D.5) and the fact that
$B_{n}^{s}(x,r)$ does not depend on $n$ we find a constant $C(r)>0$ such that

$C(r)\,{\rm
ess\,inf}_{y}\,\mu^{u}_{y}(B_{n}^{u}(y,r_{1}))\leq\mu(B_{n}(x,r))\leq
C(r)\,{\rm ess\,sup}_{y}\,\mu^{u}_{y}(B_{n}^{u}(y,r_{2})).$

Using the pointwise versions of measure entropy one deduces

$h_{\mu}(f)=h^{u}_{\mu}(f).$

#####  D.2.3. Some words about the general case

The general case is much more delicate and actually the authors of [LYI] don’t
follow such a naive pointwise approach. The most immediate difficulty is that
the hyperbolicity is not uniform and we must work inside Lyapunov charts,
which leads to important technicalities. But the true difficulty of the paper
is to understand and analyse the role of zero Lyapunov exponents.

Let us explain some of the difficulties. For a regular point $x\in M$ one may
consider a Lyapunov chart at $x$. We won’t enter into the details of the
definition here, let us just say that this is an open set $U_{x}$ which is
foliated by local unstable manifolds $W^{u}_{loc}(y)$ (which are well defined
for $\mu$-a.e. $y$). The size of these unstable manifolds depends on $y$ and
is not uniform a priori.

The situation is similar to what we saw in the uniformly hyperbolic setting.
One has a system of coordinates $T\times D^{u}$ around the point $x$, $D^{u}$
being a small unstable disc and $T$ being a small transversal to the unstable
foliation. This time $T$ is not uniformly contracted and one may think of $T$
as a center stable set. And we must analyse how $f$ acts on such sets. The
most important difficulty here is the following.

There is no canonical choice of a transverse distance on $T$.

More precisely we want to show that the separation of unstable plaques is less
than the expansion along unstable manifolds. We know that this is the case for
the expansion along a central transversal. But this apparent weaker expansion
could be a lure and be caused by the effect of a separation inside unstable
plaques. And the information that unstable holonomies are Hölder continuous,
is not enough a priori to rule out the possibility that the actual separation
of unstable plaques (the dynamics in the quotient by unstable plaques) is
stronger than the expansion along unstable manifolds. The authors treat this
difficulty by proving that

unstable holonomies inside center-unstable manifolds are Lipschitz

We don’t enter here into the technical details of the statement and refer to
[LYI, §4.2]). Before we carry on with Ledrappier–Young’s theory let us mention
that a similar difficulty appears in Hirsch–Pugh–Shub’s theory of normally
hyperbolic laminations. In [HPSbook, §7] the authors consider a diffeomorphism
$f$ with an invariant normally hyperbolic lamination and study the
neighbouring diffeomorphisms. It is not quite true that close to $f$, a
diffeomorphism $g$ has an invariant lamination which is (leaf)-conjugate to
that of $f$. When one applies the graph transform to such a $g$, the leaves of
the lamination can merge: the phenomenon of “sliding along plaques” could lead
to a branched invariant lamination (see [BI] for the definition and [CP] for
more information). This phenomenon can be avoided by requiring a technical
condition, called plaque expansivity, under which M. Hirsch, C. Pugh and M.
Shub prove that the branched lamination is a true lamination. This plaque
expansiveness is satisfied for Lipschitz foliations (see [HPSbook, Theorem
7.2.]). We don’t know examples of partially hyperbolic diffeomorphisms with a
foliation tangent to the central bundle that does not satisfy plaque
expansiveness, and we don’t know how to prove plaque expansiveness for all
such foliations.

The argument provided by F. Ledrappier and L.-S. Young works for $C^{2}$
diffeomorphisms, and this is the only argument of the paper that needs this
regularity assumption. A. Brown showed in [B] how this crucial step of
Ledrappier–Young’s proof can be carried on for $C^{1+\beta}$ diffeomorphisms.
The proof then consists in the precise analysis of how $f$ expands the
transverse distance and of dynamical balls. One may think of this part as a
sophistication of the proof given in the uniformly hyperbolic setting.

#####  D.2.4. Idea of proof of Theorem D.1

Now that we know that $h_{\mu}(f)=h_{\mu}^{u}(f)$ we may follow an argument
due to Ledrappier [L] and deduce that

$h_{\mu}(f)=\sum_{\lambda^{j}>0}m^{j}\lambda^{j}\quad\Longrightarrow\quad\mu\text{
is SRB}.$

##### Unstable jacobian

Let us define the unstable jacobian of $f$ at $x$ as the quantity

$J^{u}(x)=\left|{\rm Jac}(D_{x}f|_{E^{u}(x)})\right|.$

Using Oseledec’s and Birkhoff’s theorems we get that for $\mu$-a.e. $x\in M$

$\int_{M}J^{u}d\mu=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n}\log
J^{u}(f^{i}(x))=\sum_{\lambda^{j}>0}m^{j}\lambda^{j}.$

##### Dynamical prescription of densities

It is a well known fact that when an $f$-invariant measure $\mu$ is absolutely
continuous along an expanding foliation, the densities along unstable
manifolds are dynamically prescribed. Denote by $m^{u}_{x}$ the Riemannian
volume of $W^{u}(x)$ and assume that

$d\mu^{u}_{x}=\rho\,dm^{u}_{x},$

where $\rho$ is a positive and measurable function and $\mu^{u}_{x}$ is a
system of conditional measures of $\mu$ with respect to a measurable partition
$\xi$ associated with $W^{u}$ that we will fix from now on. Then for
$\mu_{x}^{u}$-a.e. $y,z\in W^{u}(x)$

$\frac{\rho(z)}{\rho(y)}=\prod_{i=1}^{\infty}\frac{J^{u}(f^{-i}(y))}{J^{u}(f^{-i}(z))}.$
(D.6)

In order to derive the equation above we prove the following lemma.

######  Lemma D.11.

For $\mu$-almost every $x\in M$ the map given by

$g(y)=\frac{\rho(y)}{\rho(f^{-1}(y))}J^{u}(f^{-1}(y))$ (D.7)

is constant on $\xi(x)$.

###### Proof.

Note that $\xi\prec f^{-1}\xi$ so an atom of $\xi$ is a countable union of
atoms of $f^{-1}(\xi)$. We will consider the measurable function $h$, constant
on atoms of $f^{-1}(\xi)$, by setting

$h(y)=\mu^{u}_{y}(f^{-1}\xi(y)).$

The idea now is to write the family of conditional measures with respect to
$f^{-1}\xi$ in two ways. Firstly, using that $\xi\prec f^{-1}\xi$ we have that
for every Borel set $K\mathop{\subset}M$

$\begin{split}\mu_{f^{-1}\xi(y)}(K\cap f^{-1}\xi(y))&=\frac{\mu^{u}_{y}(K\cap
f^{-1}\xi(y))}{\mu^{u}_{y}(f^{-1}\xi(y))}\\\ &=\frac{1}{h(y)}\int_{K\cap
f^{-1}\xi(y)}\rho(z)dm^{u}_{y}(z).\end{split}$ (D.8)

Secondly, by $f$-invariance of $\mu$ we have

$\mu_{f^{-1}\xi(y)}(K\cap
f^{-1}\xi(y))=\mu^{u}_{f(y)}\left(f(K)\cap\xi(f(y))\right).$ (D.9)

We get from (D.8), (D.9) and from a change of variable formula that

$\displaystyle\frac{1}{h(y)}\int_{K\cap
f^{-1}\xi(y)}\rho(z)dm^{u}_{y}(z)=\int_{f(K)\cap\xi(f(y))}\rho(z)dm^{u}_{f(y)}(z)$
$\displaystyle=$ $\displaystyle\int_{f(K\cap
f^{-1}\xi(y))}\rho(z)dm^{u}_{f(y)}(z)=\int_{K\cap
f^{-1}\xi(y)}\rho(f(z))J^{u}(z)dm^{u}_{y}(z).$

Consequently for $\mu$-almost every $y\in M$ and $m^{u}_{y}$-almost every
$z\in f^{-1}(y)$ we have

$\frac{1}{h(y)}\rho(z)=\rho(f(z))J^{u}(z).$

This proves that the function

$g\circ f=\frac{1}{h}$

is constant on atoms of $f^{-1}\xi$. Finally we have that $g$ is constant on
atoms of $\xi$. ∎

##### Reconstructing $\mu$

Ledrappier’s argument is an inductive one. Denote by $\mathcal{B}(\xi)$ the
$\sigma$-algebra whose elements are union of atoms of $\xi$. We can see it as
a $\sigma$-algebra “transverse” to $\xi$. By Rokhlin’s theorem (Theorem B.8),
a measure $\nu$ on $M$ is determined by its trace on $\mathcal{B}(\xi)$ and by
a system of conditional measures with respect to $\xi$.

We will consider a positive function $\rho$ satisfying (D.6). It is proven in
[L, Theorem 3.1. Item viii] that $\log(\rho)$ is Hölder continuous in every
atom $\xi(x)$. Hence on every atom $\xi(x)$, the function $\rho$ is uniformly
bounded away from $0$ and $\infty$, and in particular it is
$m^{u}_{x}$-integrable. In order to prove the Hölder continuity of
$\log(\rho)$, one uses that on local unstable manifolds, $z\mapsto E^{u}(z)$
is Lipschitz, that $z\mapsto D_{z}f$ is Hölder continuous, that $f^{-1}$
contracts unstable manifolds so we can apply the usual distortion controls. We
will furthermore normalize $\rho$ so that for $\mu$-almost every $x\in M$

$\int_{\xi(x)}\rho(x)dm^{u}_{x}=1,$ (D.10)

Define the probability measure $\nu$ on $M$ satisfying both conditions

  1. (1)

$\mu$ and $\nu$ coincide on $\mathcal{B}(\xi)$;

  2. (2)

the disintegration $(\nu_{x})$ of $\nu$ with respect to $\xi$ satisfies

$d\nu_{x}=\rho\,dm^{u}_{x},$

where $\rho$ satisfies (D.6) and (D.10).

We will prove by induction that $\mu$ and $\nu$ coincide on the
$\sigma$-algebra $\mathcal{B}({f^{-n}}\xi)$ for every $n\in\mathbb{N}$. Since
$\xi$ is increasing and generating this implies that $\mu=\nu$. All this
follows from the next lemma.

######  Lemma D.12.

Assume that $\int J^{u}d\mu=h_{\mu}(f,\xi)$. Then $\mu$ and $\nu$ coincide on
$\mathcal{B}({f^{-1}\xi}).$

###### Proof.

Each element $\xi(x)$ contains countably many atoms of $f^{-1}\xi$ and $\mu$
and $\nu$ coincide on $\mathcal{B}(\xi)$. Hence in order to show that $\mu$
and $\nu$ coincide on $\mathcal{B}({f^{-1}\xi})$ it is enough to prove that
$\nu_{y}(f^{-1}\xi(y))=\mu_{y}(f^{-1}\xi(y))$ almost everywhere. We consider
the derivative

$q(y)=\frac{\nu_{y}\left(f^{-1}\xi(y)\right)}{\mu_{y}\left(f^{-1}\xi(y)\right)}$

which is well-defined almost everywhere and positive. Furthermore we have
$\int q\,d\mu=1$. Using Jensen’s inequality and the concavity of the
logarithm, one has:

$\int\log(q)\,d\mu\leq\log\left(\int q\,d\mu\right)=0.$

Moreover this inequality is an equality if and only if $\log(q)=0$
$\mu$-almost everywhere, which as mentioned before, would imply that $\mu=\nu$
on $\mathcal{B}({f^{-1}\xi})$. On the one hand, we have by definition

$-\int\log\mu_{y}\left(f^{-1}\xi(y)\right)d\mu(y)=H_{\mu}(f^{-1}\xi\mid\xi)=h_{\mu}(f,\xi).$

On the other hand it is possible, thanks to the definition of $\rho$, to
compute explicitly

$-\int\log\nu_{y}\left(f^{-1}\xi(y)\right)d\mu(y)=\int\log J^{u}\,d\mu.$

By hypothesis these two quantities are equal. ∎

####  D.3. Ledrappier–Young II

The second part of Ledrappier–Young’s work focuses on finding a general
entropy formula for measures which are not SRB. The formula they find is
similar, the role of the multiplicities being replaced by some quantities
$\gamma^{j}$ representing roughly the dimension of the measure $\mu$ in the
$E^{j}$-direction. They prove the following theorem.

######  Theorem D.13 (Ledrappier–Young II).

Let $f$ be a $C^{1+\beta}$ diffeomorphism of a compact Riemannian manifold $M$
and $\mu$ be an ergodic, $f$-invariant probability measure. Let $\delta_{j}$
denote the dimension of $\mu$ along the $j$-th dimensional unstable manifolds
and

$\gamma^{j}=\delta^{j}-\delta^{j-1}$

(with $\delta_{0}=0$). Then

$h_{\mu}(f)=\sum_{\lambda^{j}>0}\gamma^{j}\lambda^{j}.$

As for Theorem D.1, F. Ledrappier and L.-S. Young proved Theorem D.13 only for
$C^{2}$ diffeomorphisms, but again, after the works of A. Brown [B] and of L.
Barreira, Ya. Pesin and J. Schmeling [BPS, Appendix] establishing the required
Lipschitz regularity of holonomies of intermediate foliations, it holds for
$C^{1+\beta}$ diffeomorphisms.

Before entering into the proof of Theorem D.13 we need to introduce some of
the objects appearing in the statement.

#####  D.3.1. Nested foliations and Hausdorff dimension

##### Unstable foliations

For a regular point $x\in M$ let

$\lambda_{1}(x)>\lambda_{2}(x)>\cdots>\lambda_{k^{u}(x)}(x)>0$

be the positive Lyapunov exponents of $x$. They are associated with a
splitting of the tangent space

$T_{x}M=E^{1}(x)\oplus E^{2}(x)\oplus\cdots\oplus E^{k^{u}(x)}(x).$

We assume that $\mu$ is ergodic so $k^{u}(x)$, $m^{j}(x)=\dim E^{j}(x)$ and
$\lambda^{j}(x)$ do not depend on the $\mu$-typical $x$. The intermediate
spaces $E^{2}(x),\ldots,E^{k^{u}}(x)$ need not to be integrable. But by
Pesin’s theory there exist $C^{2}$-immersed manifolds at $\mu$ a.e. $x$
denoted by $W^{1}(x),W^{2}(x),\ldots,W^{k^{u}}(x)$, tangent at $x$ to the
spaces

$\displaystyle E^{1}(x)$ $\displaystyle E^{1}(x)\oplus E^{2}(x)$
$\displaystyle\vdots$ $\displaystyle E^{1}(x)\oplus E^{2}(x)\oplus\cdots\oplus
E^{k^{u}}(x).$

These manifolds are dynamically determined as follows

$W^{j}(x)=\left\\{y\in M;\,\mathop{\overline{\rm
lim}}_{n\to\infty}\frac{1}{n}\log
d(f^{-n}(x),f^{-n}(y))\leq-\lambda^{j}\right\\}.$

They form (a.e.) nested foliations $W^{1},\ldots,W^{k^{u}}$, where $W^{k^{u}}$
is the unstable foliation $W^{u}$ we have already worked with. The foliation
$W^{j}$ will be referred to as the $j$-th unstable foliation.

##### Pointwise dimension of measures

Let $X$ be a metric space and $m$ be a probability measure on $X$. Recall the
following classical fact. A proof may be found in [Yo1].

######  Definition D.14.

Say the dimension of $m$, denoted by $\dim m$, is well defined and equal to
$\alpha$ if for $m$-a.e. point $x$ the following limit is well defined

$\alpha=\lim_{\mathchar 271\relax\to 0}\frac{\log\,m(B(x,\mathchar
271\relax))}{\log\mathchar 271\relax}.$

In that case the dimension of $m$ coincides with its Hausdorff dimension, i.e.

$\dim m={\rm HD}(m)=\inf_{m(Y)=1}{\rm HD}(Y).$

We can adopt this viewpoint and study the dimension of an ergodic measure
along unstable manifolds. Using Ledrappier–Strelcyn’s argument (see
Proposition D.3) one deduces for every $j$ the existence of a measurable
partition $\xi^{j}$ subordinate to the $j$-th unstable manifold $W^{j}$. We
can moreover ask

$\xi^{1}\succ\xi^{2}\succ\ldots\succ\xi^{k^{u}}.$

For $j\in\\{1,\ldots,k^{u}\\}$ we denote by $d^{j}$ the distance in $j$-th
unstable leaves for the induced Riemannian structure. The corresponding balls
are denoted by $B^{j}(x,r)$ and the dynamical balls (by analogy with (D.4) in
Section D.1.3) are denoted by $B^{j}_{n}(x,r)$, $r>0$ and $n\in\mathbb{N}$.

We can consider for every $j$ a system $(\mu^{j}_{x})_{x\in M}$ of conditional
measures of $\mu$ associated with $\xi^{j}$. We define for $\mu$-a.e. $x\in M$

$\underline{\delta}^{j}(x,\xi^{j})=\mathop{\underline{\rm lim}}_{\mathchar
271\relax\to 0}\frac{\log\mu(B^{j}(x,\mathchar 271\relax))}{\log\mathchar
271\relax},$ $\overline{\delta}^{j}(x,\xi^{j})=\mathop{\overline{\rm
lim}}_{\mathchar 271\relax\to 0}\frac{\log\mu(B^{j}(x,\mathchar
271\relax))}{\log\mathchar 271\relax}.$

F. Ledrappier and L.-S. Young prove in [LYII, Proposition 7.3.1] the
following.

######  Proposition D.15.

The numbers $\underline{\delta}^{j}=\underline{\delta}^{j}(x,\xi^{j})$ and
$\overline{\delta}^{j}=\overline{\delta}^{j}(x,\xi^{j})$ don’t depend on
$\xi^{j}$ nor on the $\mu$-typical $x$. Moreover

$\underline{\delta}^{j}=\overline{\delta}^{j}.$

If $\delta^{j}$ is the common value then for $\mu$-a.e. $x\in M$ and every
system of conditional measures $(\mu_{x}^{j})_{x\in M}$ associated with a
measurable partition $\xi^{j}$ subordinate to $W^{j}$ we have

$\delta^{j}=\lim_{\mathchar 271\relax\to 0}\frac{\log\mu(B^{j}(x,\mathchar
271\relax))}{\log\mathchar 271\relax}.$

The number $\delta^{j}$ is called the dimension of $\mu$ on the $j$-th
unstable foliation.

##### Pointwise entropies

F. Ledrappier and L.-S. Young also define a pointwise version of the entropy
along the $j$-th unstable foliation. Considering a partition $\xi_{j}$
subordinate to $W^{j}$ and a system of conditional measures
$(\mu^{j}_{x})_{x\in M}$ we define

$\underline{h}^{j}_{\mu}(f,x,\xi^{j})=\lim_{r\to 0}\mathop{\underline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{j}(B_{n}^{j}(x,r)),$

and

$\overline{h}_{\mu}(f,x,\xi^{j})=\lim_{r\to 0}\mathop{\overline{\rm
lim}}_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{j}(B_{n}^{j}(x,r)).$

They proved a result analogue to Theorem D.9:

$\underline{h}^{j}_{\mu}(f,x,\xi^{j})=\overline{h}^{j}_{\mu}(f,x,\xi^{j})=\lim_{r\to
0}\lim_{n\to\infty}-\frac{1}{n}\log\,\mu_{x}^{j}(B^{j}_{n}(x,r)),$

and the common value is

$H_{\mu}(\xi^{j}\mid f\xi^{j})=h^{j}_{\mu}(f).$

#####  D.3.2. Idea of proof for Theorem D.13

##### Conformal case

Before explaining the idea of the proof, let us mention the simplest case.

######  Theorem D.16 (The conformal case).

Let $f:M\to M$ be a $C^{2}$ mapping and $\mu$ an ergodic $f$-invariant
probability measure. Assume that $f$ has a unique Lyapunov exponent
$\lambda>0$, $\mu$-a.e. Then

$h_{\mu}(f)=\lambda\dim(\mu).$

###### Proof.

Let us give the main idea. The rigorous proof uses Pesin’s theory and in
particular, Mañé’s argument.

Let $n\in\mathbb{N}$ and $\mathchar 271\relax>0$. Locally, $f$ “looks like” an
expansion by $e^{\lambda}$ and the dynamical ball $B_{n}(x,r)$ looks like a
ball $B(x,re^{-\lambda n})$ (this affirmation is the one that needs Pesin’s
theory to be made rigorous). So if $\mathchar 271\relax=re^{-\lambda n}$ we
have

$-\frac{1}{n}\log\mu(B_{n}(x,r))\sim\lambda\frac{\log\mu(B(x,\mathchar
271\relax))}{\log\mathchar 271\relax}.$

The formula follows. ∎

Of course if $f$ has various Lyapunov exponents, a dynamical ball looks like
an ellipsoid whose eccentricity tends to infinity and the previous argument
doesn’t work as easily. The precise formula is then given by Ledrappier–Young.

#####  D.3.3. Global strategy

The proof of Theorem D.13 follows three steps.

  1. (1)

$h^{1}=\lambda^{1}\delta^{1}$;

  2. (2)

$h^{j}-h^{j-1}=\lambda^{j}(\delta^{j}-\delta^{j-1})=\lambda^{j}\gamma^{j}$;

  3. (3)

$h^{k^{u}}=h^{u}=h_{\mu}(f)$.

The first case is analogous to the conformal case. In restriction to the first
unstable manifolds, there is only one Lyapunov exponent and $f$ looks like an
expansion by $e^{\lambda^{1}}$. Here again one has to use Pesin’s theory to
make this idea rigorous.

In order to consider the second case, one has to collapse $W^{j-1}$ inside
$W^{j}$ and to consider “quotient dynamics” on the quotient space
$W^{j}/W^{j-1}$ (one rather works with quotient partitions
$\xi^{j}/\xi^{j-1}$). Once again one of the main technical issues is that
there is no canonical “transverse distance” on the quotient, and one has to
prove and use the fact that $W^{j-1}$-holonomies are Lipschitz inside $W^{j}$.
Once we manage to deal with these important technicalities, we see that the
quotient dynamics induced by $f$ on $\xi^{j}/\xi^{j-1}$ has a unique Lyapunov
exponent (this is $\lambda^{j}$) and that the corresponding entropy and
dimension are respectively $h^{j}-h^{j-1}$ and $\delta^{j}-\delta^{j-1}$. The
situation is one more time analogous to the conformal case.

The third case is treated by Theorem D.10. Summing those equalities, the
second Ledrappier–Young’s theorem follows.

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Sébastien Alvarez
CMAT, Facultad de Ciencias, Universidad de la República
Igua 4225 esq. Mataojo. Montevideo, Uruguay.
<EMAIL_ADDRESS>
Mario Roldán
Departamento de Matemática, Universidade Federal de Santa Catarina
Campus Universitário Trindade,
Florianópolis - SC - Brazil CEP 88.040-900
<EMAIL_ADDRESS>
### Index

  * action
    * affine Example 10.5
    * Anosov Definition 1.3
    * exotic Example 10.10, Example 10.12
    * finite Definition 10.4
    * group §1.1
    * isometric Example 10.9
    * projective Example 10.7
    * volume-preserving §1.1
  * Anosov
    * action Definition 1.3
    * diffeomorphism Definition 1.1
    * structural stability §1.2
  * Cartan flow §13.2, §21
  * coarse Lyapunov
    * exponent §7.4.2
      * fiberwise §14.2
    * manifold §7.4.3
      * fiberwise §15.1
    * subspace §7.4.2
  * conjecture
    * Furstenberg Conjecture 2.3
    * Margulis §21
    * Zimmer §1.3.3, §11.2, Conjecture 11.3, Conjecture
  * conjugacy §1.2
    * semi- §20
  * entropy of a partition §8.1.2
    * conditional §D.1.2, §8.1.2
  * entropy of a transformation §8.1.3
    * conditioned on a foliation Definition 8.3
    * conditioned on an orbit §9.1
    * fiberwise §8.2
    * unstable Definition D.7, §8.3.3
  * ergodic decomposition Theorem B.11, Definition 6.5
  * Følner sequence Remark 18.2
  * group
    * action §1.1
    * amenable Remark 18.2
  * Hopf argument Theorem C.4
  * invariance principle
    * Avila–Viana Remark 15.4, Remark 9.6
    * Ledrappier Remark 9.6
    * low dimensional fibers Theorem 15.1
    * nonresonant roots §15.1
  * Lie group §10.1
    * (semi-)simple §10.1
    * higher-rank §10.2
    * Iwasawa decomposition §10.2
    * lattice §1.3, §10.1
      * cocompact §10.1
      * nonuniform §10.1
    * parabolic subgroup Remark 10.8
    * rank §10.2
    * root subgroup §13.1
    * roots §13.1
      * nonresonant §15.1
      * resonant §15.1
  * linear cocycle
    * cohomologous §12.1
    * constant §12.1
    * measurable §12.1
    * superrigidity §12.2
  * Lyapunov exponent §7.1, §7.3
    * coarse §7.4.2
    * fiberwise §14.2, item 3
    * functionals Definition 3.1, §7.3
    * top §7.1
      * fiberwise §18.1
  * Lyapunov manifold §3.2
    * coarse §7.4.3
  * Margulis superrigidity §12.3
    * compact codomain §12.3
  * Margulis–Ruelle inequality Appendix D, §8.4
  * measure(s)
    * conditional §B.5, Definition B.2, §3.3, Definition 8.1
    * disintegration Definition B.2
    * equivalent §5.1
    * ergodic Remark 2.4
    * leaf-wise §3.3
    * physical §9
    * proportional §3.3, §5.1
    * SRB Appendix D, Definition 8.5
    * support §5.1
  * partition
    * increasing item 2, §8.3.3
    * join §B.3, §8.1.2
    * measurable Definition B.7, §8.1.1
    * measurable hull §C.1, §8.6.1
    * partial order §B.3, §8.3.2
    * Pinsker Definition C.1, §8.6.2
      * $\pi$-partition trick Lemma 6.6
    * subordinate §D.1.1, §3.3, Definition 8.2
  * Pesin
    * entropy formula Appendix D, §8.4
    * set Remark D.5
    * unstable manifold §7.4.1
  * rigidity §1.5
    * global item 2, §1.3.5
    * local item 1, Definition 1.2
    * measure item 4
    * structural stability §1.2
  * strong property (T) §16.2
  * suspension §14.1, §6.1, §6.4
  * theorem
    * Abramov–Rokhlin Theorem 15.7, §8.2
    * Birkhoff’s ergodic Theorem B.12
    * Brin–Katok Theorem D.8
    * Furstenberg Theorem A.1
    * Katok–Spatzier Theorem 2.8
    * Ledrappier Theorem 9.3
    * Ledrappier–Strelcyn Proposition D.3, §8.4
    * Ledrappier–Young Theorem C.3, Theorem D.1, Theorem D.10, Theorem D.13, Theorem 8.7
    * Margulis superrigidity §12.3
    * Oseledec Theorem 7.1
      * higher-rank Theorem 7.4
    * Ratner measure classification and equidistribution §18.2
    * Rokhlin disintegration Theorem B.8
    * Rudolph Theorem 2.5
    * subadditive ergodic §7.1
    * Zimmer cocycle superrigidity §12.2
  * uniform subexponential growth of derivatives Definition 16.1, §7.2
  * unstable jacobian §D.2
  * Zimmer
    * cocycle superrigidity §1.3.2, §12.2
    * conjecture §1.3.3, §11.2, Conjecture 11.3
    * program §1.3.2