$\displaystyle=\tilde{\nu}^{i}_{x^{\prime}}\left(B^{E^{i}}_{1}(0)\right)=1.$

In particular, we have that
$(T_{v})_{*}\tilde{\nu}^{i}_{x}=(T_{v})_{*}\tilde{\nu}^{i}_{x^{\prime}}=(\Phi_{x}^{-1}\circ\Phi_{x^{\prime}})_{*}\tilde{\nu}^{i}_{x^{\prime}}\,\propto\,\tilde{\nu}^{i}_{x}$
with explicit coefficient of proportionality:
$\displaystyle(T_{v})_{*}\tilde{\nu}^{i}_{x}=\dfrac{1}{\tilde{\nu}^{i}_{x}(B^{E^{i}}_{1}(v))}\tilde{\nu}^{i}_{x}.$

####  6.4. Overcoming lack of ergodicity: the $\pi$-partition trick

The proof of Lemma 6.3 seems to fail if the measure $\tilde{\mu}$ is not
$\phi_{t}$-ergodic. Indeed, we used that the assignment
$x\mapsto\tilde{\nu}^{i}_{x}$ was $\phi_{t}$-invariant to conclude that the
assignment $x\mapsto\tilde{\nu}^{i}_{x}$ was constant in order to conclude
that $\tilde{\nu}^{i}_{x}=\tilde{\nu}^{i}_{x^{\prime}}$ for
$\tilde{\mu}^{i}_{x}$-typical $x^{\prime}\in W^{i}(x)$ in (6.1).

We recall the following constructions and definitions. See also Theorem B.11
in Appendix B.4 and [MR1086631, Section 3.5].

######  Definition 6.5.

Let $f\colon X\to X$ be a Borel map of a metric space $X$ preserving a Borel
probability measure $\mu$. Then, there exists a measurable partition
$\mathcal{E}$ of $(X,\mu)$ such that—writing $\\{\mu^{\mathcal{E}}_{x}\\}$ for
a family of conditional measures of $\mu$ relative to $\mathcal{E}$ (see
Definition 8.1)—for $\mu$-a.e. $x$ the measure $\mu^{\mathcal{E}}_{x}$ is an
ergodic, $f$-invariant Borel probability measure. The partition $\mathcal{E}$
is called the ergodic decomposition or the partition into ergodic components
of $\mu$ with respect to $f$. The measures $\\{\mu^{\mathcal{E}}_{x}\\}$ are
called the ergodic components of $\mu$.

To illustrate the most extreme defect when ergodicity fails, for a typical
$x$, it could be that the conditional measure along $W^{i}(x)$ of the
$\phi_{t}$-ergodic component $\tilde{\mu}^{\mathcal{E}}_{x}$ of $\tilde{\mu}$
containing $x$ is an atom at the point $x$. Then, the only point
$x^{\prime}\in W^{i}(x)$ for which one could conclude that
$\tilde{\nu}^{i}_{x}=\tilde{\nu}^{i}_{x^{\prime}}$ would be $x^{\prime}=x$.

We now correct the proof of Lemma 6.3. This requires tools and notation
discussed in Section 8.6 below which we encourage the reader to read first.

Examining the proof of Lemma 6.3, we did not fully use that the assignment
$x\mapsto\tilde{\nu}^{i}_{x}$ was constant. Rather, we used that the
assignment $x\mapsto\tilde{\nu}^{i}_{x}$ was constant along the support of
$\tilde{\mu}^{i}_{x}$ in $W^{i}(x)$. From this we concluded that
$\tilde{\nu}^{i}_{x}=\tilde{\nu}^{i}_{x^{\prime}}$ for $\tilde{\mu}$-typical
$x$ and $\tilde{\mu}^{i}_{x}$-typical $x^{\prime}$ in $W^{i}(x)$.

Recall that if $f\colon N\to\mathbb{R}$ is a $\phi_{t}$-invariant, measurable
function then $f$ is constant on almost every $\phi_{t}$-ergodic component of
$\tilde{\mu}$. Thus, the proof of Lemma 6.3 above works if we establish that
almost every $\phi_{t}$-ergodic component of $\tilde{\mu}$ is “saturated” by
full $W^{i}$-manifolds. The precise statement appears in the following lemma,
known as the “$\pi$-partition trick.” We refer to Section 8.6 and Appendix C
for details of the $\pi$-partition and measurable hulls. From entropy
considerations in Lemma 3.6, we have that the partition of $N$ into full
$W^{i}$-leaves is not measurable. We let $\Xi^{i}$ denote the measurable hull
of the partition of $(N,\tilde{\mu})$ into full $W^{i}$-leaves (see Section
8.6.1). Also see Section 8.3.2 for the definition of the partial order on the
space of partitions.

######  Lemma 6.6 ($\pi$-partition trick).

$\Xi^{i}$ is finer than the partition of $(N,\tilde{\mu})$ into
$\phi_{t}$-ergodic components.

Note that if the $W^{i}$-leaves were expanded (or contracted) by $\phi_{t}$,
then the conclusion of Lemma 6.6 would follow from Proposition 8.12 below.
However, we chose $\phi_{t}$ precisely so that it neither expands nor
contracts $W^{i}$-leaves.

It follows from Lemma 6.6 that almost every $\phi_{t}$-ergodic component
contains full $W^{i}$-leaves and hence the proof of Lemma 6.3 works by
replacing $\tilde{\mu}$ with a $\phi_{t}$-ergodic component of $\tilde{\mu}$.

We complete the proof of Theorem 2.8 by giving the proof of Lemma 6.6. For
$\mathbf{s}\in\mathbb{R}^{2}$ and $x\in N$ recall that $W^{s}_{\mathbf{s}}(x)$
and $W^{u}_{\mathbf{s}}(x)$ denote the stable and unstable manifolds,
respectively, for the 1-parameter flow $\alpha(t\mathbf{s})$. We let
$\Xi^{u}_{\mathbf{s}}$ denote the measurable hull (see 8.6.1 below) of the
partition of $(N,\tilde{\mu})$ into full $W^{u}_{\mathbf{s}}$-leaves.
Similarly $\Xi^{s}_{\mathbf{s}}$ denotes the measurable hull of the partition
of $(N,\tilde{\mu})$ into full $W^{s}_{\mathbf{s}}$-leaves.

Given $\mathbf{s}\in\mathbb{R}^{2}$, let $\mathcal{E}_{\mathbf{s}}$ denote the
measurable partition of $(N,\tilde{\mu})$ into ergodic components of the
1-parameter flow $\alpha(t\mathbf{s})$. Similarly, let $\pi_{\mathbf{s}}$
denote the Pinsker partition (see 8.6.2 below) for the 1-parameter flow
$\alpha(t\mathbf{s})$ on $(N,\tilde{\mu})$. As stated in Proposition 8.12
below, for any $\mathbf{s}\in\mathbb{R}^{2}$ we have

$\mathcal{E}_{\mathbf{s}}\prec\Xi^{s}_{\mathbf{s}}.$

From Proposition 8.13 below, we have for any $\mathbf{s}\in\mathbb{R}^{2}$
that

$\Xi^{u}_{\mathbf{s}}=\pi_{\mathbf{s}}=\Xi^{s}_{\mathbf{s}}.$

(See 8.3.2 for definition of the partial order on space of partitions.)

With the above notation, the conclusion of Lemma 6.6 states, for our fixed
$\mathbf{s}_{0}$, that

$\mathcal{E}_{\mathbf{s}_{0}}\prec\Xi^{i}.$

Given the abstract ergodic theoretic facts above, the proof of Lemma 6.6 is
remarkably straightforward.

Figure 3. Choice of $s_{1}$

###### Proof of Lemma 6.6.

Recall we chose $\mathbf{s}_{0}\neq(0,0)$ so that
$\lambda^{i}(\mathbf{s}_{0})=0$ and $\lambda^{j}(\mathbf{s}_{0})\neq 0$ for
each $j\neq i$. Pick $\mathbf{s}_{1}\in\mathbb{R}^{2}$ close to
$\mathbf{s}_{0}$ with the following properties (see Figure 3):

  1. (1)

$\lambda^{j}(\mathbf{s}_{1})\neq 0$ for every $1\leq j\leq 3$;

  2. (2)

$\lambda^{i}(\mathbf{s}_{1})<0$;

  3. (3)

for each $j\neq i$, the numbers $\lambda^{j}(\mathbf{s}_{1})$ and
$\lambda^{j}(\mathbf{s}_{0})$ have the same sign.

Since $\lambda^{i}(\mathbf{s}_{1})<0$, it follows that $W^{i}(x)\subset
W^{s}_{\mathbf{s}_{1}}(x)$ for every $x\in N$. This immediately implies

$\Xi^{s}_{\mathbf{s}_{1}}\prec\Xi^{i}.$

Also, note that $W^{u}_{\mathbf{s}_{1}}(x)=W^{u}_{\mathbf{s}_{0}}(x)$ for all
$x\in N$ whence $\Xi^{u}_{\mathbf{s}_{0}}=\Xi^{u}_{\mathbf{s}_{1}}$. We then
obtain the following string of refinements and equalities:

$\mathcal{E}_{\mathbf{s}_{0}}\prec\Xi^{s}_{\mathbf{s}_{0}}=\pi_{\mathbf{s}_{0}}=\Xi^{u}_{\mathbf{s}_{0}}=\Xi^{u}_{\mathbf{s}_{1}}=\pi_{\mathbf{s}_{1}}=\Xi^{s}_{\mathbf{s}_{1}}\prec\Xi^{i}.$

In symbols, this is exactly what we needed to prove. ∎

Lemma 6.6 completes the proof of Theorem 2.8. We end this section with some
technical remarks on the proof of Lemma 6.6.

######  Remark 6.7.

In the proof of Lemma 6.6, to choose $\mathbf{s}_{1}$ satisfying condition
(3), we heavily used the fact that there is no $\lambda^{j}$ with
$\lambda^{j}=-c\lambda^{i}$ for any $c>0$; that is, we are using that the
action has no Lyapunov exponents that are negatively proportional to
$\lambda^{i}$. Indeed, if $\lambda^{j}=-c\lambda^{i}$ then, for any choice of
$\mathbf{s}_{1}$ such that $\lambda^{i}(\mathbf{s}_{1})<0$, the sign of
$\lambda^{j}$ changes from zero at $\mathbf{s}_{0}$ to positive at
$\mathbf{s}_{1}$. On the other hand, if $\lambda^{j}$ and $\lambda^{i}$ were
positively proportional, we can adapt the proof by grouping all exponents
positively proportional to $\lambda^{i}$ together into a single coarse
Lyapunov exponent (see Section 7.4.2) and then study the leaf-wise measures
along higher-dimensional coarse Lyapunov manifolds (see Section 7.4.3.)

For an explicit example of a higher-rank action where the proof of Lemma 6.6
(and consequently Theorem 2.8) fails, let

$A=\left(\begin{array}[]{cc}2&1\\\ 1&1\end{array}\right)\quad\text{and}\quad
B=\left(\begin{array}[]{cc}2&3\\\ 3&5\end{array}\right).$

We have eigenvalues

$\chi^{1}_{A}>1>\chi^{2}_{A},\quad\quad\chi^{1}_{B}>1>\chi^{2}_{B}>0$

and both $L_{A}\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ and
$L_{B}\colon\mathbb{T}^{2}\to\mathbb{T}^{2}$ are Anosov. Consider

$L_{A}\times\mathrm{Id}\colon\mathbb{T}^{4}\to\mathbb{T}^{4},\quad\quad\mathrm{Id}\times
L_{B}\colon\mathbb{T}^{4}\to\mathbb{T}^{4}.$

Clearly $L_{A}\times\mathrm{Id}$ and $\mathrm{Id}\times L_{B}$ commute and so
generate a $\mathbb{Z}^{2}$-action by automorphisms of $\mathbb{T}^{4}$. Note
that while the generators $L_{A}\times\mathrm{Id}$ and $\mathrm{Id}\times
L_{B}$ are not Anosov, the $\mathbb{Z}^{2}$-action contains Anosov
diffeomorphisms such as $L_{A}\times L_{B}$.

Let $\mu_{1}$ be any ergodic, $L_{A}$-invariant measure on $\mathbb{T}^{2}$
and let $\mu_{2}$ be any ergodic, $L_{B}$-invariant measure on
$\mathbb{T}^{2}$. Note that we can pick $\mu_{1}$ different from Lebesgue with
$h_{\mu_{1}}(L_{A})>0$. Then $\mu_{1}\times\mu_{2}$ is an ergodic,
$\mathbb{Z}^{2}$-invariant measure on $\mathbb{T}^{4}$ that is not Lebesgue
and has

$h_{\mu_{1}\times\mu_{2}}(L_{A}\times\mathrm{Id})>0.$

The conclusion of Theorem 2.8 thus fails for this $\mathbb{Z}^{2}$-action on
$\mathbb{T}^{4}$. The proof degenerates in a number of places.

  1. (1)

The $\mathbb{Z}^{2}$-action on $\mathbb{T}^{4}$ has four Lyapunov exponent
functionals:

$(n_{1},n_{2})\mapsto n_{1}\lambda^{1}_{A},\quad n_{1}\lambda^{2}_{A},\quad
n_{2}\lambda^{1}_{B},\quad n_{2}\lambda^{2}_{B}.$

We note that the functionals

$(n_{1},n_{2})\mapsto n_{1}\lambda^{1}_{A},\quad\quad(n_{1},n_{2})\mapsto
n_{1}\lambda^{2}_{A}$

are negatively proportional. Similarly

$(n_{1},n_{2})\mapsto n_{2}\lambda^{1}_{B},\quad\quad(n_{1},n_{2})\mapsto
n_{2}\lambda^{2}_{B}$

are negatively proportional.222In the literature, negatively proportional
exponents such as $(n_{1},n_{2})\mapsto n_{1}\lambda^{1}_{A}$ and
$(n_{1},n_{2})\mapsto n_{1}\lambda^{2}_{A}$ are often referred to as a
symplectic pair. The presence of negatively proportional Lyapunov exponents
makes it impossible to choose $\mathbf{s}_{1}$ in the proof of Lemma 6.6 above
with the desired properties. In fact, the conclusion of 6.6 is false for this
example.

  2. (2)

Consider the map
$h\colon\mathbb{T}^{4}=\mathbb{T}^{2}\times\mathbb{T}^{2}\to\mathbb{T}^{2}$
given by $h(x,y)=x$. Then $h$ semiconjugates the $\mathbb{Z}^{2}$-action
generated by $L_{A}\times\mathrm{Id}$ and $\mathrm{Id}\times L_{B}$ with the
$\mathbb{Z}$-action generated by $L_{A}$. For any $L_{A}$-invariant measure
$\mu$ on $\mathbb{T}^{2}$, we can find a $\mathbb{Z}^{2}$-invariant measure on
$\mathbb{T}^{4}$ projecting to $\mu$ under $h$. In the language of
[MR1406432], $L_{A}$ is a rank-1 factor.

To state the general version ([MR1406432, Theorem 5.1]) of Theorem 2.8, Katok
and Spatzier impose additional hypotheses on the $\mathbb{Z}^{2}$-action to
rule out the defects discussed above. For the action in Example 2.6, neither
of these defects occurs.

The primary obstruction to the rigidity in this example is the presence of
rank-1 factors. For genuinely higher-rank actions with negatively proportional
pairs, other tools developed in [MR2029471] can be used to overcome the
failure of Lemma 6.6.

$\bigast$6.5. Algebraic construction of the suspension space $N$. We outline
the construction in Proposition 6.1. Recall our fixed commuting matrices $A$
and $B$ are jointly diagonalizable: there is $Q\in\mathrm{GL}(3,\mathbb{R})$
with

$Q^{-1}AQ=\left(\begin{array}[]{ccc}e^{\lambda^{1}_{A}}&0&0\\\
0&e^{\lambda^{2}_{A}}&0\\\
0&0&e^{\lambda^{3}_{A}}\end{array}\right),\quad\quad
Q^{-1}BQ=\left(\begin{array}[]{ccc}e^{\lambda^{1}_{B}}&0&0\\\
0&e^{\lambda^{2}_{B}}&0\\\ 0&0&e^{\lambda^{3}_{B}}\end{array}\right).$

Also recall our Lyapunov exponent functionals
$\lambda^{j}\colon\mathbb{R}^{2}\to\mathbb{R}$, given by

$\lambda^{j}(t_{1},t_{2})=t_{1}\lambda^{j}_{A}+t_{2}\lambda^{j}_{B}.$

Given $\mathbf{t}=(t_{1},t_{2})\in\mathbb{R}^{2}$ let
$M^{\mathbf{t}}\in\mathrm{GL}(3,\mathbb{R})$ be the interpolation matrix

$M^{\mathbf{t}}=Q\left(\begin{array}[]{ccc}e^{\lambda^{1}(t_{1},t_{2})}&0&0\\\
0&e^{\lambda^{2}(t_{1},t_{2})}&0\\\
0&0&e^{\lambda^{3}(t_{1},t_{2})}\end{array}\right)Q^{-1}.$

Note that for $\mathbf{t}=(n_{1},n_{2})\in\mathbb{Z}^{n}$,
$M^{\mathbf{t}}=A^{n_{1}}B^{n_{2}}\in\mathrm{GL}(3,\mathbb{Z})$. However, for
$\mathbf{t}\notin\mathbb{Z}^{3}$, we expect
$M^{\mathbf{t}}\notin\mathrm{GL}(3,\mathbb{Z})$; in particular,
$M^{\mathbf{t}}$ does not define an action on the torus $\mathbb{T}^{3}$.

For $\mathbf{t}\in\mathbb{R}^{2}$ define the “twisted” lattice subgroup
$\Lambda_{\mathbf{t}}\subset\mathbb{R}^{3}$ by

$\Lambda_{\mathbf{t}}=M^{\mathbf{t}}\mathbb{Z}^{3}.$

Note that if $\mathbf{m}\in\mathbb{Z}^{2}$ then $\Lambda_{\mathbf{m}}$ is the
standard integer lattice $\mathbb{Z}^{3}$. For $\mathbf{t}\in\mathbb{R}^{2}$
define a “twisted torus” $T_{\mathbf{t}}$ by

$T_{\mathbf{t}}=\mathbb{R}^{3}/\Lambda_{\mathbf{t}}.$

Note that if $\mathbf{m}\in\mathbb{Z}^{2}$ then $T_{\mathbf{m}}$ is the
standard torus $\mathbb{T}^{3}$; also if
$\mathbf{t}^{\prime}=\mathbf{t}+\mathbf{m}$ where
$\mathbf{m}\in\mathbb{Z}^{2}$ then
$\Lambda_{\mathbf{t}}=\Lambda_{\mathbf{t}^{\prime}}$ whence
$T_{\mathbf{t}}=T_{\mathbf{t}^{\prime}}$.

Consider $\mathbb{R}^{2}\times\mathbb{R}^{3}$. Let $\mathbb{Z}^{3}$ act on
$\mathbb{R}^{2}\times\mathbb{R}^{3}$ as follows: given
$(\mathbf{t},x)\in\mathbb{R}^{2}\times\mathbb{R}^{3}$ and
$\mathbf{n}\in\mathbb{Z}^{3}$ define

$(\mathbf{t},x)\cdot\mathbf{n}=(\mathbf{t},x+M^{\mathbf{t}}\mathbf{n}).$

Let $\tilde{N}$ be the quotient of $\mathbb{R}^{2}\times\mathbb{R}^{3}$ by
this action. Note that $\tilde{N}$ is a fiber-bundle over $\mathbb{R}^{2}$
whose fiber over $\mathbf{t}$ is exactly the twisted torus $T_{\mathbf{t}}.$

Consider the following $\mathbb{Z}^{2}$-action on $\tilde{N}$: given
$\mathbf{m}\in\mathbb{Z}^{2}$ and
$(\mathbf{t},x+\Lambda_{\mathbf{t}})\in\tilde{N}$

$(\mathbf{t},x+\Lambda_{\mathbf{t}})\cdot\mathbf{m}=(\mathbf{t}+\mathbf{m},x+\Lambda_{\mathbf{t}}).$

Also consider the following $\mathbb{R}^{2}$-action on $\tilde{N}$: given
$\mathbf{s}\in\mathbb{R}^{2}$ and
$(\mathbf{t},x+\Lambda_{\mathbf{t}})\in\tilde{N}$

$\mathbf{s}\cdot(\mathbf{t},x+\Lambda_{\mathbf{t}})=(\mathbf{s}+\mathbf{t},M^{\mathbf{s}}x+\Lambda_{\mathbf{t}+\mathbf{s}}).$

Let $N$ be the quotient manifold $\tilde{N}/\mathbb{Z}^{2}$ where the quotient
is by the $\mathbb{Z}^{2}$-action described above. As the $\mathbb{R}^{2}$\-
and $\mathbb{Z}^{2}$-actions on $\tilde{N}$ commute, the
$\mathbb{R}^{2}$-action on $\tilde{N}$ descends to an $\mathbb{R}^{2}$-action
on $N$ which we denote by $\tilde{\alpha}$.

Note that $N$ is a $5$-dimensional manifold which fibers over the torus
$\mathbb{T}^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ and has $3$-dimensional fibers
where the fiber over $\mathbf{t}+\mathbb{Z}^{2}\in\mathbb{T}^{2}$ is
$T_{\mathbf{t}}.$ Also note that $N$ inherits a Riemannian metric from
$\mathbb{R}^{2}\times\mathbb{R}^{3}$. Given $x\in N$ the tangent space
$T_{x}N$ decomposes as

$T_{x}N=\mathbb{R}^{2}\oplus E^{1}\oplus E^{2}\oplus E^{3}$

where $E^{i}$ are the joint eigenspaces of $A$ and $B$ enumerated as before.

Each $E^{i}$ acts on $N$ as follows: given
$p=(\mathbf{t}+\mathbb{Z}^{2},x+\Lambda_{\mathbf{t}})\in N$ and $v\in E^{i}$

$p+v=(\mathbf{t}+\mathbb{Z}^{2},x+\Lambda_{\mathbf{t}})+v=(\mathbf{t}+\mathbb{Z}^{2},x+v+\Lambda_{\mathbf{t}}).$

Note that the orbit $W^{i}(p)=\\{p+v:v\in E^{i}\\}$ is contained in the fiber
through $p$. Moreover, for any
$p=(\mathbf{t}+\mathbb{Z}^{2},x+\Lambda_{\mathbf{t}})\in N$, any
$p^{\prime}=p+v$ for $v\in E^{i}$, and any
$\mathbf{s}=(s_{1},s_{2})\in\mathbb{R}^{2}$ we have

$\tilde{\alpha}(\mathbf{s})(p^{\prime})=\tilde{\alpha}(\mathbf{s})(p)+e^{s_{1}\lambda^{i}_{A}+s_{2}\lambda^{i}_{B}}v=\tilde{\alpha}(\mathbf{s})(p)+e^{\lambda^{i}(\mathbf{s})}v.$

##  Part II Primer: smooth ergodic theory for $\mathbb{Z}^{d}$-actions

We present background material and a number of tools from the theory of
nonuniformly hyperbolic dynamics that will be used in Part III and explain a
number of facts and constructions that were used in Part I. We will be
particularly interested in the relation between entropy, conditional measures,
and Lyapunov exponents for single diffeomorphisms and for actions of higher-
rank abelian groups.

###  7\. Lyapunov exponents and Pesin manifolds

####  7.1. Lyapunov exponents for diffeomorphisms

Let $f\colon M\to M$ be a $C^{1}$ diffeomorphism of a compact manifold $M$.
Let $\mu$ be an ergodic, $f$-invariant Borel probability measure. We recall
Oseledec’s Theorem [MR0240280]; see also [MR1073779, MR0571089].

######  Theorem 7.1 (Oseledec [MR0240280]).

There are

  1. (1)

a measurable set $\Lambda$ with $\mu(\Lambda)=1$;

  2. (2)

numbers $\lambda^{1}>\lambda^{2}>\dots>\lambda^{p}$;

  3. (3)

a $\mu$-measurable, $Df$-invariant splitting
$T_{x}M=\bigoplus_{i=1}^{p}E^{i}(x)$ defined for $x\in\Lambda$

such that for every $x\in\Lambda$

  1. (a)

for every $v\in E^{i}(x)\smallsetminus\\{0\\}$

$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\|D_{x}f^{n}(v)\|=\lambda^{i};$

  2. (b)

if $Jf$ denotes the Jacobian determinant of $f$ then

$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log|{J}f^{n}|=\sum_{i=1}^{p}m_{i}\lambda^{i}$

where $m_{i}=\dim E^{i}(x)$;

  3. (c)

for every $i\neq j$ we have

$\displaystyle\lim_{n\to\pm\infty}\frac{1}{n}\log\Big{(}\sin\angle\Big{(}E^{i}(f^{n}(x)),E^{j}(f^{n}(x))\Big{)}\Big{)}=0.$

The numbers $\lambda^{i}$ are called the Lyapunov exponents of $f$ with
respect to $\mu$ and the subspaces $E^{i}(x)$ are called the Oseledec’s
subspaces. Above, $m^{i}$ denotes the almost-surely constant value of $\dim
E^{i}(x)$, called the multiplicity of $\lambda^{i}$.

Given any $f$-invariant measure $\mu$ on $M$ (which may be nonergodic) the
average top Lyapunov exponent of $f$ with respect to $\mu$ is

$\lambda_{\mathrm{top}}(f,\mu)=\inf_{n\geq
1}\frac{1}{n}\int\log\|{D_{x}f^{n}}\|\ d\mu(x).$ (7.1)

Since $\mu$ is $f$-invariant, the sequence $n\mapsto\int\log\|{D_{x}f^{n}}\|\
d\mu(x)$ is subadditive and the infimum in (7.1) can be replaced by a limit.

By the subadditive ergodic theorem, the functions

$x\mapsto\frac{1}{n}\log\|{D_{x}f^{n}}\|$

converge a.e. to an invariant, integrable function with integral
$\lambda_{\mathrm{top}}(f,\mu)$; see [MR0121828, MR0254907] and [MR3289050,
Chapter 3]. If $\mu$ is ergodic, we have in the notation of Theorem 7.1 that
$\lambda_{\mathrm{top}}(f,\mu)=\lambda^{1}$. If $\mu$ is not ergodic, let
$\\{\mu^{\mathcal{E}}_{x}\\}$ denote the ergodic decomposition (see Definition
6.5) of $\mu$ and let
$\lambda^{1}_{x}>\lambda_{x}^{2}>\dots>\lambda^{p(x)}_{x}$ denote the Lyapunov
exponents of $f$ with respect to the ergodic invariant measure
$\mu^{\mathcal{E}}_{x}$. Then we have

$\lambda_{\mathrm{top}}(f,\mu)=\int\lambda^{1}_{x}\ d\mu(x).$

####  7.2. Lyapunov exponents and (sub)exponential growth of derivatives

Let $M$ be a compact manifold and equip $TM$ with a background Riemannian
metric and associated norm. Let $f\colon M\to M$ be a $C^{1}$ diffeomorphism.
We say $f\colon M\to M$ has uniform subexponential growth of derivatives if
for all $\epsilon>0$ there is a $C_{\epsilon}>0$ such that

$\|Df^{n}\|:=\sup_{x\in
M}\|D_{x}f^{n}\|<C_{\epsilon}e^{\epsilon|n|}\quad\quad\text{for all
$n\in\mathbb{Z}.$}$

Note that we allow that $C_{\epsilon}\to\infty$ as $\epsilon\to 0$.

######  Proposition 7.2.

A diffeomorphism $f\colon M\to M$ has uniform subexponential growth of
derivatives if and only if for any $f$-invariant Borel probability measure
$\mu$, all Lyapunov exponents of $f$ with respect to $\mu$ are zero.

That is, $f\colon M\to M$ has uniform subexponential growth of derivatives if
and only if
$\lambda_{\mathrm{top}}(f,\mu)=\lambda_{\mathrm{top}}(f^{-1},\mu)=0$ for every
$f$-invariant Borel probability measure $\mu$.

###### Proof.

We show that vanishing of all Lyapunov exponents for all $f$-invariant
probability measures implies that $f$ has uniform subexponential growth of
derivatives; the converse is clear.

Suppose that $f\colon M\to M$ fails to have uniform subexponential growth of
derivatives. Then there is an $\epsilon>0$ and sequences of iterates
$n_{j}\in\mathbb{Z}$ with $|n_{j}|\to\infty$, base points $x_{j}\in M$, and
unit vectors $v_{j}\in T_{x_{j}}M$ such that

$\|D_{x_{j}}f^{n_{j}}v_{j}\|\geq e^{\epsilon|n_{j}|}.$ (7.2)

Replacing $f$ with $f^{-1}$, we may assume without loss of generality that
$n_{j}\to\infty.$

Let $UM\subset TM$ denote the unit-sphere bundle. We represent an element of
$UM$ by a pair $(x,v)$ where $v\in T_{x}M$ with $\|v\|=1$. Note that $UM$ is
compact. Note also that $Df\colon TM\to TM$ induces a map $Uf\colon U\to U$
given by the renormalized derivative:

$Uf(x,v):=\left(f(x),\frac{D_{x}f(v)}{\|D_{x}f(v)\|}\right).$

Define $\Phi\colon UM\to\mathbb{R}$ as follows: given $(x,v)\in UM$, let

$\Phi(x,v):=\log\|D_{x}f(v)\|.$

By the chain rule, we have

$\log\|D_{x}f^{n}(v)\|=\sum_{j=0}^{n-1}\Phi(Uf^{j}(x,v)).$

For each $j$, let $\nu^{j}$ denote the empirical measure along the orbit
segment

$(x_{j},v_{j}),Uf(x_{j},v_{j}),\dots,Uf^{n_{j}-1}(x_{j},v_{j})$

in $UM$ given by

$\nu^{j}=\frac{1}{n_{j}}\sum_{k=0}^{n_{j}-1}\delta_{Uf^{k}(x_{j},v_{j})}.$

From (7.2) we have for every $j$ that

$\int\Phi\ d\nu^{j}\geq\epsilon.$

######  Claim 7.3.

Let $\nu$ be any weak-$*$ subsequential limit of $\\{\nu^{j}\\}$. Then

  1. (a)

$\nu$ is $Uf$-invariant;

  2. (b)

$\displaystyle\int\Phi\ d\nu\geq\epsilon$.

###### Proof.

Conclusion (a) follows as in the proof of the Krylov-Bogolyubov theorem: if
$\phi\colon M\to\mathbb{R}$ is any (bounded) continuous function then

$\lim_{j\to\infty}\left|\int\phi\ d\nu^{j}-\int\phi\circ f\
d\nu^{j}\right|\leq\lim_{j\to\infty}\frac{2\|\phi\|_{C^{0}}}{n_{j}}=0$ (7.3)

showing that $\nu$ is $f$-invariant. Conclusion (b) follows from continuity of
$\Phi$ and weak-$*$ convergence. ∎

From 7.3(b), we may replace $\nu$ with an ergodic component (see Definition
6.5) $\nu^{\prime}$ of $\nu$ such that $\int\Phi\ d\nu^{\prime}\geq\epsilon$.

Take $\mu$ to be the push-forward of $\nu^{\prime}$ under the natural
projection $UM\to M$. Then $\mu$ is an $f$-invariant, ergodic measure on $M$.
Let $\\{\nu^{\prime}_{x}\\}$ denote a family of conditional measures of
$\nu^{\prime}$ for the partition of $UM$ into fibers over $M$. By the
pointwise ergodic theorem, for $\mu$-a.e. $x\in M$ and $\nu^{\prime}_{x}$-a.e.
$v\in UM(x)$ we have

$\lim_{n\to\infty}\frac{1}{n}\log\|D_{x}f^{n}(v)\|=\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\Phi(Uf^{j}(x,v))=\int\Phi\
d\nu^{\prime}\geq\epsilon.$

On the other hand,

$\displaystyle\lambda_{\mathrm{top}}(f,\mu)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\int\log\|{D_{x}f^{n}}\|\ d\mu(x)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\int\sup_{v\in
UM(x)}\sum_{j=0}^{n-1}\Phi(Uf^{j}(x,v))\ d\mu(x)$
$\displaystyle\geq\lim_{n\to\infty}\int\int\frac{1}{n}\sum_{j=0}^{n-1}\Phi(Uf^{j}(x,v))\
d\nu_{x}(v)\ d\mu(x)$ $\displaystyle=\int\Phi\ d\nu^{\prime}\geq\epsilon.$

Above, the inequality follows from comparing the maximal growth with the
average growth (averaged by $\nu_{x}^{\prime}$.) It follows that the largest
Lyapunov exponent of $f$ with respect to $\mu$ is at least $\epsilon>0$. ∎

####  7.3. Lyapunov exponents for nonuniformly hyperbolic
$\mathbb{Z}^{d}$-actions

How does the theory of Lyapunov exponents change for actions of more general
abelian groups? We state a version of Oseledec’s theorem for actions of
$\mathbb{Z}^{d}$ which can easily be extended to actions of
$\mathbb{R}^{\ell}\times\mathbb{Z}^{k}$. One should think of the following as
a non-stationary version of the joint-diagonalizability or joint-Jordan-
normal-form for commuting matrices as exploited in Section 3.1.

Let $M$ be a compact manifold, let
$\alpha\colon\mathbb{Z}^{d}\to\mathrm{Diff}^{1}(M)$ be a
$\mathbb{Z}^{d}$-action, and let $\mu$ be an ergodic, $\alpha$-invariant
measure.

######  Theorem 7.4 (Higher-rank Oseledec’s theorem (see [AWB-GLY-P1])).

There are

  1. (1)

a measurable set $\Lambda$ with $\mu(\Lambda)=1$;

  2. (2)

linear functionals
$\lambda^{1},\lambda^{2},\dots,\lambda^{p}\colon\mathbb{R}^{d}\to\mathbb{R}$;

  3. (3)

a $\mu$-measurable, $D\alpha$-invariant splitting
$T_{x}M=\bigoplus_{i=1}^{p}E^{i}(x)$ defined for $x\in\Lambda$

such that for every $x\in\Lambda$

  1. (a)

for every $v\in E^{i}(x)\smallsetminus\\{0\\}$

$\lim_{|n|\to\infty}\dfrac{\log\|D_{x}\alpha(n)(v)\|-\lambda^{i}(n)}{|n|}=0;$

  2. (b)

if $Jf$ denotes the Jacobian determinant of $f$ then

$\displaystyle\lim_{|n|\to\infty}\frac{\log|{J}\alpha(n)|-\sum_{i=1}^{p}m^{i}\lambda^{i}(n)}{|n|};$

  3. (c)

for every $i\neq j$

$\lim_{n\to\infty}\frac{1}{|n|}\log\Big{(}\sin\angle\Big{(}E^{i}(\alpha(n)(x)),E^{j}(\alpha(n)(x))\Big{)}\Big{)}=0.$

The linear functionals
$\lambda^{1},\lambda^{2},\dots,\lambda^{p}\colon\mathbb{R}^{d}\to\mathbb{R}$
are called the Lyapunov exponent functionals or simply the Lyapunov exponents
of $\mu$. In (b), $m^{i}$ is the almost-surely constant value of $\dim
E^{i}(x)$, called the multiplicity of $\lambda^{i}$. Note that (a) implies
convergence along rays: for any $n\in\mathbb{Z}^{d}$ and $v\in
E^{i}(x)\smallsetminus\\{0\\}$

$\lim_{k\to\infty}\dfrac{1}{k}\log\|D_{x}\alpha(kn)(v)\|=\lambda^{i}(n).$
(7.4)

The convergence in (a) is taken along any sequence $n\to\infty$; this is
stronger than (7.4) and is typically needed in applications.

####  7.4. Unstable manifolds and coarse Lyapunov manifolds

#####  7.4.1. Unstable subspaces and unstable manifolds for a single
diffeomorphism

Let $f\colon M\to M$ be a $C^{1}$ diffeomorphism of $M$ and let $\mu$ be an
ergodic, $f$-invariant measure. Let $\lambda^{i}$ be the Lyapunov exponents
for $f$ with respect to $\mu$. For $x\in\Lambda\subset M$, where $\Lambda$ is
as in Theorem 7.1, define

$E^{u}(x):=\bigoplus_{\lambda^{i}>0}E^{i}(x)$

to be the unstable subspace through $x$. We have that

$E^{u}(x):=\\{v\in
T_{x}M:\limsup_{n\to\infty}\frac{1}{n}\log\|D_{x}f^{-n}(v)\|<0\\}.$

We may similarly define stable and neutral (or center) subspaces through $x$,
respectively, by

$E^{s}(x):=\bigoplus_{\lambda^{i}<0}E^{i}(x)$

and

$E^{c}(x):=\bigoplus_{\lambda^{i}=0}E^{i}(x).$

We now assume that $f\colon M\to M$ is $C^{1+\beta}$ for $\beta>0$. Through
$\mu$-almost every point $x$ the set

$W^{u}(x):=\left\\{y:\limsup_{n\to\infty}\frac{1}{n}\log(d(f^{-n}(x),f^{-n}(y)))<0\right\\}$

is a connected $C^{1+\beta}$ injectively immersed manifold with
$T_{x}W^{u}(x)=E^{u}(x)$ (see [MR0458490]) called the (global) unstable Pesin
manifold of $f$ through $x$. The collection of all $W^{u}(x)$ forms a
partition of (a full measure subset of) $M$; in general, this partition does
not have the structure of a nice foliation. However, restricted to sets of
large measure the partition into local unstable manifolds has the structure of
a continuous lamination. That is, for almost every $x\in M$ and any
$\epsilon>0$ there is a neighborhood $U$ of $x$ such that, on a set $\Omega$
of relative measure $(1-\epsilon)$ in $U$, the local leaves of
$W^{u}$-manifolds form a partition of $\Omega$ by embedded
$\dim(E^{u})$-dimensional balls that vary continuously in the
$C^{1+\beta}$-topology.

Given the Lyapunov exponents $\lambda^{1}>\lambda^{2}>\dots>\lambda^{p}$ of
$\mu$, fix $j\in\\{1,\cdots,p\\}$ such that $\lambda^{j}>0$. Then, for almost
every $x$, the set

$W^{j}(x):=\left\\{y:\limsup_{n\to\infty}\frac{1}{n}\log(d(f^{-n}(x),f^{-n}(y)))\leq-\lambda^{j}\right\\}$

is again a connected, $C^{1+\beta}$ injectively immersed manifold with

$T_{x}W^{j}(x)=\bigoplus_{\lambda^{i}\geq\lambda^{j}}E^{i}(x)$

called the (global) $j$th unstable manifold through $x$. We remark that, in
general, the intermediate unstable distributions, $E^{i}(x)$ for
$\lambda^{i}>0$, do not integrate to invariant family of immersed
submanifolds.

#####  7.4.2. Coarse Lyapunov exponents and subspaces

Let $\alpha\colon\mathbb{Z}^{d}\to\mathrm{Diff}^{1}(M)$ be an action and let
$\mu$ be an ergodic, $\alpha$-invariant probability measure. We introduce
objects that play the role of unstable subspaces and unstable manifolds for
the $\mathbb{Z}^{d}$-action $\alpha$.

Given Lyapunov exponents
$\lambda^{1},\lambda^{2},\dots,\lambda^{p}\colon\mathbb{R}^{d}\to\mathbb{R}$
we say $\lambda^{i}$ and $\lambda^{j}$ are positively proportional if there is
a $c>0$ with

$\lambda^{i}=c\lambda^{j}.$

Note that this defines an equivalence relation on the linear functionals

$\lambda^{1},\lambda^{2},\dots,\lambda^{p}\colon\mathbb{R}^{d}\to\mathbb{R}.$

The positive proportionality classes are called coarse Lyapunov exponents. For
a $\mathbb{Z}$-action generated by a single diffeomorphism $f$, the coarse
Lyapunov exponents are simply the collections of positive, zero, and negative
Lyapunov exponents.

Let $\chi=\\{\lambda^{i}\\}$ be a coarse Lyapunov exponent. While the size of
$\chi(n)$ is not well defined, the sign of $\chi(n)$ is well defined. Write

$E^{\chi}(x)=\oplus_{\lambda^{i}\in\chi}E^{i}(x)$

called the corresponding coarse Lyapunov subspace.

#####  7.4.3. Coarse Lyapunov manifolds for $\mathbb{Z}^{d}$-actions

Analogous to the existence and properties of unstable Pesin manifolds for
nonuniformly hyperbolic diffeomorphisms we have the following for actions of
higher-rank abelian groups.

Let $\alpha\colon\mathbb{Z}^{d}\to\mathrm{Diff}^{1+\beta}(M)$ be an action and
let $\mu$ be an ergodic, $\alpha$-invariant probability measure. Let $\Lambda$
be as in Theorem 7.4.

######  Proposition 7.5.

For almost every $x\in\Lambda$ and for every coarse Lyapunov exponent $\chi$
there is a connected, $C^{1+\beta}$, injectively immersed manifold
$W^{\chi}(x)$ satisfying the following:

  1. (1)

$T_{x}W^{\chi}(x)=E^{\chi}(x)$;

  2. (2)

$\alpha(n)W^{\chi}(x)=W^{\chi}(\alpha(n)(x))$ for all $n\in\mathbb{Z}^{d}$;

  3. (3)

$W^{\chi}(x)$ is the set of all $y\in M$ satisfying

$\limsup_{k\to\infty}\frac{1}{k}\log d(\alpha(-kn)(y),\alpha(-kn)(x))<0\text{
for all $n\in\mathbb{Z}^{d}$ with $\chi(n)>0$}.$

The manifold $W^{\chi}(x)$ is called the coarse Lyapunov manifold through $x$
associated with the coarse Lyapunov exponent $\chi$.

To construct $W^{\chi}$-manifolds, given $n\in\mathbb{Z}^{d}$ with $\chi(n)>0$
let $W_{\alpha(n)}^{u}(x)$ denote the unstable manifold for the diffeomorphism
$\alpha(n)\colon M\to M$ through $x$. Then, for almost every $x\in M$ the
manifold $W^{\chi}(x)$ is the path component of the intersection

$\bigcap_{n\in\mathbb{Z}^{d},\chi(n)>0}W_{\alpha(n)}^{u}(x)$

containing $x$.

###  8\. Metric entropy

####  8.1. Metric entropy

General references for this subsection include [MR02172581, MR1086631].
Throughout, we take $(X,\mu)$ to be a standard probability space. That is,
$(X,\mu)$ equipped with the $\sigma$-algebra of $\mu$-measurable sets is
measurably isomorphic to an interval equipped with the Lebesgue measure and a
countable number of point masses; see for instance [MR1086631, Chapter 2].

#####  8.1.1. Measurable partitions and conditional measures

Recall that a partition $\xi$ of $(X,\mu)$ is measurable if the quotient
$(Y,\hat{\mu}):=(X,\mu)/\xi$ is a standard probability space. See also
Definition B.7 in Appendix B.3. This is a technical but crucial condition. For
more discussion and other characterizations of measurability see [1208.4550],
[MR0047744], and Appendices B and C.

A key property of measurable partitions is the existence and uniqueness of a
family of conditional measures (or a disintegration) of $\mu$ relative to this
partition. Given a partition $\xi$ of $X$, for $x\in X$ we write $\xi(x)$ for
the element of $\xi$ containing $x$.

######  Definition 8.1.

Let $\xi$ be a measurable partition of $(X,\mu)$. Then there is family of
Borel probability measure $\\{\mu^{\xi}_{x}\\}_{x\in X}$, called a family of
conditional measures of $\mu$ relative to $\xi$, with the following
properties: For almost every $x$

  1. (1)

$\mu_{x}^{\xi}$ is a Borel probability measure on $X$ with
$\mu^{\xi}_{x}(\xi(x))=1$;

  2. (2)

if $y\in\xi(x)$ then $\mu_{y}^{\xi}=\mu_{x}^{\xi}$.

Moreover, if $D\subset X$ is a Borel subset then

  1. (3)

$x\mapsto\mu_{x}^{\xi}(D)$ is measurable and

  2. (4)

$\mu(D)=\int\mu_{x}^{\xi}(D)\ d\mu(x)$.

Such a family is unique modulo $\mu$-null sets.

For construction and properties of $\\{\mu^{\xi}_{x}\\}$ see for instance
[MR0047744]. See also Appendix B.2 for further discussion.

#####  8.1.2. Conditional information and conditional entropy

Given a measurable partition $\xi$ of a standard probability space $(X,\mu)$,
write $\\{\mu_{x}^{\xi}\\}$ for a family of conditional measures of $\mu$ with
respect to the partition $\xi$. Given two measurable partitions $\eta,\xi$ of
$(X,\mu)$ the conditional information of $\eta$ relative to $\xi$ is

$I_{\mu}(\eta\mid\xi)(x)=-\log(\mu^{\xi}_{x}(\eta(x)))$

and the conditional entropy of $\eta$ relative to $\xi$ is

$H_{\mu}(\eta\mid\xi)=\int I_{\mu}(\eta\mid\xi)(x)\ d\mu(x).$

The join $\eta\vee\xi$ of two partitions $\eta$ and $\xi$ is

$\eta\vee\xi=\\{A\cap B\mid A\in\eta,B\in\xi\\}.$

The entropy of $\eta$ is $H_{\mu}(\eta)=H_{\mu}(\eta\mid\\{\emptyset,X\\})$.
Note that if $H_{\mu}(\eta)<\infty$ then $\eta$ is necessarily countable (mod
zero) and $H_{\mu}(\eta)=-\sum_{C\in\eta}\log(\mu(C))\mu(C).$

#####  8.1.3. Metric entropy of a transformation

Let $f\colon(X,\mu)\to(X,\mu)$ be an invertible, measurable, measure-
preserving transformation. Let $\eta$ be an arbitrary measurable partition of
$(X,\mu)$. We define

$\eta^{+}:=\bigvee_{i=0}^{\infty}f^{i}\eta,\quad\quad\quad\eta^{f}:=\bigvee_{i\in\mathbb{Z}}^{\infty}f^{i}\eta.$

We define the entropy of $f$ given the partition $\eta$ to be

$h_{\mu}(f,\eta):=H_{\mu}(\eta\mid f\eta^{+})=H_{\mu}(\eta^{+}\mid
f\eta^{+})=H_{\mu}(f^{-1}\eta^{+}\mid\eta^{+}).$

We define the $\mu$-metric entropy of $f$ to be
$h_{\mu}(f)=\sup\\{h_{\mu}(f,\eta)\\}$ where the supremum is taken over all
measurable partitions of $(X,\mu)$. If

$\mu=\alpha\mu_{1}+\beta\mu_{2}$

where $\alpha,\beta\in[0,1]$ satisfy $\alpha+\beta=1$ and $\mu_{1}$ and
$\mu_{2}$ are $f$-invariant Borel probability measures then

$h_{\mu}(f)=\alpha h_{\mu_{1}}(f)+\beta h_{\mu_{2}}(f).$ (8.1)

####  8.2. Entropy under factor maps

Let $(X,\mu)$ and $(Y,\nu)$ be standard probability spaces. Let $f\colon X\to
X$ and $g\colon Y\to Y$ be measure-preserving transformations. Suppose there
is a measurable map $\psi\colon X\to Y$ with

$\psi_{*}\mu=\nu$

and

$\psi\circ f=g\circ\psi.$

In this case, we say that $g\colon(Y,\nu)\to(Y,\nu)$ is a measurable factor of
$f\colon(X,\mu)\to(X,\mu)$.

We note that entropy only decreases under measurable factors: if
$g\colon(Y,\nu)\to(Y,\nu)$ is a measurable factor of
$f\colon(X,\mu)\to(X,\mu)$ then

$h_{\nu}(g)\leq h_{\mu}(f).$

The difference between the entropies $h_{\nu}(g)$ and $h_{\mu}(f)$ is captured
by the Abramov–Rokhlin theorem. Let $\zeta$ be the measurable partition of
$(X,\mu)$ into level sets of $\psi\colon X\to Y$. Note that $\zeta$ is an
$f$-invariant partition: $\zeta=\zeta^{f}$. Define the conditional entropy
$h_{\mu}(f\mid\zeta)$ of $f$ relative to $\zeta$ to be

$h_{\mu}(f\mid\zeta)=\sup_{\xi}h_{\mu}(f,\xi\vee\zeta)$

where, as usual, the supremum is over all measurable partitions $\xi$ of
$(X,\mu)$. We call $h_{\mu}(f\mid\zeta)$ the fiberwise entropy of $f$. The
Abramov–Rokhlin theorem (see [MR0140660, MR1179170, MR0476995]) states the
following:

$h_{\mu}(f)=h_{\nu}(g)+h_{\mu}(f\mid\zeta).$ (8.2)

####  8.3. Unstable entropy of a diffeomorphism

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be an
ergodic, $f$-invariant measure.

#####  8.3.1. Partitions subordinate to a foliation

For the following discussion and in most applications considered in this text,
we may take $\mathcal{F}$ to be an $f$-invariant foliation of $M$ with
$C^{1+\beta}$ leaves. More generally, we may take $\mathcal{F}$ to be, in the
terminology introduced in [AWB-GLY-P1], an $f$-invariant, tame measurable
foliation; that is, $\mathcal{F}$ a partition of a full measure set by
$C^{1+\beta}$ manifolds with the property that locally, restricting to sets of
large measure, $\mathcal{F}$ has the structure of a continuous family of
$C^{1+\beta}$ discs. The primary examples of such measurable foliations
include the partition into global $j$th unstable Pesin manifolds and the
partition into global coarse Lyapunov manifolds in the setting of
$\mathbb{Z}^{d}$-actions. Note that the partition into global leaves of a
measurable foliation is not necessarily a measurable partition; rather locally
the partition looks like a measurable family of $C^{1+\beta}$ discs.

Write $\mathcal{F}(x)$ for the leaf of $\mathcal{F}$ through $x$. We say
$\mathcal{F}$ is expanding (for $f$) if $\mathcal{F}(x)\subset W^{u}(x)$, i.e.
if $\mathcal{F}(x)$ is a subset of the global unstable manifold through $x$
for $f$ discussed in Section 7.4. As a key example, one should consider
$\mathcal{F}^{u}$, the partition of $M$ into full global unstable manifolds.

######  Definition 8.2.

We say a measurable partition $\xi$ is subordinate to $\mathcal{F}$ if

  1. (1)

$\xi(x)\subset\mathcal{F}(x)$ for $\mu$-a.e. $x$;

  2. (2)

$\xi(x)$ contains an open (in the immersed topology) neighborhood of $x$ in
$\mathcal{F}(x)$ for $\mu$-a.e. $x$;

  3. (3)

$\xi(x)$ is precompact in (the immersed topology of) $\mathcal{F}(x)$ for
$\mu$-a.e. $x$;

#####  8.3.2. Partial ordering on the set of partitions

We recall the partial order on partitions of $(M,\mu)$. Let $\xi$ and $\eta$
be partitions of the probability space $(M,\mu)$. We write

$\eta\prec\xi$

and say that $\xi$ is finer than $\eta$ (or that $\eta$ is coarser than $\xi$)
if there is a subset $X\subset M$ with $\mu(X)=1$ such that for almost every
$x$,

$\xi(x)\cap X\subset\eta(x)\cap X.$

We say $\eta=\xi$ if $\eta\prec\xi$ and $\xi\prec\eta$.

#####  8.3.3. Entropy conditioned on a foliation

We say that a partition $\xi$ is increasing if $f\xi\prec\xi$ where $f\xi$
denotes the partition $f\xi=\\{f(C)\mid C\in\xi\\}$.

######  Definition 8.3.

Given an expanding, $f$-invariant foliation $\mathcal{F}$ we define the
entropy of $f$ conditioned on $\mathcal{F}$ to be

$h_{\mu}(f\mid\mathcal{F})=h_{\mu}(f,\xi)$

where $\xi$ is any increasing, measurable partition subordinate to
$\mathcal{F}$.

There are two small claim in Definition 8.3: First we have that
$h_{\mu}(f,\xi_{1})=h_{\mu}(f,\xi_{2})$ for any two increasing partitions
$\xi_{1}$ and $\xi_{2}$ subordinate to $\mathcal{F}$; see for example
[MR819556, Lemma 3.1.2]. Second, such a partition $\xi$ always exists. This
was shown when $\mathcal{F}=\mathcal{F}^{u}$ is the partition into global
unstable Pesin manifolds for a $C^{1+\beta}$ diffeomorphism in [MR693976] (see
also discussion in [MR819556, (3.1)]) extending a construction due to Sinai
for uniformly hyperbolic dynamics [MR0197684, Sinai1968]; the proof in
[MR693976] can be adapted for general invariant expanding $\mathcal{F}$.

When $\mathcal{F}=\mathcal{F}^{u}$ is the partition into full unstable
manifolds, define the unstable metric entropy of $f$ to be

$h^{u}_{\mu}(f):=h_{\mu}(f\mid\mathcal{F}^{u}).$

The principal result (Corollary 5.3) of [MR819556] shows that for $C^{2}$
diffeomorphisms we have equality of the metric entropy of $f$ and the unstable
metric entropy of $f$:

$h_{\mu}(f)=h_{\mu}^{u}(f).$ (8.3)

For $C^{1+\beta}$-diffeomorphism without zero Lyapunov exponents equality
(8.3) was shown by Ledrappier in [MR743818]; for the general case of
$C^{1+\beta}$-diffeomorphisms, (8.3) holds from [1608.05886].

####  8.4. Entropy, exponents, and geometry of conditional measures.

(See Appendix D for further details). In this section, we consider the
relationships between metric entropy $h_{\mu}(f)$, Lyapunov exponents, and the
geometry of conditional measures along unstable manifolds.

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be an
ergodic, $f$-invariant measure. At one extreme we have the following
generalization of Lemma 3.6 characterizing invariant measures with zero
entropy.

######  Lemma 8.4.

Let $\mu$ be an ergodic, $f$-invariant measure on $M$ and let $\xi$ be a
measurable partition of $(M,\mu)$ subordinate to the partition into unstable
manifolds. The following are equivalent:

  1. (1)

$h_{\mu}(f)=0$;

  2. (2)

for $\mu$-a.e. $x$, the conditional measure $\mu^{\xi}_{x}$ has at least one
atom;

  3. (3)

for $\mu$-a.e. $x$, the conditional measure $\mu^{\xi}_{x}$ is a single atom
supported at $x$;

  4. (4)

the partition of $(M,\mu)$ into full $W^{u}$-manifolds is a measurable
partition.

###### Proof sketch.

The implications (1) $\implies$ (4) and (1) $\implies$ (3) are a consequence
of [MR819556]*Theorem B (see also [1608.05886] for $C^{1+\beta}$ setting.)
Indeed, if $h_{\mu}(f)=0$, then the Pinsker partition (see Section 8.6.2
below) is the point partition. From [MR819556, Theorem B] we have that the
Pinsker partition is the measurable hull of (and in particular is coarser
than) the partition into full unstable manifolds. As the point partition is
the finest partition, it follows that the partition into full unstable
manifolds is measurably equivalent to the point partition and (3) and (4)
follow.

The implications (4) $\implies$ (3) and (2) $\implies$ (3) follow from the
dynamics on unstable manifolds and ergodicity of the measure. For instance, to
see (4) $\implies$ (3), assume the partition of $(M,\mu)$ into full
$W^{u}$-manifolds is measurable and let $\\{\mu^{u}_{x}\\}$ denote a family of
conditional probability measures for this partition. As $\mu$ is $f$-invariant
and as the partition into full unstable leaves is $f$-invariant, we have
$f_{*}\mu^{u}_{x}=\mu^{u}_{f(x)}$ for almost every $x$.

Given $x\in M$, let $W^{u}(x,R)$ denote the metric ball of radius $R$ centered
at $x$ in the internal metric of $W^{u}(x)$. Given $\delta>0$ and $R>0$,
define the set $G_{\delta,R}$ of $(\delta,R)$-good points to be

$G_{\delta,R}:=\\{x\in M\mid\mu^{u}_{x}(W^{u}(x,R))\geq 1-\delta\\}.$

Fix $R>0$ such that $\mu(G_{\delta,R})>0$. Take a subset $G^{\prime}\subset
G_{\delta,R}$ with $\mu(G^{\prime})>0$ such that the function

$x^{\prime}\mapsto\mathrm{diam}^{u}_{f^{-n}(x^{\prime})}(f^{-n}(W^{u}(x^{\prime},R)))$

converges to $0$ uniformly on $G^{\prime}$ as $n\to\infty$ where
$\mathrm{diam}^{u}_{x}(B)$ denotes the diameter of $B\subset W^{u}(x)$ with
respect to the internal metric on $W^{u}(x)$. For almost every $x$, we have
$f^{n}(x)\in G^{\prime}$ for infinitely many $n\in\mathbb{N}$. For such $x$
and any $\epsilon>0$, there is $n_{0}\in\mathbb{N}$ such that for all $n\geq
n_{0}$ with $f^{n}(x)\in G^{\prime}$ we have

$f^{-n}(W^{u}(f^{n}(x),R))\subset W^{u}(x,\epsilon)$

whence

$\mu^{u}_{x}(W^{u}(x,\epsilon))\geq\mu^{u}_{f^{n}(x)}(W^{u}(f^{n}(x),R))\geq
1-\delta.$

Taking $\epsilon\to 0$ we have $\mu^{u}_{x}(\\{x\\})\geq 1-\delta$ and, as
$\delta$ was arbitrary, (3) follows.

Finally, the implication (3) $\implies$ (1) follows from Corollary 5.3 of
[MR819556] (see (8.3) below) and the computation of unstable entropy in
Definition 8.3. ∎

At the other extreme, we have the following definition.

######  Definition 8.5.

We say $\mu$ is an SRB measure (or satisfies the  SRB property) if, for any
measurable partition $\xi$ of $(M,\mu)$ subordinate to the partition into
unstable manifolds, for almost every $x$ the conditional measure
$\mu^{\xi}_{x}$ is absolutely continuous with respect to Riemannian volume on
$W^{u}(x)$.

We have the following summary of a number of important results.

######  Theorem 8.6.

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be an
ergodic, $f$-invariant measure. Then

  1. (1)

$h_{\mu}(f)\leq\sum_{\lambda^{i}>0}m^{i}\lambda^{i}$;

  2. (2)

if $\mu$ is absolutely continuous with respect to volume then

$h_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i};$

  3. (3)

if $\mu$ is SRB then $h_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i}$.

Theorem 8.6(1), known as the Margulis–Ruelle inequality, is proven in
[MR516310]. Theorem 8.6(2), known as the Pesin entropy formula, is shown in
[MR0466791]. Theorem 8.6(3) was established by Ledrappier and Strelcyn in
[MR693976]. In the next section, we will complete Theorem 8.6 with
Ledrappier’s Theorem, Theorem 9.3, which provides a converse to Theorem
8.6(3).

For general measures invariant under a $C^{2}$-diffeomorphism (for the case of
$C^{1+\beta}$-diffeomorphisms, see [1608.05886]), Ledrappier and Young explain
explicitly the defect from equality in Theorem 8.6(1). This captures the
intermediate geometry of measures with positive entropy (and hence non-atomic
unstable conditional measures) but entropy strictly smaller than the sum of
positive Lyapunov exponents.

Let $\delta^{i}$ denote the (almost-surely constant value of the) pointwise
dimension of $\mu$ along the $i$th unstable manifolds; see Section D.3 in
Appendix D for definition. With $\delta^{0}=0$, let

$\gamma^{i}=\delta^{i}-\delta^{i-1}.$

The coefficients $\gamma^{i}$ reflect the transverse geometry (in particular
the transverse dimension) of the measure $\mu$ inside of the $i$th unstable
manifold transverse to the collection of $(i-1)$th unstable manifolds. In
particular, we have $\gamma^{i}\leq m^{i}$ (see [MR819557, Proposition
7.3.2].)

######  Theorem 8.7 ([MR819557]).

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be an
ergodic, $f$-invariant measure. Then

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

(Note that the proof in [MR819557] required $f$ to be $C^{2}$; following
[1608.05886] and [MR1709302]*Appendix, the theorem holds when $f\in
C^{1+\beta}.)$

$\bigast$8.5. Coarse-Lyapunov entropy and entropy product structure. Consider
now $\alpha\colon\mathbb{Z}^{d}\to\mathrm{Diff}^{1+\beta}(M)$ a smooth
$\mathbb{Z}^{d}$-action on a compact manifold $M$. Let $\mu$ be an ergodic,
$\alpha$-invariant measure. Recall that a coarse Lyapunov exponent $\chi$ is a
positive-proportionality class of Lyapunov exponents of $\alpha$. For almost
every $x\in M$ there is a coarse Lyapunov subspace $E^{\chi}(x)\subset T_{x}M$
and a coarse Lyapunov manifold $W^{\chi}(x)$ tangent to $E^{\chi}(x)$ at $x$.

Let $\mathcal{F}^{\chi}$ denote the partition of $M$ into full
$W^{\chi}$-manifolds. Given $n\in\mathbb{Z}^{d}$ with $\chi(n)>0$, following
the construction from [MR693976] we can find a measurable partition $\xi$ of
$(M,\mu)$ that is subordinate to $\mathcal{F}^{\chi}$ and increasing for
$\alpha(n)$. We then define the $\chi$-entropy of $\alpha(n)$ to be

$h_{\mu}^{\chi}(\alpha(n))=h_{\mu}(\alpha(n)\mid\chi):=h_{\mu}(\alpha(n)\mid\mathcal{F}^{\chi})=h_{\mu}(\alpha(n),\xi).$

The main result of [AWB-GLY-P3] is the following “product structure of
entropy” for $\mathbb{Z}^{d}$-actions.

######  Theorem 8.8 ([AWB-GLY-P3, Corollary 13.2]).

Let $\alpha\colon\mathbb{Z}^{d}\to\mathrm{Diff}^{1+\beta}(M)$ be a smooth
$\mathbb{Z}^{d}$-action on a compact manifold $M$ and let $\mu$ be an ergodic,
$\alpha$-invariant measure. Then for any $n\in\mathbb{Z}^{d}$

$h_{\mu}(\alpha(n))=\sum_{\chi(n)>0}h_{\mu}(\alpha(n)\mid\chi).$

Fix $n\in\mathbb{Z}^{d}$ and let $f=\alpha(n)$. The formulas in Theorem 8.7
and Theorem 8.8 then look quite similar. However, the contribution of each
Lyapunov exponent $\lambda^{i}$ to the total entropy in Theorem 8.7 is a
“transverse entropy” (the coefficient $\gamma^{i}$ is a measure of “transverse
dimension”). In Theorem 8.8, the entropy of each coarse Lyapunov exponent
$\chi$ is a “tangential entropy” $h_{\mu}(\alpha(n)\mid\chi)$ obtained by
conditioning along $W^{\chi}$-manifolds. Thus, Theorem 8.7 does not
immediately imply Theorem 8.8. To show Theorem 8.8, one first shows that the
total “transverse entropy” in Theorem 8.7 contributed by all
$\lambda^{i}\in\chi$ is equal to the total conditional entropy
$h_{\mu}(\alpha(n)\mid\chi)$. This is done in [AWB-GLY-P3]. The idea is to
first establish an analogue of Theorem 8.7 for the conditional entropy
$h_{\mu}(f\mid\chi)$; this is done in [AWB-GLY-P2] where a formula of the form

$h_{\mu}(f\mid\chi)=h_{\mu}(\alpha(n)\mid\chi)=\sum_{\lambda^{i}\in\chi}\gamma^{\chi,i}_{n}\lambda^{i}(n)$

is shown. Then (following [MR1213080]) one uses that $n\mapsto
h_{\mu}(\alpha(n)\mid\chi)$ is linear on any half-cone where no coarse
Lyapunov exponent $\chi^{\prime}$ changes sign to show that the transverse
dimensions $\gamma^{\chi,i}_{n}$ of each $\lambda^{i}\in\chi$ are independent
of $n$ and coincide with the transverse dimensions $\gamma^{i}$ appearing in
Theorem 8.7 for $f=\alpha(n).$

####  8.6. Abstract ergodic theoretic constructions in smooth dynamics

(See Appendices B and C for further details.) Let $f\colon M\to M$ be a
$C^{1+\beta}$ diffeomorphism and let $\mu$ be an $f$-invariant probability
measure. We do not assume $\mu$ to be ergodic. We introduce here a number of
measurable partitions of the measure space $(M,\mu)$ associated with the
dynamics $f$:

  1. (1)

$\mathcal{E}$, the ergodic decomposition (see Definition 6.5 and Theorem B.11
in Appendix B.4);

  2. (2)

$\pi$, the Pinsker partition;

  3. (3)

$\Xi^{u}$, the measurable hull of the partition into unstable manifolds;

  4. (4)

$\Xi^{s}$, the measurable hull of the partition into stable manifolds.

In this section, we will define objects (2)–(4) above and explain the
following two assertions:

  1. (1)

$\mathcal{E}\prec\Xi^{s}$

  2. (2)

$\Xi^{s}=\pi=\Xi^{u}.$

The first assertion is a standard fact in hyperbolic dynamics (which forms the
first step in the Hopf argument for ergodicity) and is discussed in detail in
Theorem C.4 in Appendix C. The second is [MR819556, Theorem B].

#####  8.6.1. Measurable hull of a partition

Given a (possibly nonmeasurable) partition $\xi$ of $(M,\mu)$ we write
$\Xi(\xi)$ for the measurable hull of $\xi$; that is, $\Xi(\xi)$ is the finest
measurable partition with $\Xi(\xi)\prec\xi$. If $\xi$ is measurable, then we
have $\Xi(\xi)=\xi$ but in general $\Xi(\xi)$ is strictly coarser than $\xi$.
We illustrate this concept with a few examples.

######  Example 8.9.

Suppose that $\mu$ is $f$-invariant and ergodic. Let $\mathcal{O}$ be the
partition into orbits of $f$. Then $\mathcal{O}$ is not measurable (see
Example B.6 in Appendix B). The measurable hull of $\mathcal{O}$ is the
trivial partition $\Xi(\mathcal{O})=\\{M,\emptyset\\}$. For example, given a
totally irrational flow on the torus $\mathbb{T}^{2}$, the partition into flow
lines is not measurable and the measurable hull is the trivial partition.

More generally, if $\mu$ is not ergodic then the measurable hull of
$\mathcal{O}$ is $\Xi(\mathcal{O})=\mathcal{E}$, the ergodic decomposition
$(M,\mu)$. (See Definition 6.5 and Example C.2 in Appendix C.)

For the following two examples, recall Lemma 3.6 and Lemma 8.4.

######  Example 8.10.

Let $f$ be a $C^{1+\beta}$ volume-preserving Anosov diffeomorphism of a
connected manifold $M$. Let $\xi^{u}$ denote the partition of $M$ into
unstable manifolds. Then $\xi^{u}$ is not measurable (for the invariant
volume). In fact, it is known that the measurable hull of $\xi^{u}$ is again
the trivial partition $\Xi(\xi^{u})=\\{M,\emptyset\\}$.

######  Example 8.11.

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be any
ergodic, $f$-invariant probability measure. Let $\xi^{u}$ denote the partition
of $M$ into (full) unstable manifolds. Then $\xi^{u}$ is measurable if and
only if $h_{\mu}(f)=0$. In particular, if $h_{\mu}(f)>0$ then the measurable
hull of $\xi^{u}$ is strictly coarser than $\xi^{u}$.

In general, given a $C^{1+\beta}$ diffeomorphism $f\colon M\to M$ and an
ergodic, $f$-invariant probability measure $\mu$ we let $\Xi^{u}$ and
$\Xi^{s}$ denote, respectively, the measurable hulls of the partition of
$(M,\mu)$ into full unstable and stable manifolds.

We state the first relationship between the above objects in the following
proposition whose proof follows immediately from the pointwise ergodic
theorem. (See Theorem C.4, Appendix C.)

######  Proposition 8.12.

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be any
$f$-invariant probability measure. Then $\mathcal{E}\prec\Xi^{s}$ and
$\mathcal{E}\prec\Xi^{u}$.

###### Proof.

Let $\sigma(\mathcal{E})$ and $\sigma(\Xi^{s})$ denote the $\sigma$-algebras
of $\mathcal{E}$-saturated and $\Xi^{s}$-saturated sets, respectively.

Consider any continuous function $\phi\colon M\to\mathbb{R}$. Then
$\phi^{+}\colon M\to\mathbb{R}$ defined by

$\phi^{+}(x):=\limsup_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}\phi(f^{k}(x))$

is an $f$-invariant function that is constant along $W^{s}$-leaves. In
particular, the function $\phi^{+}$ is measurable with respect to
$\sigma(\mathcal{E})$ and $\sigma(\Xi^{s})$. Moreover, using that $C^{0}(M)$
is separable and dense in $L^{1}(\mu)$ and applying the pointwise ergodic
theorem, it follows that the $\sigma$-algebra $\sigma(\mathcal{E})$ is the
minimal $\sigma$-algebra for which $\phi^{+}$ is measurable for all continuous
$\phi\colon M\to\mathbb{R}$. It follows that
$\sigma(\mathcal{E})\subset\sigma(\Xi^{s})$ whence $\mathcal{E}\prec\Xi^{s}$.
∎

#####  8.6.2. The Pinsker partition

Let $f\colon(X,\mu)\to(X,\mu)$ be a measure-preserving transformation of a
standard probability space $(X,\mu)$. The Pinsker partition $\pi$ of
$f\colon(X,\mu)\to(X,\mu)$ is the finest measurable partition $(X,\mu)$ with
the following property: for any measurable partition $\xi\prec\pi$, we have

$h_{\mu}(f,\xi)=0.$

Another characterization of $\pi$ is the following: $\pi$ is the unique
$f$-invariant partition such that, if $(g,Y,\nu)$ is a measurable factor of
$(f,X,\mu)$ with zero entropy, then $(g,Y,\nu)$ is also a factor of the factor
system $(f,X,\mu)/\pi$.

Our second relationship, stated as [MR819556, Theorem B], characterizes the
Pinsker partition in smooth dynamics.

######  Proposition 8.13 ([MR819556, Theorem B]).

Let $f\colon M\to M$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be any
$f$-invariant Borel probability measure. Then we have equality of partitions

$\Xi^{u}=\pi=\Xi^{s}.$

######  Remark 8.14.

We say that a measure-preserving transformation $f\colon(X,\mu)\to(X,\mu)$ has
the K-property (or the Kolmogorov property) if the Pinsker partition $\pi$ is
the trivial partition $\pi=\\{\emptyset,X\\}$. For such systems, every non-
trivial factor has positive entropy.

Let $f\colon M\to M$ be a $C^{1+\beta}$ volume-preserving Anosov
diffeomorphism. Anosov first showed that such maps are ergodic with respect to
the invariant volume in [MR0224110]. In this setting, the analogue of
Proposition 8.13 is established in [MR0197684]; that is any set $A\in\pi$ is
equal modulo $0$ to a set fully saturated by stable manifolds and also equal
modulo $0$ to a (possibly different set) that is fully saturated by unstable
manifolds. Using the absolute continuity of the stable and unstable foliations
established by Anosov in his proof of ergodicity, one may show that any
$A\in\pi$ is equal modulo $0$ to a set that is both fully saturated by stable
manifolds and unstable manifolds. It follows that any $A\in\pi$ is null or
conull. In particular, this shows that volume-preserving Anosov
diffeomorphisms have the K-property. This explains the conclusion in Example
8.10 that $\Xi^{u}$ is the trivial partition. See, for example, [MR2630044]
for a modern discussion of absolute continuity and the $K$-property in
uniformly (partially) hyperbolic settings.

###  9\. Entropy, invariance, and the SRB property

In dissipative (i.e. non-volume-preserving) dynamical systems, ergodic SRB
measures $\mu$ without zero Lyapunov exponents provide examples of physical
measures: there is a set $B$ of positive Lebesgue measure such that for any
continuous function $\phi$, the forward time average of $\phi$ along the orbit
of points in $B$ converges to $\int\phi\ d\mu$. In applications and specific
examples, a recurring problem is to establish the existence of physical and
SRB measures. We pose a related question that arises naturally in the settings
considered in this text:

######  Question 9.1.

Given a diffeomorphism $f\colon M\to M$ and an $f$-invariant measure $\mu$,
how do you verify that $\mu$ is an SRB measure?

Seemingly unrelated, consider a group $G$ acting smoothly on a manifold $M$.
We pose the following:

######  Question 9.2.

Given a Borel probability measure $\mu$ on $M$ and a subgroup $H\subset G$,
how do you verify that $\mu$ is $H$-invariant?

One method to answer both of these questions is given in Theorem 9.3 and
Theorem 9.5 below.

####  9.1. Ledrappier’s theorem

(See Appendix D for further details.) We outline one approach that solves both
Question 9.1 and 9.2 in a number of settings. We discuss other approaches
towards verifying the existence of SRB measures below.

We recall Section 8.3 where the notion of unstable entropy was introduced. The
main result (Corollary 5.3) of [MR819556] shows for a $C^{2}$ (see
[1608.05886] for the $C^{1+\beta}$ case) diffeomorphism $f\colon M\to M$
preserving an ergodic probability measure $\mu$ that the metric entropy of $f$
and the unstable metric entropy of $f$ coincide:

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

Using this fact, Ledrappier gave a geometric characterization of all measures
satisfying equality $h_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i}$ in the
Margulis–Ruelle inequality, giving a converse of Theorem 8.6(3).

######  Theorem 9.3 (Ledrappier’s Theorem [MR743818]).

Let $f$ be a $C^{1+\beta}$ diffeomorphism and let $\mu$ be an ergodic,
$f$-invariant, Borel probability measure. Then $\mu$ is SRB if and only if

$h^{u}_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i}.$ (9.1)

In the proof of Theorem 9.3, Ledrappier actually proves something much
stronger than the SRB property: if
$h^{u}_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i}$ then the leaf-wise
measures $\mu^{u}_{x}$ of $\mu$ along unstable manifolds are equivalent to the
Riemannian volume with a Hölder continuous density. That is, if $m^{u}_{x}$
the Riemannian volume along $W^{u}(x)$ then for a.e. $x$ there is a Hölder
continuous, nowhere vanishing function $\rho\colon W^{u}(x)\to(0,\infty)$ with

$\mu^{u}_{x}=\rho\ m^{u}_{x}.$ (9.2)

In particular, the leaf-wise measure $\mu^{u}_{x}$ has full support in
$W^{u}(x)$. Moreover, Ledrappier explicitly computes the density function
$\rho$; see (D.6) in Appendix D and [MR819556, Corollary 6.1.4].

We make use of the explicit formula for the density $\rho$ in the following
setup. Consider a Lie group $G$ and a smooth, locally free action of $G$ on a
manifold $M$. We denote the action by $g\cdot x$ for $g\in G$ and $x\in M$.
Consider a Lie subgroup $H\subset G$ and $s\in G$ that normalizes $H$. Let
$f\colon M\to M$ be the diffeomorphism given by $s$; that is $f(x)=s\cdot x$.
Let $\mu$ be an ergodic, $f$-invariant Borel probability measure and suppose
that the orbit $H\cdot x$ is contained in the unstable manifold $W^{u}(x)$ for
$\mu$-almost every $x$.

Since $s$ normalizes $H$, the partition of $M$ into $H$-orbits is preserved by
$f$; in particular, the partition into $H$-orbits is a subfoliation of the
partition into unstable manifolds. Given a Borel probability measure $\mu$ on
$M$ and a measurable partition $\xi$ subordinate to the partition into
$H$-orbits we can define conditional measures $\mu^{\xi}_{x}$ of $\mu$. Given
$x\in M$ (using that the action is locally free) we can push forward the left-
Haar measure on $H$ onto the orbit $H\cdot x$ via the parametrization $H\cdot
x=\\{h\cdot x:h\in H\\}.$

######  Lemma 9.4.

$\mu$ is $H$-invariant if and only if for any measurable partition $\xi$
subordinate to the partition into $H$-orbits and $\mu$-a.e. $x$ the
conditional measure $\mu^{\xi}_{x}$ coincides—up to normalization—with the
restriction of the left-Haar measure on $H\cdot x$ to $\xi(x)$.

Similar to the definition of metric entropy of $f$ conditioned on unstable
manifolds, we can define the metric entropy of $f$ conditioned on $H$-orbits,
written $h_{\mu}(f\mid H)$, by

$h_{\mu}(f\mid H):=h_{\mu}(f,\xi)$

where $\xi$ is any increasing, measurable partition $\xi$ subordinate to
$H$-orbits. Let $\lambda^{i}$, $E^{i}(x)$, and $m^{i}$ be as in 7.1 for the
dynamics of $f$ and the measure $\mu$. We define the multiplicity of
$\lambda^{i}$ relative to $H$ to be (the almost surely constant value of)

$m^{i,H}=\dim(E^{i}(x)\cap T_{x}(H\cdot x)).$

Generalizing Theorem 8.6(1) we have (see for instance [AWB-GLY-P2])

$h_{\mu}(f\mid H)\leq\sum_{\lambda^{i}>0}\lambda^{i}m^{i,H}.$ (9.3)

From the proof of Theorem 9.3, (in particular, the explicit formula for the
density function $\rho$ in (9.2); see (D.6) in Appendix D and proof of
Proposition 5.8) we have the following.

######  Theorem 9.5.

With the above setup, the following are equivalent:

  1. (1)

$h_{\mu}(f\mid H)=\sum_{\lambda^{i}>0}\lambda^{i}m^{i,H}$;

  2. (2)

for any measurable partition $\xi$ subordinate to the partition into
$H$-orbits and almost every $x$, $\mu^{\xi}_{x}$ is absolutely continuous with
respect to the Riemannian volume on the $H$-orbit $H\cdot x$;

  3. (3)

$\mu$ is $H$-invariant.

The proof is only a slightly more complicated version of the proof of
Proposition 5.8. Note that as Theorem 9.5 only concerns the entropy and
dynamics inside $H$-orbits, the result holds for $C^{1}$ or even $C^{0}$
actions since the dynamics permuting $H$-orbits is affine and hence
$C^{\infty}$. See for instance [MR2191228] where related entropy results are
shown for $C^{0}$ actions of Lie groups.

A possible critique of Theorem 9.3 is that in examples it seems nearly
impossible to verify equality in (9.1) without first knowing that the measure
is SRB. However, in a number of settings of group actions on manifolds, it
turns out one can, in fact, verify equality in (9.1) (or typically, equality
in Theorem 9.5(1)) and thus derive the SRB property or gain additional
invariance of the measure only from entropy considerations. This is one key
idea in this text, the papers [AWBFRHZW-latticemeasure, 1608.04995], and also
appears as a main tool in [MR3814652, MR1253197].

######  Remark 9.6.

The statement and proof of Theorem 9.3, especially the reformulation in
Theorem 9.5, is very similar to the invariance principle for fiberwise
disintegrations of measures invariant under skew products. The earliest
version of this invariance principle is due to Ledrappier [MR850070] for
projectivized linear cocycles. Avila–Viana extended this to cocycles taking
values in the group of $C^{1}$ diffeomorphisms in [MR2651382]. See Proposition
15.5 for a related invariance principle in the setting of actions of lattices
on manifolds.

####  9.2. Approaches to Questions 9.1 and 9.2

Although not the main focus of this text, we summarize a number of alternative
approaches towards approaching Questions 9.1 and 9.2 that arise in various
dynamical settings.

SRB property from dynamical hypotheses. In the setting of uniformly hyperbolic
dynamics, SRB measures are known to exist for Anosov diffeomorphisms, Anosov
flows, and Axiom A attractors. See [MR0442989, MR0380889]. In the setting of
partially hyperbolic diffeomorphisms, under suitable conditions on the central
dynamics SRB measures are known to exist; related results hold for dynamics
with a dominated splitting. See for example [MR1743717, MR1749677, MR1757000,
MR3712997].

SRB measures via detailed analysis. For specific families of examples
exhibiting nonuniform hyperbolicity, tools of parameter exclusion, normal
forms, and detailed analysis can be used to show the existence of an SRB
measure. See for example, [MR1701385, MR1087346, MR799250, MR630331,
MR1218323, MR1835392, MR3098967, MR2005855]. General hypotheses that can be
verified in a number of examples are given in [MR1637655, MR1824198] which
guarantee the existence of SRB measures. See the survey article [MR1933431]
for more background.

Verifying equality in the entropy formula. As discussed above, the culmination
of the results of [MR693976, MR743818, MR819556] characterizes SRB measures
exactly as those for which the equality
$h_{\mu}(f)=\sum_{\lambda^{i}>0}m^{i}\lambda^{i}$ holds. Similarly, equality
in Theorem 9.5(1) holds if and only if the measure $\mu$ is invariant under
the action of the subgroup $H$. This approach—verifying equality in the
entropy formula to obtain invariance of a measure—has been exploited in
particular in [MR1253197, MR3814652, AWBFRHZW-latticemeasure, 1608.04995].

Shearing and translation invariance in a homogeneous structure. A common tool
to obtain invariance or absolute continuity properties of leaf-wise measures
is to manufacture a shear of leaf-wise measures along leaves of a foliation.
That is, given an invariant measure $\mu$ and an affine foliation
$\mathcal{F}$ with family of normalized leave-wise measures
$\\{\mu^{\mathcal{F}}_{x}\\}$, for a $\mu$-typical $x$ one may be able to use
the dynamics to construct approximations of translations along the support of
$\mu^{\mathcal{F}}_{x}$ in the leaf $\mathcal{F}(x)$ that preserve the measure
$\mu^{\mathcal{F}}_{x}$ up to normalization. Taking a limit, one has that
$\mu^{\mathcal{F}}_{x}$ is preserved up to normalization under some
translations which gives strong information (see Proposition 5.6) on the
geometry of $\mu^{\mathcal{F}}_{x}$. Additional dynamical arguments can then
often establish translation invariance of the leaf-wise measures
$\mu^{\mathcal{F}}_{x}$. Manufacturing translation invariance of leaf-wise
measures along their support in an affine foliation $\mathcal{F}$ is a main
tool used to establish Ratner’s measure classification results in [MR1075042,
MR1262705] and [MR1253197]. This was also one of the main steps (see Lemma
6.3) in the proof of Theorem 2.8.

In a setting similar to that of Theorem 2.8, for higher-rank diagonal actions
on semisimple homogeneous spaces (see Section 13.2), the high entropy method
[MR1989231, MR2191228] and low entropy method [MR2195133, MR2366231] provide
mechanisms to obtain translation invariance of leaf-wise measures, culminating
in the landmark paper [MR2247967]. Another mechanism to obtain translation
invariance of leaf-wise measures appears in [MR2831114, MR3037785] and is used
to establish measure rigidity results for stationary measure for affine random
walks. In [MR3814652], a mechanism inspired by [MR2831114] is used to obtain
invariance for certain affine actions of $\mathrm{SL}(2,\mathbb{R})$.

This approach, and specifically the method presented in Part I from
[MR1406432], has been adapted to establish measure rigidity in a number of
nonlinear settings including [MR2261075] and [MR2811602]. In non-linear
settings, unstable manifolds $W^{u}(x)$ are $C^{2}$ injectively immersed
copies of $\mathbb{R}^{k}$ for some $k$. Although there might be no natural
notion of translation, relative to certain coordinate systems
$H_{x}\colon\mathbb{R}^{k}\to W^{u}(x)$ obtained from normal forms of the
dynamics along unstable manifolds, leaf-wise measures $\mu^{u}_{x}$ are
absolutely continuous if and only if their images
$(H_{x}^{-1})_{*}\mu^{u}_{x}$ in these coordinates are translation invariant
in $\mathbb{R}^{k}$. In a number of non-linear settings including [MR2261075,
MR2811602, 1506.06826] absolute continuity properties of a measure $\mu$ along
unstable foliations is shown by establishing translation invariance of the
leaf-wise measures $(H_{x}^{-1})_{*}\mu^{u}_{x}$ in these coordinates.

##  Part III Smooth lattice actions and new results in the Zimmer program

The main goal of this part will be to understand properties and to classify
smooth actions of certain countable groups $\Gamma$ on compact manifolds. The
main results of this section are Theorem 11.4 and Theorem 11.6. We give their
proofs after introducing some terminology and motivation.

###  10\. Smooth lattice actions

We give some background on lattices in semisimple Lie groups and a number of
examples of smooth actions of lattices on manifolds. References with
additional details for this and the next section include [MR3307755,
MR2655311, MR1920389, MR1648087, MR1090825, MR2807830].

####  10.1. Lattices in semisimple Lie groups

Recall that a Lie algebra $\mathfrak{g}$ is simple if it is non-abelian and
has no non-trivial ideal. A Lie algebra $\mathfrak{g}$ is semisimple if it is
the direct sum $\mathfrak{g}=\oplus_{i=1}^{\ell}\mathfrak{g}_{i}$ of simple
Lie algebras $\mathfrak{g}_{i}$; this is equivalent to the fact that
$[\mathfrak{g},\mathfrak{g}]=\mathfrak{g}.$ We say a Lie group $G$ is simple
(resp. semisimple) if its Lie algebra $\mathfrak{g}$ is simple (resp.
semisimple). The main example for this text is the simple Lie group
$G=\mathrm{SL}(n,\mathbb{R})$.

Let $G$ be a connected semisimple Lie group with finite center. Semisimple Lie
groups are unimodular and hence admit a bi-invariant measure, called the Haar
measure, which is unique up to normalization. A lattice in $G$ is a discrete
subgroup $\Gamma\subset G$ with finite co-volume. That is, if $D$ is a
measurable fundamental domain for the right-action of $\Gamma$ on $G$ then $D$
has finite volume. If the quotient $G/\Gamma$ is compact, we say that $\Gamma$
is a cocompact lattice. If $G/\Gamma$ has finite volume but is not compact we
say that $\Gamma$ is nonuniform. The quotient manifold $G/\Gamma$ by the right
action of $\Gamma$ admits a left-action by $G$ and the Haar measure on $G$
descends to a finite, $G$-invariant measure on $G/\Gamma$ which we normalize
to be a probability measure.

######  Example 10.1.

The standard example of a lattice in $G=\mathrm{SL}(n,\mathbb{R})$ is
$\Gamma=\mathrm{SL}(n,\mathbb{Z})$. Note that $\mathrm{SL}(n,\mathbb{Z})$ is
not cocompact in $\mathrm{SL}(n,\mathbb{R})$. However,
$\mathrm{SL}(n,\mathbb{R})$ and more general simple and semisimple Lie groups
possess both nonuniform and cocompact lattices. (See for example [MR3307755,
Sections 6.7, 6.8] for examples and constructions.)

######  Example 10.2.

In the case $G=\mathrm{SL}(2,\mathbb{R})$, the fundamental group of any finite
area hyperbolic surface is a lattice in $G$. In particular, the fundamental
group of a compact hyperbolic surface is a cocompact lattice in $G$. This can
be seen by identifying the fundamental group of $S$ with the deck group of the
hyperbolic plane
$\mathbb{H}=\mathrm{SO}(2,\mathbb{R})\backslash\mathrm{SL}(2,\mathbb{R})$. For
instance, the free group $\Gamma=F_{2}$ on two generators is a lattice in $G$
as can be seen by giving the punctured torus
$S=\mathbb{T}^{2}\smallsetminus\\{\mathrm{pt}\\}$ a hyperbolic metric.

See [MR3307755] for further details on constructions and properties of
lattices in Lie groups.

####  10.2. Rank of $G$

Every semisimple matrix group admits an Iwasawa decomposition $G=KAN$ where
$K$ is compact, $A$ is a simply connected free abelian group of
$\mathbb{R}$-diagonalizable elements, and $N$ is unipotent. For general
semisimple Lie groups with finite center, we have a similarly defined Iwasawa
decomposition $G=KAN$ where the images of $A$ and $N$ under the adjoint
representation are, respectively, $\mathbb{R}$-diagonalizable and unipotent.
See for instance [MR1920389] for details. The dimension of $A$ is the rank of
$G$. We call such a group $A$ a maximal split Cartan subgroup.

In the case of $G=\mathrm{SL}(n,\mathbb{R})$, the standard choice of $K$, $A$,
and $N$ are

$K=\mathrm{SO}(n,\mathbb{R}),\quad\quad
A=\left\\{\mathrm{diag}(e^{t_{1}},e^{t_{2}},\dots,e^{t_{n}}):t_{1}+\dots+t_{n}=0\right\\},$

and $N$ the group of upper-triangular matrices with all diagonal entries equal
to $1$. Note that, as elements in $\mathrm{SL}(n,\mathbb{R})$ have determinant
$1$, we have

$\mathrm{diag}(e^{t_{1}},e^{t_{2}},\dots,e^{t_{n}})\in\mathrm{SL}(n,\mathbb{R})$

if and only if $t_{1}+\dots+t_{n}=0$. Thus $A\simeq\mathbb{R}^{n-1}$ and the
rank of $\mathrm{SL}(n,\mathbb{R})$ is $n-1$.

We say that a simple Lie group $G$ is higher-rank if its rank is at least $2$.
We will say that a lattice $\Gamma$ in a higher-rank simple Lie group $G$ is a
higher-rank lattice. In particular, $G=\mathrm{SL}(n,\mathbb{R})$ and its
lattices are higher-rank when $n\geq 3$.

In Example 10.9 below, we present an example of a cocompact lattice $\Gamma$
in the group $G=\mathrm{SO}(n,n)$ when $n\geq 4$. The group $\mathrm{SO}(n,n)$
has rank $n$ and thus $\Gamma$ is a higher-rank, cocompact lattice.

For further examples, see Table 1 for calculations of rank for various matrix
groups and [MR1920389, VI.4] for examples of Iwasawa decompositions for
various matrix groups.

####  10.3. Standard actions of lattices in Lie groups

We present a number of standard examples of “algebraic” actions of lattices in
Lie groups. We also discuss in Example 10.10 some modifications of algebraic
actions and constructions of more exotic actions.

######  Example 10.3 (Finite actions).

Let $\Gamma^{\prime}$ be a finite-index normal subgroup of $\Gamma$. Then
$F=\Gamma/\Gamma^{\prime}$ is finite. Suppose the finite group $F$ acts on a
manifold $M$. Since $F$ is a quotient of $\Gamma$ we naturally obtain a
$\Gamma$-action on $M$.

Note that an action of a finite group preserves a volume simply by averaging
any volume form by the action.

######  Definition 10.4.

An action $\alpha\colon\Gamma\to\mathrm{Diff}(M)$ is finite or almost trivial
if it factors through the action of a finite group. That is, $\alpha$ is
finite if there is a finite-index normal subgroup
$\Gamma^{\prime}\subset\Gamma$ such that
${\alpha}{\restriction_{{\Gamma^{\prime}}}}$ is the identity.

We remark that by a theorem of Margulis [MR515630], if $\Gamma$ is a lattice
in a higher-rank, simple Lie group with finite center then all normal
subgroups of $\Gamma$ are either finite or of finite-index.

######  Example 10.5 (Affine actions).

Let $\Gamma=\mathrm{SL}(n,\mathbb{Z})$ (or any finite-index subgroup of
$\mathrm{SL}(n,\mathbb{Z})$). Let
$M=\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$ be the $n$-dimensional torus.
We have a natural action $\alpha\colon\Gamma\to\mathrm{Diff}(\mathbb{T}^{n})$
given by

$\alpha(\gamma)(x+\mathbb{Z}^{n})=\gamma\cdot x+\mathbb{Z}^{n}$

for any matrix $\gamma\in\mathrm{SL}(n,\mathbb{Z})$.

To generalize this example to other lattices, let
$\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be any lattice and let
$\rho\colon\Gamma\to\mathrm{SL}(d,\mathbb{Z})$ be any representation. Then we
have a natural action $\alpha\colon\Gamma\to\mathrm{Diff}(\mathbb{T}^{d})$
given by

$\alpha(\gamma)(x+\mathbb{Z}^{d})=\rho(\gamma)\cdot x+\mathbb{Z}^{d}.$

Note that these examples preserve a volume form, namely, the Lebesgue measure
on $\mathbb{T}^{d}$. Also note that these actions are non-isometric.

######  Remark 10.6.

Both constructions in Example 10.5 give actions
$\alpha\colon\Gamma\to\mathrm{Diff}(\mathbb{T}^{d})$ that have global fixed
points. That is, the coset of $0$ in $\mathbb{T}^{d}$ is a fixed point of
$\alpha(\gamma)$ for every $\gamma\in\Gamma$.

The construction can be modified further to obtain genuinely affine actions
without global fixed points. Given a lattice
$\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ and a representation
$\rho\colon\Gamma\to\mathrm{SL}(d,\mathbb{Z})$, there may exist non-trivial
elements $c\in H^{1}_{\rho}(\Gamma,\mathbb{T}^{d})$; that is,
$c\colon\Gamma\to\mathbb{T}^{d}$ is a function with

$c(\gamma_{1}\gamma_{2})=\rho(\gamma_{1})c(\gamma_{2})+c(\gamma_{1})$ (10.1)

and such that there does not exist any $\eta\in\mathbb{T}^{d}$ with

$c(\gamma)=\rho(\gamma)\eta-\eta$ (10.2)

for all $\gamma\in\Gamma$. (Equation (10.1) says that $c$ is a cocycle with
coefficients in the $\Gamma$-module $\mathbb{T}^{d}$; (10.2) says $c$ is not a
coboundary.) We may then define
$\tilde{\alpha}\colon\Gamma\to\mathrm{Diff}(\mathbb{T}^{d})$ by

$\tilde{\alpha}(\gamma)(x+\mathbb{Z}^{d})=\rho(\gamma)\cdot
x+c(\gamma)+\mathbb{Z}^{d}.$

Equation (10.1) ensures that $\tilde{\alpha}$ is an action and (10.2) ensures
that $\tilde{\alpha}$ is not conjugate to the action $\alpha$.

In the above construction, any cocycle $c\colon\Gamma\to\mathbb{T}^{d}$ is
necessarily cohomologous to a torsion-valued (that is,
$\mathbb{Q}^{d}/\mathbb{Z}^{d}$-valued) cocycle. This follows from Margulis’s
result (see [MR1090825, Theorem 3 (iii)]) on the vanishing of
$H^{1}_{\rho}(\Gamma,\mathbb{R}^{d}).$ In particular, $\tilde{\alpha}$ and
$\alpha$ are conjugate when restricted to a finite-index subgroup of $\Gamma$.
See [MR1236179] for more details.

######  Example 10.7 (Projective actions).

Let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be any lattice. Then $\Gamma$ has
a natural linear action on $\mathbb{R}^{n}$. The linear action of $\Gamma$ on
$\mathbb{R}^{n}$ induces an action of $\Gamma$ on the sphere $S^{n-1}$ thought
of as the set of unit vectors in $\mathbb{R}^{n}$: we have
$\alpha\colon\Gamma\to\mathrm{Diff}(S^{n-1})$ given by

$\alpha(\gamma)(x)=\frac{\gamma\cdot x}{\|\gamma\cdot x\|}.$

Alternatively we could act on the space of lines in $\mathbb{R}^{n}$ and
obtain an action of $\Gamma$ on the $(n-1)$-dimensional real projective space
$\mathbb{R}P^{n-1}$. This action does not preserve a volume; in fact there is
no invariant probability measure for this action. Additionally, these actions
are not isometric for any Riemannian metric.

######  Remark 10.8 (Actions on boundaries).

Example 10.7 generalizes to actions of lattices $\Gamma$ in $G$ acting on
boundaries of $G$. Given a semisimple Lie group $G$ with Iwasawa decomposition
$G=KAN$, let $M=K\cap C_{G}(A)$ be the centralizer of $A$ in $K$. A closed
subgroup $Q\subset G$ is parabolic if it is conjugate to a group containing
$MAN$. When $G=\mathrm{SL}(n,\mathbb{R})$ we have that $M$ is a finite group
and any parabolic subgroup $Q$ is conjugate to a group containing all upper
triangular matrices. See [MR1920389, Section VII.7] for further discussion on
the structure of parabolic subgroups.

Given a semisimple Lie group $G$, a (finite-index subgroup of a) proper
parabolic subgroup $Q\subset G$, and a lattice $\Gamma\subset G$, the coset
space $M=G/Q$ is compact and $\Gamma$ acts on $M$ naturally as

$\alpha(\gamma)(xQ)=\gamma xQ.$

These actions never preserve a volume form or any Borel probability measure
and are not isometric.

In Example 10.7, the action on the projective space $\mathbb{R}P^{n-1}$ can be
seen as the action on $\mathrm{SL}(n,\mathbb{R})/Q$ where $Q$ is the parabolic
subgroup

$Q=\left\\{\left(\begin{array}[]{cccc}*&*&\cdots&*\\\ 0&*&\cdots&*\\\
\vdots&\vdots&\ddots&\vdots\\\ 0&*&\cdots&*\end{array}\right)\right\\}.$

######  Example 10.9 (Isometric actions).

Another important family of algebraic actions are isometric actions obtained
from embeddings of cocompact lattices in Lie groups into compact groups.

##### Isometric actions of cocompact lattices in split orthogonal groups of
type $D_{n}$ ($n\geq 4$)

For $n\geq 4$, consider the quadratic form in ${2n}$ variables

$Q(x_{1},\dots,x_{n},y_{1},\dots,y_{n})=x_{1}^{2}+\dots
x_{n}^{2}-\sqrt{2}(y_{1}^{2}+\dots+y_{n}^{2}).$

Let

$B=\mathrm{diag}\left(1,\dots,1,-\sqrt{2},\dots,-\sqrt{2}\right)\in\mathrm{GL}(2n,\mathbb{R})$

be the matrix such that $Q(x)=x^{T}Bx$ for all $x\in\mathbb{R}^{2n}$ and let

$G=\mathrm{SO}(Q)=\\{g\in\mathrm{SL}(2n,\mathbb{R})\mid g^{T}Bg=B\\}$

be the special orthogonal group associated with $Q$. We have that

$\mathrm{SO}(Q)\simeq\mathrm{SO}(n,n)$

is a Lie group of rank $n$ with restricted root system of type $D_{n}$ when
$n\geq 4$.333For $n=1$, $\mathrm{SO}(1,1)$ is a one-parameter group and for
$n=2$, $\mathrm{SO}(2,2)$ is not simple (it is double covered by
$\mathrm{SL}(2,\mathbb{R})\times\mathrm{SL}(2,\mathbb{R})$). For $n=3$,
$\mathrm{SO}(3,3)$ is double covered by $\mathrm{SL}(4,\mathbb{R})$.

Let $\mathbb{K}=\mathbb{Q}[\sqrt{2}]$ and let $\mathbb{Z}[\sqrt{2}]$ be the
ring of integers in $\mathbb{K}$. Let

$\Gamma=\\{g\in\mathrm{SL}(2n,\mathbb{Z}[\sqrt{2}])\mid g^{T}Bg=B\\}.$

Then $\Gamma$ is a cocompact lattice in $G$. (See for example [MR3307755],
Proposition 5.5.8 and Corollary 5.5.10.)

Let $\tau\colon\mathbb{K}\to\mathbb{K}$ be the nontrivial Galois automorphism,
$\tau(\sqrt{2})=-\sqrt{2}$, and let $\tau$ act coordinate-wise on matrices
with entries in $\mathbb{K}$. Given $\gamma\in\Gamma$ we have
$\tau(\gamma)=\mathrm{Id}$ if and only if $\gamma=\mathrm{Id}$. Moreover, as
$\tau^{2}=\mathrm{Id}$ we have

$\tau(\gamma)\in\mathrm{SO}(\tau(Q)):=\\{g\in\mathrm{SL}(2n,\mathbb{R})\mid
g^{T}\tau(B)g=\tau(B)\\}\simeq\mathrm{SO}(2n).$

In particular, the map $\gamma\to\tau(\gamma)$ gives a representation
$\Gamma\to\mathrm{SO}(2n)$ with infinite image into the compact group
$\mathrm{SO}(2n)$.

As $\mathrm{SO}(2n)$ is the isometry group of the sphere
$S^{2n-1}=\mathrm{SO}(2n)/\mathrm{SO}(2n-1)$ we obtain an action of $\Gamma$
by isometries on a manifold of dimension $2n-1$.

##### Isometric actions of cocompact lattices in $\mathrm{SL}(n,\mathbb{R})$

A more complicated construction can be used to build cocompact lattices
$\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ that possess infinite-image
representations $\pi\colon\Gamma\to\mathrm{SU}(n)$ (see discussion in
[MR3307755, Sections 6.7, 6.8] as well as [MR3307755, Warning 16.4.3].) In
this case, one obtains isometric actions of certain cocompact lattices
$\Gamma$ in $\mathrm{SL}(n,\mathbb{R})$ on the $(2n-2)$-dimensional
homogeneous space

$\mathrm{SU}(n)/\mathrm{S}(\mathrm{U}(1)\times\mathrm{U}(n-1)).$

######  Example 10.10 (Modifications of standard examples and exotic actions).

Beyond the “algebraic actions” discussed in Examples 10.5–10.9, it is possible
to modify certain algebraic constructions to construct genuinely new actions;
these actions might not be conjugate to algebraic actions and may exhibit much
weaker rigidity properties. One such construction starts with the standard
action of (finite-index subgroups of) $\mathrm{SL}(n,\mathbb{Z})$ on
$\mathbb{T}^{n}$ and creates a non-volume-preserving action by blowing-up
fixed points or finite orbits of the action. In [MR1380646, Section 4], Katok
and Lewis showed this example can be modified to obtain volume-preserving,
real-analytic actions of $\mathrm{SL}(n,\mathbb{Z})$ that are not $C^{0}$
conjugate to an affine action; moreover, these actions are not locally
$C^{1}$-rigid. In [MR2716615, MR2154667, MR2342012], constructions of non-
locally $C^{1}$-rigid, ergodic, volume-preserving actions of any lattice in a
simple Lie group are constructed by more general blow-up constructions.

Another example due to Stuck [MR1406436] demonstrates that it is impossible to
fully classify all lattice actions. Let $P\subset\mathrm{SL}(n,\mathbb{R})$ be
the group of upper triangular matrices. There is a non-trivial homomorphism
$\rho\colon P\to\mathbb{R}$. Now consider any flow (i.e. $\mathbb{R}$-action)
on a manifold $M$ and view the flow as a $P$-action via the image of $\rho$.
Then $G$ acts on the induced space $N=(G\times M)/P$ and the restriction
induces a non-volume-preserving, non-finite action of $\Gamma$. This example
shows—particularly in the non-volume-preserving-case—that care is needed in
order to formulate any precise conjectures that assert that every action
should be “of an algebraic origin.” Note, however, that we obtain a natural
map $N\to G/P$ that intertwines $\Gamma$-actions; in particular, this action
has an “algebraic action” as a factor.

We refer to [MR2807830, Sections 9 and 10] for more detailed discussion and
references to modifications of algebraic actions and exotic actions.

####  10.4. Actions of lattices in rank-1 groups

Actions by lattices in higher-rank Lie groups are expected to be rather
constrained. Although Example 10.10 shows there exists exotic, genuinely “non-
algebraic” actions of such groups, these actions are built from modifying
algebraic constructions or factor over algebraic actions. For lattices in
rank-one Lie groups such as $\mathrm{SL}(2,\mathbb{R})$, the situation is very
different. There exist natural actions that have no algebraic origin and the
algebraic actions of such groups seem to exhibit far less rigidity (for
example Example 10.12 which is not locally rigid) than those above.

######  Example 10.11 (Actions of free groups).

Let $G=\mathrm{SL}(2,\mathbb{R})$. The free group $\Gamma=F_{2}$ is isomorphic
to a lattice in $G$. (For instance, the fundamental group of the punctured
torus is isomorphic to $F_{2}$; more explicitly, $\mathrm{SL}(2,\mathbb{Z})$
contains a copy of $F_{2}$ as an index 12 subgroup.) Let $M$ be any manifold
and let $f,g\in\mathrm{Diff}(M)$. Then $f$ and $g$ generate an action of
$\Gamma$ on $M$ which in general is not of an algebraic origin and does not
exhibit any local rigidity. In particular, there is no expectation that any
rigidity phenomena should hold for actions of all lattices in
$\mathrm{SL}(2,\mathbb{R})$.

For the next example, recall Definitions 1.1 and 1.3 of Anosov actions.

######  Example 10.12 (Non-standard Anosov actions of
$\mathrm{SL}(2,\mathbb{Z})$).

Consider the standard action $\alpha_{0}$ of $\mathrm{SL}(2,\mathbb{Z})$ on
the 2 torus $\mathbb{T}^{2}$ as constructed in Example 10.5. In [MR1154597,
Example 7.21], Hurder presents an example of a 1-parameter family of
deformations
$\alpha_{t}\colon\mathrm{SL}(2,\mathbb{Z})\to\mathrm{Diff}(\mathbb{T}^{2})$ of
$\alpha_{0}$ with the following properties:

  1. (1)

Each $\alpha_{t}$ is a real-analytic, volume-preserving action;

  2. (2)

For $t>0$, $\alpha_{t}$ is not topologically conjugate to $\alpha_{0}$, (even
when restricted to a finite-index subgroup of $\mathrm{SL}(2,\mathbb{Z})$.)

Moreover, since $\alpha_{0}$ is an Anosov action and since the Anosov property
is an open property we have that

  1. (3)

each $\alpha_{t}$ is an Anosov action.

This shows that even affine Anosov actions of $\mathrm{SL}(2,\mathbb{Z})$ fail
to exhibit local rigidity properties and that there exist genuinely exotic
Anosov actions of $\mathrm{SL}(2,\mathbb{Z})$. This is in stark contrast to
the affine Anosov actions of higher-rank lattices which are known to be
locally rigid by [MR1632177, Theorem 15].

In contrast, it is expected that all Anosov actions of higher-rank lattices
are smoothly conjugate to affine actions as in Example 10.5 or Remark 10.6 (or
analogous constructions in infra-nilmanifolds). See Question 11.1(6) below.
Recent progress towards this conjecture appears in [BRHW1].

######  Remark 10.13.

There are a number of rank-1 Lie groups whose lattices are known to exhibit
some rigidity properties relative to linear representations. For instance,
Corlette established superrigidity and arithmeticity of lattices in certain
rank-1 simple Lie groups in [MR1147961]. In particular, Corlette establishes
superrigidity for lattices in $\mathrm{Sp}(n,1)$ and $F_{4}^{-20}$, the
isometry groups of quaternionic hyperbolic space and the Cayley plane. It
seems plausible that lattices in certain rank-1 Lie groups would exhibit some
rigidity properties for actions by diffeomorphisms; currently, there do not
seem to be any results in this direction.

###  11\. Actions in low dimension and Zimmer’s conjecture

####  11.1. Motivating questions

For actions by lattices in rank-1 groups, we have seen that it is easy to
construct exotic actions of free groups and Example 10.12 shows there are
exotic Anosov actions of $\mathrm{SL}(2,\mathbb{Z})$ on tori.

However, for actions of lattices in higher-rank, simple Lie groups, the
situation is expected to be far more rigid. In particular, the examples from
the previous section lead to a number of more precise questions and
conjectures. For concreteness, fix $n\geq 3$ and let
$G=\mathrm{SL}(n,\mathbb{R})$. Let $\Gamma\subset G$ be a lattice. Recall the
action of $\Gamma$ on $S^{n-1}$ and the volume-preserving Anosov action of
$\Gamma=\mathrm{SL}(n,\mathbb{Z})$ on $\mathbb{T}^{n}$.

######  Questions 11.1.

Consider the following questions:

  1. (1)

Is there a non-finite action of $\Gamma$ on a manifold of dimension at most
$n-2$?

  2. (2)

If the answer to (1) is unknown, does every action of $\Gamma$ on a manifold
of dimension at most $n-2$ preserve a volume form?

  3. (3)

Is there a non-finite, volume-preserving action of $\Gamma$ on a manifold of
dimension at most $n-1$?

  4. (4)

Is every non-finite action of $\Gamma$ on an $n$-torus of the type considered
in Example 10.5? What about volume-preserving actions? That is, if
$\alpha\colon\Gamma\to\mathrm{Diff}(\mathbb{T}^{n})$ is a non-finite action is
$\alpha$ smoothly conjugate to an affine action as in Example 10.5 (or as in
Remark 10.6)?

  5. (5)

Are the only non-finite actions of $\Gamma$ on a connected $(n-1)$-manifold
those considered in Example 10.7? That is, if
$\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ is a non-finite action is $M$
either $S^{n-1}$ or $\mathbb{R}P^{n-1}$ and is $\alpha$ smoothly conjugate to
the projective action?

Motivated by various conjectures on the classification of Anosov
diffeomorphisms and Question 11.1(4), we also pose the following.

  1. (6)

Is every (volume-preserving) Anosov action of $\Gamma$ of the type considered
in Example 10.5? That is, if $\alpha\colon\Gamma\to\mathrm{Diff}(M)$ is an
Anosov action is $M$ a (infra-)nilmanifold and is $\alpha$ smoothly conjugate
to an affine action as in Example 10.5 (or as in Remark 10.6)?

11.1(1) and (3) are referred to as Zimmer’s conjecture, discussed in the next
section. Question 11.1(2) is irrelevant given a negative answer to Question
11.1(1) but motivated the result stated in Theorem 11.4 below and was natural
to conjecture before an answer to Question 11.1(1) was known. It may be that
answering Question 11.1(2) is possible in dimension ranges where Conjecture
11.3(1) below is expected to hold but is not yet known.

####  11.2. Zimmer’s conjecture for actions by lattices in
$\mathrm{SL}(n,\mathbb{R})$

Recall Example 10.5 and Example 10.7. For lattices in
$\mathrm{SL}(n,\mathbb{R})$, Zimmer’s conjecture asserts that these are the
minimal dimensions in which non-finite actions can occur. We have the
following precise formulation.

######  Conjecture 11.2.

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice. Let
$M$ be a compact manifold.

  1. (1)

If $\dim(M)<n-1$ then any homomorphism $\Gamma\rightarrow\mathrm{Diff}(M)$ has
finite image.

  2. (2)

In addition, if $\mathrm{vol}$ is a volume form on $M$ and if $\dim(M)=n-1$
then any homomorphism $\Gamma\rightarrow\mathrm{Diff}_{\mathrm{vol}}(M)$ has
finite image.

We are intentionally vague about the regularity in 11.2 (and 11.3 below).
Zimmer originally stated 11.2(2) for the case of $C^{\infty}$ volume-
preserving actions; see [MR682830, MR934329, MR900826]. 11.2(1) for
$C^{\infty}$ actions first appears in [MR1666834]. Most evidence for the
conjecture requires the action to be at least $C^{1}$. It is possible the
conjecture holds for actions by homeomorphisms; see for instance [MR1198459,
MR2807834, MR3150210] for a partial list of results in this directions. The
results we discuss below require the action to be at least $C^{1+\beta}$ as we
use tools nonuniformly hyperbolic dynamics though some of our results still
hold for actions by $C^{1}$ diffeomorphisms (see Theorem 11.7 below.)

$\bigast$11.3. Zimmer’s conjecture for actions by lattices in other Lie
groups. To formulate Zimmer’s conjecture for lattices in general Lie groups,
to each simple, non-compact Lie group $G$ we associate 3 positive integers
$d_{0}(G),d_{\mathrm{rep}}(G),d_{\mathrm{cmt}}(G)$ defined roughly as follows:

  1. (1)

$d_{0}(G)$ is the minimal dimension of $G/H$ as $H$ varies over proper closed
subgroups $H\subset G$. (We remark that $H$ is necessarily a parabolic
subgroup in this case.)

  2. (2)

$d_{\mathrm{rep}}(G)$ is the minimal dimension of a non-trivial linear
representation of (the Lie algebra) of $G$.

  3. (3)

$d_{\mathrm{cmt}}(G)$ is the minimal dimension of a non-trivial homogeneous
space of a compact real form of $G$.

See Table 1 where we compute the above numbers for a number of matrix groups,
(split) real forms of exceptional Lie algebras, and complex matrix groups. We
also include another number $r(G)$ which is defined in [AWBFRHZW-
latticemeasure, 1608.04995] and arises from certain dynamical arguments444A
precise definition that is equivalent to that in [AWBFRHZW-latticemeasure,
1608.04995] is that $r(G)$ is $d_{0}(G^{\prime})$ where $G^{\prime}$ is the
largest $\mathbb{R}$-split simple subgroup in $G$.; this number gives the
bounds appearing in the most general result, Theorem 19.2 below, towards
solving Conjecture 11.3. For complete tables of values of
$d_{\mathrm{rep}}(G)$, $d_{\mathrm{cmt}}(G)$, and $d_{0}(G)$, we refer to
[Cantat].

$G$ |  restricted root system | rank | $d_{\mathrm{rep}}(G)$ | $d_{\mathrm{cmt}}(G)$ | $d_{0}(G)$ | $r(G)$
---|---|---|---|---|---|---
$\mathrm{SL}(n,\mathbb{R})$ | $A_{n-1}$ | $n-1$ | $n$ | $2n-2$ | $n-1$ | $n-1$
$\mathrm{SO}(n,n+1)$ | $B_{n}$ | $n$ | $2n+1$ | $2n$ | $2n-1$ | $2n-1$
$\mathrm{Sp}(2n,\mathbb{R})$ | $C_{n}$ | $n$ | $2n$ | $4n-4$ | $2n-1$ | $2n-1$
$\mathrm{SO}(n,n)$ | $D_{n}$ | $n$ | $2n$ | $2n-1$ | $2n-2$ | $2n-2$
$E_{I}$ | $E_{6}$ | $6$ | $27$ | $26$ | $16$ | $16$
$E_{V}$ | $E_{7}$ | $7$ | $56$ | $54$ | $27$ | $27$
$E_{VIII}$ | $E_{8}$ | $8$ | $248$ | $112$ | $57$ | $57$
$F_{1}$ | $F_{4}$ | $4$ | $26$ | $16$ | $15$ | $15$
$G$ | $G_{2}$ | $2$ | $7$ | $6$ | $5$ | $5$
$\mathrm{SL}(n,\mathbb{C})$ | $A_{n-1}$ | $n-1$ | $2n$ | $2n-2$ | $2n-2$ | $n-1$
$\mathrm{SO}(2n,\mathbb{C})$ | $D_{n}$ | $n$ | $4n$ | $2n-1$ | $4n-4$ | $2n-2$
$\mathrm{SO}(2n+1,\mathbb{C})$ | $B_{n}$ | $n$ | $4n+2$ | $2n$ | $4n-2$ | $2n-1$
$\mathrm{Sp}(2n,\mathbb{C})$ | $C_{n}$ | $n$ | $4n$ | $4n-4$ | $4n-2$ | $2n-1$
$\mathrm{SO}(p,q)$ $p<q$ | $B_{p}$ | $p$ | $p+q$ | $p+q-1$ | $p+q-2$ | $2p-1$
Table 1. Numerology appearing in Zimmer’s conjecture for various groups. See
also [Cantat] for more complete tables. See Theorem 19.2 where the number
$r(G)$ appears and [AWBFRHZW-latticemeasure, 1608.04995] or Footnote 4 for
definition.

Given the examples in Section 10.3 and the integers
$d_{\mathrm{rep}}(G),d_{\mathrm{cmt}}(G)$, and $d_{0}(G)$ defined above, it is
natural to conjecture the following.

######  Conjecture 11.3 (Zimmer’s Conjecture).

Let $G$ be a connected, simple Lie group with finite center. Let
$\Gamma\subset G$ be a lattice. Let $M$ be a compact manifold and
$\mathrm{vol}$ a volume form on $M$. Then

  1. (1)

if $\dim(M)<\min\\{d_{\mathrm{rep}}(G),d_{\mathrm{cmt}}(G),d_{0}(G)\\}$ then
any homomorphism $\alpha\colon\Gamma\rightarrow\mathrm{Diff}(M)$ has finite
image;

  2. (2)

if $\dim(M)<\min\\{d_{\mathrm{rep}}(G),d_{\mathrm{cmt}}(G)\\}$ then any
homomorphism $\alpha\colon\Gamma\rightarrow\mathrm{Diff}_{\mathrm{vol}}(M)$
has finite image;

  3. (3)

if $\dim(M)<\min\\{d_{0}(G),d_{\mathrm{rep}}(G)\\}$ then for any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}(M)$, the image $\alpha(\Gamma)$
preserves a Riemannian metric;

  4. (4)

if $\dim(M)<d_{\mathrm{rep}}(G)$ then for any homomorphism
$\alpha\colon\Gamma\rightarrow\mathrm{Diff}_{\mathrm{vol}}(M)$, the image
$\alpha(\Gamma)$ preserves a Riemannian metric.

####  11.4. Recent results in the Zimmer program

The following two recent results address 11.1(1)–(3) above. In the remainder
of this part, we outline their proofs (at times, specializing to the case of
$C^{\infty}$ actions and the case of $G=\mathrm{SL}(3,\mathbb{R})$.) We also
refer the reader to the excellent article by Serge Cantat [Cantat] that
presents (in French) a complete proof of Theorem 11.6.

Before an answer to 11.1(1) and (3) were known, the author together with
Federico Rodriguez Hertz and Zhiren Wang studied Question 11.1(2) and were
able to show that all such actions preserve some probability measure.

######  Theorem 11.4 ([AWBFRHZW-latticemeasure, Theorem 1.6]).

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice. Let
$M$ be a manifold with $\dim(M)<n-1$. Then, for any $C^{1+\beta}$ action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$, there exists an
$\alpha$-invariant Borel probability measure.

For actions on the circle, an analogue of Theorem 11.4 is shown in [MR1703323,
Theorem 3.1] for actions by homeomorphisms.

In the critical dimension, $\dim(M)=n-1$, the projective action on
$\mathbb{R}P^{n-1}$ discussed in Example 10.7 gives an example of an action
that does not preserve any Borel probability measure. If $\alpha$ is an action
of $\Gamma$ on a space $X$, we say that a Borel probability measure $\mu$ is
nonsingular for $\alpha$ if the measure class of $\mu$ is preserved by the
action. In particular, any smooth volume on $\mathbb{R}P^{n-1}$ is nonsingular
for the projective action. In [AWBFRHZW-latticemeasure, Theorem 1.7], it is
shown that all non-measure-preserving actions on manifolds of the critical
dimension $(n-1)$ have the projective action on $\mathbb{R}P^{n-1}$ equipped
with a smooth volume as a measurable factor. Precisely, for any action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1+\beta}(M)$ where $\dim(M)=n-1$ it is
shown that either

  1. (1)

there exists an $\alpha$-invariant Borel probability measure $\mu$ on $M$; or

  2. (2)

there exists a Borel probability measure $\mu$ on $M$ that is nonsingular for
the action $\alpha$; moreover the action $\alpha$ on $(M,\mu)$ is measurably
isomorphic to a finite extension of the projective action in Example 10.7 and
the image of $\mu$ factors to a smooth volume form on $\mathbb{R}P^{n-1}$.

This gives strong evidence for a positive answer to Question 11.1(5) which we
pose as a formal conjecture.

######  Conjecture 11.5.

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice, let
$M$ be a closed $(n-1)$-dimensional manifold, and let
$\alpha\colon\Gamma\to\mathrm{Diff}^{\infty}(M)$ be an action with infinite
image. Then, either $M=S^{n-1}$ or $M=\mathbb{R}P^{n-1}$ and the action
$\alpha$ is $C^{\infty}$ conjugate to the projective action on either
$S^{n-1}$ or $\mathbb{R}P^{n-1}$ in Example 10.7.

Returning to actions on manifolds below the critical dimensions in Zimmer’s
conjecture, the author with David Fisher and Sebastian Hurtado recently
answered 11.1(1) and (3) for actions by cocompact lattices in
$\mathrm{SL}(n,\mathbb{R})$ in [1608.04995].

######  Theorem 11.6 ([1608.04995, Theorem 1.1]).

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a cocompact
lattice. Let $M$ be a compact manifold.

  1. (1)

If $\dim(M)<n-1$ then any homomorphism $\Gamma\rightarrow\mathrm{Diff}^{2}(M)$
has finite image.

  2. (2)

In addition, if $\mathrm{vol}$ is a volume form on $M$ and if $\dim(M)=n-1$
then any homomorphism $\Gamma\rightarrow\mathrm{Diff}^{2}_{\mathrm{vol}}(M)$
has finite image.

The proof of Theorem 11.6 uses ideas and results from [AWBFRHZW-
latticemeasure], particularly the proof of Theorem 11.4, as ingredients. Thus,
while Theorem 11.4 follows trivially from Theorem 11.6, we include the proof
of Theorem 11.4 below as key ideas (namely, Theorem 15.1, 15.1′, and
Proposition 15.5) will be needed in the proof of Theorem 11.6.

Remarks on Theorem 11.6. We give a number of remarks on extensions of Theorem
11.6. See also the discussion in Section 18.5.

  1. (1)

Recently, the author, together with David Fisher and Sebastian Hurtado,
announced in [1710.02735] that the conclusion of Theorem 11.6 holds for
actions of $\mathrm{SL}(n,\mathbb{Z})$ for $n\geq 3$. The result for general
lattices in $\mathrm{SL}(n,\mathbb{Z})$ as well as analogous results for
lattices in other higher-rank simple Lie groups, has been announced [BFHWM].
This establishes 11.2 for actions by $C^{2}$ (and even $C^{1+\beta}$
diffeomorphisms). See Theorem 19.2.

The results for actions of $\mathrm{SL}(n,\mathbb{Z})$ and of general
nonuniform lattices use many of the ideas presented in this text but also
require a number of new techniques (including the structure of arithmetic
groups, reduction theory, and ideas from [MR1828742]) which will not be
discussed in this text.

  2. (2)

We state Theorem 11.6 for actions by $C^{2}$ diffeomorphisms though the proof
can be adapted for actions by $C^{1+\beta}$ actions. Our proof below will
assume the action is by $C^{\infty}$ diffeomorphisms to simplify certain
Sobolev space arguments.

  3. (3)

The result for actions by lattices in general Lie groups is stated in Theorem
19.2 below. In particular, parts (1) and (2) of Conjecture 11.3 hold for
$C^{1+\beta}$ actions by lattices in simple Lie groups that are non-
exceptional, split real forms. For $C^{1+\beta}$ actions by lattices in simple
Lie groups that are exceptional, split real forms, Conjecture 11.3(1) holds.

  4. (4)

D. Damjanovich and Z. Zhang observed that the proof of Theorem 11.6 can be
adapted to the setting of actions by $C^{1}$-diffeomorphisms. Together with
the author, they have announced the following theorem.

######  Theorem 11.7 ([BDZ]).

Let $\Gamma\subset G$ be a lattice in a higher-rank simple Lie group $G$ with
finite center. Let $M$ be a compact manifold.

    1. (a)

If $\dim(M)<\mathrm{rank}(G)$ then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{1}(M)$ has finite image.

    2. (b)

In addition, if $\mathrm{vol}$ is a volume form on $M$ and if
$\dim(M)=\mathrm{rank}(G)$ then any homomorphism
$\Gamma\rightarrow\mathrm{Diff}^{1}_{\mathrm{vol}}(M)$ has finite image.

For actions by lattices in other higher-rank groups, there is a gap between
what is known for $C^{1}$ versus $C^{1+\beta}$-actions. Indeed, our number
$r(G)$ in Theorem 19.2 always satisfies $r(G)\geq\mathrm{rank}(G)$ and is a
strict inequality unless $G$ has restricted root system of type $A_{n}$.

###  12\. Superrigidity and heuristics for Conjecture 11.2

The original conjecture (for actions by lattices in
$\mathrm{SL}(n,\mathbb{R})$) posed by Zimmer was Conjecture 11.2(2) (see for
example [MR934329, Conjecture II]). Conjecture 11.2(1) was formulated later
and first appears in print in [MR1666834, Conjecture I]. The reason Zimmer
posed his conjecture as Conjecture 11.2(2) is that the strongest evidence for
the conjecture—Zimmer’s cocycle superrigidity theorem—requires the action to
preserve some Borel probability measure. Zimmer’s cocycle superrigidity
theorem also provides strong evidence for local and global rigidity
conjectures related to Questions 11.1(4)–(6) and is typically used in proofs
of results towards solving such conjectures.

In this section, we state a version of Zimmer’s cocycle superrigidity theorem
and some consequences. We also state a version of Margulis’s superrigidity
theorem (for linear representations). We also give some heuristics for
Zimmer’s conjecture that follow from the superrigidity theorems. General
references for this section include [MR1090825, MR776417, MR3307755].

####  12.1. Cocycles over group actions

Consider a standard probability space $(X,\mu)$. Let $G$ be a locally compact
topological group and let $\alpha\colon G\times X\to X$ be a measurable action
of $G$ by $\mu$-preserving transformations. In particular, $\alpha(g)$ is a
$\mu$-preserving, measurable transformation of $X$ for each $g\in G$. Below,
we will always assume the measure $\mu$ is ergodic for this action.

A $d$-dimensional measurable linear cocycle over $\alpha$ is a measurable map

$\mathcal{A}\colon G\times X\to\mathrm{GL}(d,\mathbb{R})$

satisfying for a.e. $x\in X$ the cocycle condition: for all $g_{1},g_{2}\in
G$,

$\mathcal{A}(g_{1}g_{2},x)=\mathcal{A}\left(g_{1},\alpha(g_{2})(x)\right)\mathcal{A}(g_{2},x).$
(12.1)

If $e$ is the identity element of $G$, then (12.1) implies that

$\mathcal{A}(e,x)=\mathcal{A}(e,x)\mathcal{A}(e,x)$

whence $\mathcal{A}(e,x)=\mathrm{Id}$ for a.e. $x$.

We say two cocycles $\mathcal{A},\mathcal{B}\colon G\times
X\to\mathrm{GL}(d,\mathbb{R})$ are (measurably) cohomologous if there is a
measurable map $\Phi\colon X\to\mathrm{GL}(d,\mathbb{R})$ such that for a.e.
$x$ and every $g\in G$,

$\mathcal{B}(g,x)=\Phi(\alpha(g)(x))^{-1}\mathcal{A}(g,x)\Phi(x).$ (12.2)

We say a cocycle $\mathcal{A}\colon G\times X\to\mathrm{GL}(d,\mathbb{R})$ is
constant if $\mathcal{A}(g,x)$ is independent of $x$, that is, if
$\mathcal{A}\colon G\times X\to\mathrm{GL}(d,\mathbb{R})$ coincides with a
representation $\pi\colon G\to\mathrm{GL}(d,\mathbb{R})$ on a set of full
measure.

As a primary example, let $\alpha\colon G\to\mathrm{Diff}^{1}_{\mu}(M)$ be an
action of $G$ by $C^{1}$ diffeomorphisms of a compact manifold $M$ preserving
some Borel probability measure $\mu$. Although the tangent bundle $TM$ may not
be a trivial bundle, we may choose a Borel measurable trivialization
$\Psi\colon TM\to M\times\mathbb{R}^{d}$ of the vector-bundle $TM$ where
$d=\dim(M)$. We have that $\Psi$ factors over the identity map on $M$ and,
writing $\Psi_{x}\colon T_{x}\to\mathbb{R}^{d}$ for the identification of the
fiber over $x$ with $\mathbb{R}^{d}$, we moreover assume that $\|\Psi_{x}\|$
and $\|\Psi_{x}^{-1}\|$ are uniformly bounded in $x$.

Fix such a trivialization $\Psi$ and define $\mathcal{A}$ to be the derivative
cocycle relative to this trivialization:

$\mathcal{A}(g,x)=D_{x}\alpha(g)$

where, we view $D_{x}\alpha(g)$ as an element of $\mathrm{GL}(d,\mathbb{R})$
transferring the fiber $\\{x\\}\times\mathbb{R}^{d}$ to
$\\{\alpha(g)(x)\\}\times\mathbb{R}^{d}$ via the measurable trivialization
$\Psi$. To be precise, if $\Psi\colon TM\to M\times\mathbb{R}^{d}$ is the
measurable vector-bundle trivialization then

$\mathcal{A}(g,x):=\Psi(\alpha(g)(x))D_{x}\alpha(g)\Psi(x)^{-1}.$

In this case, the cocycle relation (12.1) is simply the chain rule. Note that
if we choose another Borel measurable trivialization $\Psi^{\prime}\colon
TM\to M\times\mathbb{R}^{d}$ then we obtain a cohomologous cocycle
$\mathcal{A}^{\prime}$. Indeed, we have

$\mathcal{A}^{\prime}(g,x)=\Psi^{\prime}(\alpha(g)(x))\Psi(\alpha(g)(x))^{-1}\mathcal{A}(g,x)\Psi(x)\Psi^{\prime}(x)^{-1}$

so we may take $\Phi(x)=\Psi(x)\Psi^{\prime}(x)^{-1}$ in (12.2).

We have the following elementary fact which we frequently use in the case of
volume-preserving actions.

######  Claim 12.1.

Let $\alpha\colon G\to\mathrm{Diff}^{1}_{\mathrm{vol}}(M)$ be an action by
volume-preserving diffeomorphisms. Then, for any $\alpha$-invariant measure
$\mu$, the derivative cocycle $\mathcal{A}$ is cohomologous to a
$\mathrm{SL}^{\pm}(d,\mathbb{R})$-valued cocycle.

Above, $\mathrm{SL}^{\pm}(d,\mathbb{R})$ is the subgroup of
$\mathrm{GL}(d,\mathbb{R})$ defined by $\det(A)=\pm 1$.

####  12.2. Cocycle superrigidity

We formulate the statement of Zimmer’s cocycle superrigidity theorem when $G$
is either $\mathrm{SL}(n,\mathbb{R})$ or a lattice subgroup of
$\mathrm{SL}(n,\mathbb{R})$ for $n\geq 3$. Note that the version formulated by
Zimmer (see [MR776417]) had a slightly weaker conclusion. We state the
stronger version formulated and proved in [MR2039990].

######  Theorem 12.2 (Cocycle superrigidity [MR776417, MR2039990]).

For $n\geq 3$, let $G$ be either $G=\mathrm{SL}(n,\mathbb{R})$ or let $G$ be a
lattice in $\mathrm{SL}(n,\mathbb{R})$. Let $\alpha\colon
G\to\mathrm{Aut}(X,\mu)$ be an ergodic, measurable action of $G$ by
$\mu$-preserving transformations of a standard probability space $(X,\mu)$.
Let $\mathcal{A}\colon G\times X\to\mathrm{GL}(d,\mathbb{R})$ be a
bounded,555Here, bounded means that for every compact $K\subset G$, the map
$K\times X\to\mathrm{GL}(d,\mathbb{R})$ given by
$(g,x)\mapsto\mathcal{A}(g,x)$ is bounded. More generally, we may replace the
boundedness hypothesis with the hypothesis that the function
$x\mapsto\sup_{g\in K}\log\|\mathcal{A}(g,x)\|$ is $L^{1}(\mu)$. See
[MR2039990]. measurable linear cocycle over $\alpha$.

Then there exist

  1. (1)

a linear representation
$\rho\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$;

  2. (2)

a compact subgroup $K\subset\mathrm{GL}(d,\mathbb{R})$ that commutes with the
image of $\rho$;

  3. (3)

a $K$-valued cocycle $\mathcal{C}\colon G\times X\to K$;

  4. (4)

and a measurable function $\Phi\colon X\to\mathrm{GL}(d,\mathbb{R})$

such that for a.e. $x\in X$ and every $g\in G$

$\mathcal{A}(g,x)=\Phi(\alpha(g)(x))^{-1}\rho(g)\mathcal{C}(g,x)\Phi(x).$
(12.3)

In particular, Theorem 12.2 states that any bounded measurable linear cocycle
$\mathcal{A}\colon G\times X\to\mathrm{GL}(d,\mathbb{R})$ over the action
$\alpha$ is cohomologous to the product of constant cocycle $\rho\colon
G\to\mathrm{SL}(d,\mathbb{R})$ and a compact-valued cocycle $\mathcal{C}\colon
G\times X\to K\subset\mathrm{GL}(d,\mathbb{R})$.

####  12.3. Superrigidity for linear representations

Zimmer’s cocycle superrigidity theorem is an extension of Margulis’s
superrigidity theorem for linear representations. We formulate a version of
this theorem for linear representations of lattices in
$\mathrm{SL}(n,\mathbb{R})$.

######  Theorem 12.3 (Margulis superrigidity [MR1090825]).

For $n\geq 3$, let $\Gamma$ be a lattice in $\mathrm{SL}(n,\mathbb{R})$. Given
a representation $\rho\colon\Gamma\to\mathrm{GL}(d,\mathbb{R})$ there are

  1. (1)

a linear representation
$\hat{\rho}\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$;

  2. (2)

a compact subgroup $K\subset\mathrm{GL}(d,\mathbb{R})$ that commutes with the
image of $\hat{\rho}$

such that

$\hat{\rho}(\gamma)\rho(\gamma)^{-1}\in K$

for all $\gamma\in\Gamma$.

That is, $\rho=\hat{\rho}\cdot c$ is the product of the restriction of a
representation

$\hat{\rho}\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$

to $\Gamma$ and a compact-valued representation $c\colon\Gamma\to K$. Moreover
the image of $\hat{\rho}$ and $c$ commute.

In the case that $\Gamma$ is nonuniform, one can show that all compact-valued
representations $c\colon\Gamma\to K$ have finite image. See for instance the
discussion in [MR3307755, Section 16.4], especially [MR3307755, Exercise
16.4.1].

For certain cocompact $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$, there exists
compact-valued representations $c\colon\Gamma\to\mathrm{SU}(n)$ with infinite
image. (See discussion in Example 10.9.) The next theorem, characterizing all
homomorphisms from lattices in $\mathrm{SL}(n,\mathbb{R})$ into compact Lie
groups, shows that representations into $\mathrm{SU}(n)$ are more-or-less the
only such examples. The proof uses the $p$-adic version of Margulis’s
superrigidity theorem and some algebra. See [MR1090825, Theorem VII.6.5] and
[MR3307755, Corollary 16.4.2].

######  Theorem 12.4.

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice. Let
$K$ be a compact Lie group and $\pi\colon\Gamma\to K$ a homomorphism.

  1. (1)

If $\Gamma$ is nonuniform then $\pi(\Gamma)$ is finite.

  2. (2)

If $\Gamma$ is cocompact and $\pi(\Gamma)$ is infinite then there is a closed
subgroup $K^{\prime}\subset K$ with

$\pi(\Gamma)\subset K^{\prime}\subset K$

and the Lie algebra of $K^{\prime}$ is of the form
$\mathrm{Lie}(K^{\prime})=\mathfrak{su}(n)\times\dots\times\mathfrak{su}(n)$.

The appearance of $\mathfrak{su}(n)$ in (2) of Theorem 12.4 is due to the fact
that $\mathfrak{su}(n)$ is the compact real form of
$\mathfrak{sl}(n,\mathbb{R})$, the Lie algebra of $\mathrm{SL}(n,\mathbb{R})$.
For a cocompact lattice $\Gamma$ in $\mathrm{SO}(n,n)$ as in Example 10.9, the
analogue of Theorem 12.4 states that

$\mathrm{Lie}(K^{\prime})=\mathfrak{so}(2n)\times\dots\times\mathfrak{so}(2n).$

####  12.4. Heuristic evidence for Conjecture 11.2

We present a number of heuristics that motivate the conclusions of Conjectures
11.2 and 11.3.

#####  12.4.1. Analogy with linear representations

Note that if $d<n$, there is no non-trivial representation
$\hat{\rho}\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$;
moreover, by a dimension count, there is no embedding of $\mathfrak{su}(n)$ in
$\mathfrak{sl}(d,\mathbb{R})$. We thus immediately obtain as corollaries of
Theorems 12.3 and 12.4 the following.

######  Corollary 12.5.

For $n\geq 3$, let $\Gamma$ be a lattice in $G=\mathrm{SL}(n,\mathbb{R})$.
Then, for $d<n$, the image of any representation
$\rho\colon\Gamma\to\mathrm{GL}(d,\mathbb{R})$ is finite.

Conjecture 11.2 can be seen as a “nonlinear” analogue of this corollary. That
is, we aim to prove the same result when the linear group
$\mathrm{GL}(d,\mathbb{R})$ is replaced by certain diffeomorphism groups
$\mathrm{Diff}(M)$.

#####  12.4.2. Invariant measurable metrics

For $n\geq 3$, let $\Gamma$ be a lattice in $G=\mathrm{SL}(n,\mathbb{R})$ and
consider a measure-preserving action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1}_{\mu}(M)$ where $M$ is a compact
manifold of dimension at most $d\leq n-1$ and $\mu$ is an arbitrary Borel
probability measure on $M$ preserved by $\alpha$. The derivative cocycle of
the action $\alpha$ is then $\mathrm{GL}(d,\mathbb{R})$-valued. Since there
are no representations
$\rho\colon\mathrm{SL}(n,\mathbb{R})\to\mathrm{SL}(d,\mathbb{R})$ for $d<n$,
Theorem 12.2 implies that the derivative cocycle is cohomologous to a compact-
valued cocycle. In particular, we have the following:

######  Corollary 12.6.

For $\Gamma,M,\mu$ and $\alpha\colon\Gamma\to\mathrm{Diff}^{1}_{\mu}(M)$ as
above

  1. (1)

$\alpha$ preserves a ‘$\mu$-measurable Riemannian metric,’ namely there is a
$\mu$-measurable, $\alpha$-invariant, positive-definite symmetric two-form on
$TM$;

  2. (2)

for any $\epsilon>0$ and $\gamma\in\Gamma$, the set of $x\in M$ such that

$\liminf_{n\to\infty}\frac{1}{n}\log\|D_{x}\alpha(\gamma^{n})\|\geq\epsilon$

has zero $\mu$-measure.

For (1), suppose the derivative cocycle is cohomologous to a $K$-valued
cocycle for some compact group $K\subset\mathrm{GL}(d,\mathbb{R})$. One may
then pull-back any $K$-invariant inner product on $\mathbb{R}^{d}$ to $T_{x}M$
via the map $\Phi(x)$ in Theorem 12.2 to an $\alpha(\Gamma)$-invariant inner
product. Conclusion (2) follows from Poincaré recurrence to sets on which the
function $\Phi\colon M\to\mathrm{GL}(d,\mathbb{R})$ in Theorem 12.2 has
bounded norm and conorm. Note from (2) that all Lyapunov exponents for
individual elements of the action must vanish.

From Corollary 12.6, given $n\geq 3$ and a lattice $\Gamma$ in
$G=\mathrm{SL}(n,\mathbb{R})$, we have that every action
$\alpha\colon\Gamma\to\mathrm{Diff}^{1}_{\mathrm{vol}}(M)$ preserves a
Lebesgue-measurable Riemannian metric $g$ whenever $M$ is a compact manifold
of dimension at most $n-1$. Suppose one could show that $g$ was continuous or
$C^{\ell}$. As we discuss in Step 3 of Section 16 below, this combined with
Theorem 12.4 implies the image $\alpha(\Gamma)$ is finite. Thus, Conjecture
11.2(2) follows if one can promote the measurable invariant metric $g$
guaranteed by Corollary 12.6 of Theorem 12.2 to a continuous Riemannian
metric.

The discussion in the previous paragraphs suggests the following variant of
Conjecture 11.2(2) might hold:

> For $n\geq 3$, if $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ is a lattice and
> if $\mu$ is any fully supported Borel probability measure on a compact
> manifold $M$ of dimension at most $(n-1)$ then any homomorphism
>
> $\Gamma\rightarrow\mathrm{Diff}_{\mu}(M)$
>
> has finite image.

Our method of proof of Conjecture 11.2(2) does not establish this conjecture.
However, the conjecture would follow (even allowing for $\mu$ to have partial
support) if the global rigidity result in 11.5 holds.

#####  12.4.3. Actions with discrete spectrum

Upgrading the measurable invariant Riemannian metric in Corollary 12.6 to a
continuous Riemannian metric in the above heuristic seems quite difficult and
is not the approach we take in the proof of Theorem 11.6. In [MR743815],
Zimmer was able to upgrade the measurable metric to a continuous metric for
volume-preserving actions that are very close to isometries. This result now
follows from the local rigidity of isometric actions in [MR1779610,
MR2198325].

Zimmer later established a much stronger result in [MR1147291] which provides
very strong evidence for the volume-preserving cases in 11.3. Using the
invariant, measurable metric discussed above and that higher-rank lattices
have Property (T), Zimmer showed that any volume-preserving action appearing
in 11.3 has discrete spectrum. In particular, this result implies that (the
ergodic components of) all volume-preserving actions appearing in 11.3 are
measurably isomorphic to isometric actions.

###  13\. Structure theory of $\mathrm{SL}(n,\mathbb{R})$ and Cartan flows on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$

Let $G=\mathrm{SL}(n,\mathbb{R})$ and let $\Gamma\subset G$ be a lattice.
Recall we write $G=KAN$ for the Iwasawa decomposition where

$K=\mathrm{SO}(n,\mathbb{R}),\quad\quad
A=\\{\mathrm{diag}(e^{t_{1}},e^{t_{2}},\dots,e^{t_{n}}):t_{1}+\dots+t_{n}=0\\},$

and $N$ is the group of upper triangular matrices with 1s on the diagonal.

We will be interested in certain subgroups of $G$ and how they capture
dynamical information of the action of the Cartan subgroup $A$ on the
homogeneous space $G/\Gamma$.

####  13.1. Roots and root subgroups

We consider the following linear functionals

$\beta^{i,j}\colon A\to\mathbb{R}$

given as follows: for $i\neq j$,

$\beta^{i,j}\left(\mathrm{diag}(e^{t_{1}},e^{t_{2}},\dots,e^{t_{n}})\right)=t_{i}-t_{j}.$

The linear functionals $\beta^{i,j}$ are the roots of $G$.

Associated to each root $\beta^{i,j}$ is a 1-parameter unipotent subgroup
$U^{i,j}\subset G$. For instance, in $G=\mathrm{SL}(3,\mathbb{R})$ we have the
following 1-parameter flows

$u^{1,2}(t)=\left(\begin{array}[]{ccc}1&t&0\\\ 0&1&0\\\
0&0&1\end{array}\right),\hskip
6.99997ptu^{1,3}(t)=\left(\begin{array}[]{ccc}1&0&t\\\ 0&1&0\\\
0&0&1\end{array}\right),\hskip
6.99997ptu^{2,3}(t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&t\\\
0&0&1\end{array}\right),$ $u^{2,1}(t)=\left(\begin{array}[]{ccc}1&0&0\\\
t&1&0\\\ 0&0&1\end{array}\right),\hskip
6.99997ptu^{3,1}(t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
t&0&1\end{array}\right),\hskip
6.99997ptu^{3,2}(t)=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\
0&t&1\end{array}\right).$

We let $U^{i,j}$ denote the associated 1-parameter unipotent subgroups of $G$:

$U^{i,j}:=\\{u^{i,j}(t):t\in\mathbb{R}\\}.$ (13.1)

The groups $U^{i,j}$ have the property that conjugation by $s\in A$ dilates
their parametrization by $e^{\beta^{i,j}(s)}$:

$su^{i,j}(t)s^{-1}=u^{i,j}(e^{\beta^{i,j}(s)}t).$ (13.2)

In particular, if $g^{\prime}=u^{i,j}(t)\cdot g$ is in the $U^{i,j}$-orbit of
$g$ and $s\in A$ then

$s\cdot g^{\prime}=u^{i,j}(e^{\beta^{i,j}(s)}t)\cdot s\cdot g.$

####  13.2. Cartan flows

For concreteness, consider $G=\mathrm{SL}(3,\mathbb{R})$ and let $\Gamma$ be a
lattice in $\mathrm{SL}(3,\mathbb{R})$ such as $\mathrm{SL}(3,\mathbb{Z})$.
Let $X$ denote the coset space $X=G/\Gamma$. This is an $8$ dimensional
manifold (which is noncompact when $\Gamma$ is a nonuniform lattice such as
$\mathrm{SL}(3,\mathbb{Z})$.) $G$ acts on $X$ on the left: given $g\in G$ and
$x=g^{\prime}\Gamma\in X$ we have

$g\cdot x=gg^{\prime}\Gamma\in X.$

The Cartan subgroup $A\subset G$ is the subgroup of diagonal matrices with
positive entries

$A:=\left\\{\left(\begin{array}[]{ccc}e^{t_{1}}&0&0\\\ 0&e^{t_{2}}&0\\\
0&0&e^{t_{3}}\end{array}\right):t_{1}+t_{2}+t_{3}=0\right\\}.$

The group $A$ is isomorphic to $\mathbb{R}^{2}$, for instance, via the
embedding

$(s,t)\mapsto\mathrm{diag}(e^{s},e^{t},e^{-s-t}).$

We consider the action $\alpha\colon A\times X\to X$ of $A$ on $X$ given by

$\alpha(s)(x)=sx.$

For $x\in X$ let $W^{i,j}(x)$ be the orbit of $x$ under the 1-parameter group
$U^{i,j}$:

$W^{i,j}(x)=\\{u^{i,j}(t)x:t\in\mathbb{R}\\}\\}.$

For $s\in A$, we claim that the $s$-action on $X$ dilates the natural
parametrization of each $W^{i,j}(x)$ by exactly $\beta^{i,j}(s)$. Indeed, if
$x\in X$ and if $x^{\prime}=u^{i,j}(v)\cdot x\in W^{i,j}(x)$ then for $s\in A$
we have

$\displaystyle\alpha(s)(x^{\prime})$ $\displaystyle=su^{i,j}(v)x$
$\displaystyle=su^{i,j}(v)s^{-1}sx$
$\displaystyle=u^{i,j}(v^{\prime})\alpha(s)(x)$

where, using (13.2), we have that have

$v^{\prime}=e^{\beta^{i,j}(s)}v.$

In particular, we interpret the functionals $\beta^{i,j}$ as the (non-zero)
Lyapunov exponents for the $A$-action on $X$ (with respect to any
$A$-invariant measure). Note that the zero functional is a Lyapunov exponent
of multiplicity two corresponding to the $A$-orbits. The tangent spaces to
each $W^{i,j}(x)$ as well as the tangent space to the orbit $A\cdot x$ gives
the $A$-invariant splitting guaranteed by Theorem 7.4. Note that no two roots
$\beta^{i,j}$ are positively proportional and hence are their own coarse
Lyapunov exponents for the action (see Section 7.4.2).

###  14\. Suspension space and fiberwise exponents

We now begin the proofs of Theorem 11.4 and Theorem 11.6 with a technical but
crucial construction. Here, we induce from an action $\alpha$ of a lattice
$\Gamma$ on a manifold $M$ to an action of $G=\mathrm{SL}(n,\mathbb{R})$ on an
auxiliary manifold denoted by $M^{\alpha}$. The properties of the $G$-action
on $M^{\alpha}$ mimic the properties of the $\Gamma$-action on $M$. However,
for a number of reasons it is much more convenient to study the $G$-action on
$M^{\alpha}$. The construction is parallel to the construction described in
Section 6.1.

####  14.1. Suspension space and induced $G$-action

Fix $G=\mathrm{SL}(n,\mathbb{R})$ and let $\Gamma\subset G$ be a lattice. Let
$M$ be a compact manifold and let $\alpha\colon\Gamma\to\mathrm{Diff}(M)$ be
an action.

On the product $G\times M$ consider the right $\Gamma$-action

$(g,x)\cdot\gamma=(g\gamma,\alpha(\gamma^{-1})(x))$

and the left $G$-action

$a\cdot(g,x)=(ag,x).$

Define the quotient manifold $M^{\alpha}:=(G\times M)/\Gamma$. As the
$G$-action on $G\times M$ commutes with the $\Gamma$-action, we have an
induced left $G$-action on $M^{\alpha}$. For $g\in G$ and $x\in M^{\alpha}$ we
denote this action by $g\cdot x$ and denote the derivative of the
diffeomorphism $x\mapsto g\cdot x$ at $x\in M^{\alpha}$ $D_{x}g\colon
T_{x}M^{\alpha}\to T_{g\cdot x}M^{\alpha}$.

We write

$\pi\colon M^{\alpha}\to\mathrm{SL}(n,\mathbb{R})/\Gamma$

for the natural projection map. Note that $M^{\alpha}$ has the structure of a
fiber-bundle over $\mathrm{SL}(n,\mathbb{R})/\Gamma$ induced by the map $\pi$
with fibers diffeomorphic to $M$. The $G$-action permutes the $M$-fibers of
$M^{\alpha}$. We let $F=\ker(D\pi)$ be the fiberwise tangent bundle: for $x\in
M^{\alpha}$, $F(x)\subset T_{x}M^{\alpha}$ is the $\dim(M)$-dimensional
subspace tangent to the fiber through $x$.

Equip $M^{\alpha}$ with a continuous Riemannian metric. For convenience, we
moreover assume the restriction of the metric to $G$-orbits coincides under
push-forward by the projection $\pi\colon
M^{\alpha}\to\mathrm{SL}(n,\mathbb{R})/\Gamma$ with the metric on
$\mathrm{SL}(n,\mathbb{R})/\Gamma$ induced by a right-invariant (and left
$K$-invariant) metric on $G$. (We note that if $\Gamma$ is cocompact,
$M^{\alpha}$ is compact and all metrics are equivalent. In the case that
$\Gamma$ is not cocompact, some additional care is needed to ensure the metric
is well behaved in the fibers. We will not discuss the technicalities of this
case here.)

We outline the construction of such a metric. Fix a $C^{\infty}$ Riemannian
metric $\langle\cdot,\cdot\rangle$ on $TM$. Passing to a finite index
subgroup, we may assume that $K\backslash G/\Gamma$ is a manifold. When
$\Gamma$ is cocompact in $G$, this manifold is compact. (More generally, when
$\Gamma$ is nonuniform, $K\backslash G/\Gamma$ has a compactification as real-
analytic manifold with corners; see [MR2126641].) Let
$\\{\hat{\psi}_{i},i=1,\dots,m\\}$ be a finite, $C^{\infty}$ partition of
unity of the locally symmetric space $K\backslash G/\Gamma$ subordinate to
finitely many coordinate charts. Lift each $\hat{\psi}_{i}$ to a $K$-invariant
function defined on $G/\Gamma$. For each $i$, we select at compactly supported
$\psi_{i}\colon G\to[0,1]$ such that $\psi_{i}(g)=\hat{\psi}_{i}(g\Gamma)$,
the map $g\mapsto g\Gamma$ is injective on the support of $\psi_{i}$, and the
support of each $\psi_{i}$ intersects a fixed compact fundamental domain
containing the identity. Write $\psi_{i,\gamma}\colon G\to[0,1]$ for the
function

$\psi_{i,\gamma}(g)=\psi_{i}(g\gamma^{-1}).$

The supports satisfy
$\mathrm{supp}(\psi_{i,\gamma})\cap\mathrm{supp}(\psi_{i,\gamma^{\prime}})=\emptyset$
whenever $\gamma\neq\gamma^{\prime}$ and the collection
$\\{\psi_{i,\gamma}\mid i\in\\{1,\dots,m\\},\gamma\in\Gamma\\}$ is a partition
of unity on $G$. Given $v,w\in\\{g\\}\times T_{x}M$ set

$\langle
v,w\rangle_{g,x}:=\sum_{i=1}^{m}\sum_{\gamma\in\Gamma}\phi_{i,\gamma}(g)\langle
D_{x}\alpha(\gamma)(v),D_{x}\alpha(\gamma)(w)\rangle_{x}.$

Equip $T_{(g,x)}(G\times M)=T_{g}G\times T_{x}M$ with the product of the left
$K$-invariant, right $\Gamma$-invariant metric on $G$ and $\langle
v,w\rangle_{g,x}.$ Note that this metric is $\beta$-Hölder continuous if
$\alpha$ is an action by $C^{1+\beta}$ diffeomorphisms. We then verify that
$\Gamma$ acts by isometries and thus the metric descends to a metric on
$M^{\alpha}$. Indeed, writing $\|\cdot\|_{g,x}$ for the norm associated to
$\langle\cdot,\cdot\rangle_{g,x}$, for $v\in\\{g\hat{\gamma}\\}\times T_{x}M$
we have

$\displaystyle\|v\|_{g\hat{\gamma},x}^{2}$
$\displaystyle=\sum_{i=1}^{m}\sum_{\gamma\in\Gamma}\phi_{i,\gamma}(g\hat{\gamma})\|D_{x}\alpha(\gamma)(v)\|_{0}^{2}$
$\displaystyle=\sum_{i=1}^{m}\sum_{\gamma\in\Gamma}\phi_{i,\gamma\hat{\gamma}^{-1}}(g)\|D_{x}\alpha(\gamma)(v)\|_{0}^{2}$
$\displaystyle=\sum_{i=1}^{m}\sum_{\gamma\in\Gamma}\phi_{i,\gamma\hat{\gamma}^{-1}}(g)\|D_{x}\alpha(\gamma\hat{\gamma}^{-1}\hat{\gamma})(v)\|_{0}^{2}$
$\displaystyle=\sum_{i=1}^{m}\sum_{\gamma\in\Gamma}\phi_{i,\gamma\hat{\gamma}^{-1}}(g)\|D_{\alpha(\hat{\gamma})(x)}\alpha(\gamma\hat{\gamma}^{-1},\alpha(\hat{\gamma})(x))D_{x}\alpha(\hat{\gamma},x)(v)\|_{0}^{2}$
$\displaystyle=\|D_{x}\alpha(\hat{\gamma})v\|_{g,\alpha(\hat{\gamma})(x)}^{2}.$

####  14.2. Fiberwise Lyapunov exponents

Recall that $A\subset G$ is the subgroup

$A=\\{\mathrm{diag}(e^{t_{1}},e^{t_{2}},\dots,e^{t_{n}})\\}\simeq\mathbb{R}^{n-1}.$

The $G$-action on $M^{\alpha}$ restricts to an $A$-action on $M^{\alpha}$. Let
$\mu$ be any ergodic, $A$-invariant Borel probability measure on $M^{\alpha}$.
The $G$-action (and hence the $A$-action) permutes the fibers of $M^{\alpha}$
and hence the derivatives of the $G$\- and $A$-actions preserve the fiberwise
tangent subbundle $F\subset TM^{\alpha}$.

We equip $A\simeq\mathbb{R}^{n-1}$ with a norm $|\cdot|$. We may restrict
Theorem 7.4 to the $A$-invariant subbundle $F\subset TM^{\alpha}$ and obtain
Lyapunov exponent functionals for the fiberwise derivative cocycle. We thus
obtain

  1. (1)

an $A$-invariant set $\Lambda\subset M^{\alpha}$ with $\mu(\Lambda)=1$;

  2. (2)

linear functionals
$\lambda^{F}_{1,\mu},\lambda^{F}_{2,\mu},\dots,\lambda^{F}_{p,\mu}\colon
A\to\mathbb{R}$; and

  3. (3)

a $\mu$-measurable, $A$-invariant splitting
$F(x)=\bigoplus_{i=1}^{p}E_{i}^{F}(x)$ defined for $x\in\Lambda$

such that for every $x\in\Lambda$ and $v\in E^{F}_{i}(x)\smallsetminus\\{0\\}$

$\lim_{|a|\to\infty}\dfrac{\log\|D_{x}a(v)\|-\lambda^{F}_{i,\mu}(a)}{|a|}=0.$

In particular, for any $a\in A$ and $v\in F(x)\smallsetminus\\{0\\}$ we have

$\lim_{k\to\infty}\dfrac{1}{k}\log\|D_{x}a^{k}(v)\|=\lambda^{F}_{i,\mu}(a).$

A coarse fiberwise Lyapunov exponent $\chi^{F}_{\mu}$ is a positive
proportionality class of fiberwise Lyapunov exponents.

###  15\. Invariance principle and Proof of Theorem 11.4

####  15.1. Proof of Theorem 11.4

Given the constructions in Section 14 and Ledrappier’s theorem as formulated
in Theorem 9.5 (see also Proposition 5.8), we are now in a position to prove