functionals on $\mathbb{R}^{k}$ up to positive scalar multiple (i.e. elements
of $(\mathbb{R}^{k})^{*}/\sim$, where $f\sim g$ if and only if $f=\lambda g$
for some $\lambda>0$). For each $\alpha\in\Delta$, we let $[\alpha]$ denote
the a subset of linear functionals belonging to the class $\alpha$. We
abusively let $\alpha$ denote the “smallest” element of $[\alpha]$. That is,
we write:

$[\alpha]=\left\\{\alpha=c_{1}\alpha,c_{2},c_{3}\alpha,\dots,c_{\ell}\alpha\right\\},\mbox{
where }1=c_{1}<c_{2}<\dots<c_{\ell}$

for some $\ell=\ell(a)$. Given such an $[\alpha]$, an element
$a\in\mathbb{R}^{k}$, and a graded vector space
$V=\bigoplus_{i=1}^{\ell}E_{i}$, there exists a uniquely defined isomorphism
$a_{*}:V\to V$, by letting $a_{*}|_{E_{i}}$ be defined by scalar
multiplication by $e^{c_{i}\alpha(a)}$. We call each $E_{i}$ an Oseledets
space and $a_{*}$ the graded homothety induced by $a$.

Given a collection $a_{1},\dots,a_{n}\in\mathbb{R}^{k}$, let
$\Delta^{-}(\left\\{a_{i}\right\\})=\left\\{\chi\in\Delta:\chi(a_{i})<0\mbox{
for every }i=1,\dots,n\right\\}$ (we similarly define $\Delta^{+}$ as the set
of weights with positive evaluations on every $\chi\in\Delta$). A subset of
weights $\Phi\subset\Delta$ is called stable if there exists
$a\in\mathbb{R}^{k}$ such that $\Phi\subset\Delta^{-}(a)$.

We recall the following definitions from [48, Section 5.7]. Let
$a\in\mathbb{R}^{k}$ and let $\Phi$ be a stable subset of weights. We
introduce a (not necessarily unique) order on the set $\Phi$. Choose
$\mathbb{R}^{2}\cong V\subset\mathbb{R}^{k}$ which contains $a$ and for which
$\gamma_{1}|_{V}$ is proportional to $\gamma_{2}|_{V}$ if and only if
$\gamma_{1}$ is proportional to $\gamma_{2}$ for all
$\gamma_{1},\gamma_{2}\in\Phi$ (such choices of $V$ are open and dense). Fix
some nonzero $\chi\in V^{*}$ such that $\chi(a)=0$ ($\chi$ is not necessarily
a weight). Then $\beta|_{V}\in V^{*}\cong\mathbb{R}^{2}$ for every
$\beta\in\Phi$ and $\Phi|_{V}=\left\\{\beta|_{V}:\beta\in\Phi\right\\}$ is
contained completely on one side of the line spanned by $\chi$.

######  Definition 10.1.

The ordering $\beta<\gamma$ if and only if
$\angle(\chi,\beta|_{V})<\angle(\chi,\gamma|_{V})$ is called the circular
ordering of $\Phi$ (induced by $\chi$ and $V$) and is a total order on $\Phi$.
If the set $\Phi$ is understood, we let $|\alpha,\beta|=|\alpha,\beta|_{\Phi}$
denote the set of weights $\gamma\in\Phi$ such that
$\alpha\leq\gamma\leq\beta$.

While each $\beta\in\Delta$ is only defined up to positive scalar multiple,
this is still well-defined since the circular ordering on $\mathbb{R}^{2}$ is
invariant under orientation-preserving linear maps.

######  Definition 10.2.

If $\alpha,\beta\in\Delta$, let $\Sigma(\alpha,\beta)\subset\Delta$ (called
the $\alpha,\beta$-cone) be the set of $\gamma\in\Delta$ such that
$\gamma=\sigma\alpha+\tau\beta$ for some $\sigma,\tau>0$. We may identify
$\Sigma(\alpha,\beta)$ as a subset of the first quadrant of $\mathbb{R}^{2}$
by using the coordinates $(\sigma,\tau)$. The canonical circular ordering on
$\Sigma(\alpha,\beta)\cup\left\\{\alpha,\beta\right\\}$ is the
counterclockwise order in the first quadrant.

The Weyl chambers of $\Delta$ are the connected components of
$\mathbb{R}^{k}\setminus\bigcup_{\alpha\in\Delta}\ker\alpha$. An element
$a\in\mathbb{R}^{k}$ is called regular if $a$ belongs to a Weyl chamber. The
following lemma first appeared in [48, Lemma 5.32], and the proof is identical
(it is purely linear algebra).

######  Lemma 10.3.

Let $\alpha,\beta\in\Delta$ be linearly independent and
$\mathfrak{C}_{1},\dots,\mathfrak{C}_{m}$ be the Weyl chambers such that
$\alpha$ and $\beta$ are both negative on every $\mathcal{W}_{i}$. For each
such chamber, choose an arbitrary $a_{j}\in\mathfrak{C}_{j}$. Then
$\Sigma(\alpha,\beta)\cup\left\\{\alpha,\beta\right\\}=\Delta^{-}(\left\\{a_{i}\right\\})$.

As noted at the start of this section, we axiomatize the key structures
obtained from the dynamics of an Anosov $\mathbb{R}^{k}\times M$-action. Most
of the following properties were deduced in the previous section for smooth
Anosov actions. The reason for formulating the definition in the topological
setting is twofold: first, it allows us to identify which properties of smooth
systems are important and reference those properties quickly. Second, it
highlights the breadth of the geometric approach we employ. In particular, we
will be able to obtain a purely topological rigidity result without the use of
derivatives from these conditions.

######  Definition 10.4.

We suppose we have a continuous, locally free action of $\mathbb{R}^{k}\times
M$ on $X$ as described above, sets $\Delta$ and $[\alpha]$ as defined above,
and the properties (TA-1)-(TA-7) as described below. We call such an action a
leafwise homogeneous topological Anosov action of $\mathbb{R}^{k}\times M$.

  1. (TA-1)

The $\mathbb{R}^{k}$ action has a dense orbit.

  2. (TA-2)

For all $\alpha\in\Delta$, there is a nilpotent group $N_{\alpha}$ with
actions $N_{\alpha}\curvearrowright X$. These actions have locally bi-
Lipschitz evaluation maps onto their images and are globally Hölder.

  3. (TA-3)

For all $\alpha\in\Delta$ and $g\in\mathbb{R}^{k}\times M$, there exist
semisimple automorphisms $g_{*}:N_{\alpha}\to N_{\alpha}$ such that for all
$u\in N_{\alpha}$

$gug^{-1}\cdot x=(g_{*}u)\cdot x.$

Assume also that the $\operatorname{Lie}(N_{\alpha})$ automorphism induced by
$g_{*}$ is a graded homothety for $g\in\mathbb{R}^{k}$. (Note: we will use the
same notation $g_{*}$ for both the automorphism of the Lie group and Lie
algebra. Since these operate on different spaces, the context will make clear
which is intended.)

  4. (TA-4)

The set of $\mathbb{R}^{k}\times M$-closed orbits is dense.

  5. (TA-5)

If $a_{1},\dots,a_{m}\in\mathbb{R}^{k}$ is a list of regular elements, and
$\left\\{\chi_{1},\dots,\chi_{r}\right\\}$ is a circular ordering of
$\Delta^{-}(\left\\{a_{i}\right\\})$, the evaluation map from
$C_{(\chi_{1},\dots,\chi_{r})}$ is injective (recall the definition of
$C_{(\chi_{1},\dots,\chi_{r})}$ from Definition 6.3). Its image is denoted
$W^{s}_{(a_{1},\dots,a_{m})}(x)$.

  6. (TA-6)

If $a_{1},\dots,a_{m}\in\mathbb{R}^{k}$ are regular elements, then
$W^{s}_{(a_{1},\dots,a_{m})}(x)$ as defined in (TA-5) is exactly the set of
points $y\in X$ such that $d({a_{i}}^{t}x,{a_{i}}^{t}y)\to 0$ for every
$i=1,\dots,m$.

  7. (TA-7)

For any ordering $\bar{\beta}=(\beta_{1},...,\beta_{n})$ of $\Delta$ which
lists every weight exactly once, there exist arbitrarily small open sets
$U\subset C_{\bar{\beta}}\times(\mathbb{R}^{k}\times M)$ containing the
trivial path such that for every $x\in X$, the restriction of the evaluation
map at $x$ from $U\to X$ is onto a neighborhood of $x$.

We say that the action is genuinely higher rank if it also satisfies:

  8. (TA-8)

For every $\alpha\in\Delta$, $(\ker\alpha\times M)\cdot x$ is dense for some
$x\in X$.

We say that the action has SRB measures if it also satisfies:

  9. (TA-9)

for every pair $\alpha,\beta\in\Delta$, there exists a fully supported (not
necessarily ergodic) measure $\mu$ which is invariant under $\ker\beta\times
M$ and has absolutely continuous disintegrations along the
$N_{\alpha}$-foliation.

######  Remark 10.5.

Several properties listed here may follow from others. In particular, we
believe that properties (TA-4), (TA-6) and (TA-9) can be deduced from the
other conditions of the definition under a transitivity assumptions. Since the
proofs are straightforward when working with smooth systems, we add them as
additional conditions here. In Section 13 we deduce that the extension
constructed in Section 9 of a smooth Anosov action satisfying conditions
(FA-1) and (FA-2) is leafwise homogeneous topological Anosov action with SRB
measures. See also Section 4.6 and Remark 14.6 of [48].

######  Remark 10.6.

While many examples of smooth Anosov $\mathbb{R}^{k}\times M$ satisfy
assumptions (TA-1)-(TA-9), some do not. In particular, actions with nontrivial
Jordan blocks will fail to satisfy (TA-3) and actions with rank one factors
will fail to satisfy (TA-8). Among homogeneous actions, these are the actions
which fail to satisfy the conditions and we believe them to be the only ways
smooth Anosov $\mathbb{R}^{k}\times M$-actions fail to satisfy the conditions
(although conjecturally they are all still homogeneous). In particular,
$\mathbb{R}^{k}\times M$ actions which are restrictions of actions of
semisimple groups are all leafwise homogeneous topological Anosov actions (see
Proposition 13.1).

Note that the genuinely higher rank assumptions force that $k\geq 2$ unless we
are discussing trivial actions (when $\Delta=\emptyset$ and the
$\mathbb{R}^{k}\times M$ action is transitive). Our main (extremal) technical
result of the next few sections follows. We require an additional assumption
of integral Lyapunov coefficients, defined after Definition 11.2. This will
always hold for $C^{2}$-actions by Lemma 13.3.

######  Theorem 10.7.

Let $\alpha$ be a genuinely higher-rank leafwise homogeneous topological
Anosov action with integral Lyapunov coefficients and SRB measures. Then there
exists a Lie group $H$, an embedding of $\mathbb{R}^{k}\times M$ into $H$, a
cocompact lattice $\Lambda\subset H$, and a homeomorphism $\phi:X\to
H/\Lambda$ which conjugates the $\mathbb{R}^{k}\times M$-action to the natural
actions by left translation on $H/\Lambda$ by $\mathbb{R}^{k}\times M$.

Throughout the remainder of Section 10, we assume that $\mathbb{R}^{k}\times
M\curvearrowright X$ is a genuinely higher-rank leafwise homogeneous
topological Anosov action unless otherwise stated.

###  10.1. Basic dynamical properties

######  Lemma 10.8.

Suppose that $\mathbb{R}^{k}\times M\curvearrowright X$ is a leafwise
homogeneous topological Anosov action. There exists $\varepsilon_{0}>0$ such
that if $d(ax,ay)<\varepsilon_{0}$ for all $a\in\mathbb{R}^{k}$, then
$y\in(\mathbb{R}^{k}\times M)\cdot x$.

###### Proof.

Choose $\varepsilon_{0}$ as in so that if $d(x,y)<\varepsilon_{0}$, then $y$
is in the image of the maps as in (TA-7). Then fix a regular
$a\in\mathbb{R}^{k}$ and write $y=g\rho_{1}\rho_{2}x$, where $\rho_{1}$ is a
product of the stable legs of $a$ listed in a cirular ordering, and $\rho_{2}$
is a product of the unstable legs of $a$, listed in a circular ordering. Since
$d(a^{t}x,a^{t}y)<\varepsilon_{0}$ for $t\to\pm\infty$, by (TA-5), the
endpoints and basepoints of $\rho_{1}$ and $\rho_{2}$ expand and contract in
opposite behaviors. Therefore, unless $\rho_{1}$ and $\rho_{2}$ are both
trivial, the points $x$ and $y$ will eventually separate by at least
$\varepsilon_{0}$, a contradiction. ∎

######  Lemma 10.9.

For every $x\in X$, $W^{\alpha}(x):=N_{\alpha}x$ consists of the set of points
$y\in X$ such that $d(a^{t}x,a^{t}y)\to 0$ as $t\to\infty$ for all $a$ such
that $\alpha(a)<0$. Furthermore,
$W^{\alpha}(x)=\bigcap_{\alpha(a)<0}W^{s}_{a}(x)$, and
$W^{s}_{(a_{1},\dots,a_{m})}(x)=\bigcap W^{s}_{a_{i}}(x)$.

###### Proof.

The proof is immediate from (TA-6). Indeed, notice that given any coarse
Lyapunov exponent $\alpha$ we may choose a collection
$\left\\{a_{1},\dots,a_{m}\right\\}$ such that
$\Delta^{-}(\left\\{a_{1},\dots,a_{m}\right\\})=[\alpha]$. Indeed, given any
collection which already contains $[\alpha]$, if
$\Delta^{-}(\left\\{a_{1},a_{2},\dots,a_{m}\right\\})\cap[\chi]\not=\emptyset$
for some $\chi\in\Delta\setminus[\alpha]$, we may remove a weight $\chi$ by
adding $a$ such that $\alpha(a)<0$ and $\chi(a)>0$. Again applying (TA-6) to
each $W^{s}_{a_{i}}(x)$ independently, it is clear that their intersection
must be $W^{\alpha}(x)$. ∎

We now describe a basic operation that is critical to our analysis of the way
the group actions of $N_{\alpha}$ interact with one another: the geometric
commutator. This is in contrast to the infinitesimal commutator, which only
exist when the space a has a smooth structure and the actions of $N_{\alpha}$
are known to be at least $C^{1}$. Even in the smooth setting, our foliations
have smooth leaves but are only Hölder transversally, so we instead use a
coarser version of the commutator which works even for leafwise homogeneous
topological Anosov actions.

Recall that if $\alpha$ and $\beta$ are coarse Lyapunov exponents,
$\Sigma(\alpha,\beta)$ is the set of exponents which can be written as
$\sigma\alpha+\tau\beta$, where $\sigma,\tau>0$ (Definition 10.2). Notice that
while $\alpha$ and $\beta$ are only coarse exponents, the set $\Sigma$ is
still well-defined since we consider all positive linear combinations. Given a
group $\mathcal{G}$ and elements $g,h\in\mathcal{G}$, we use the following
convention for group commutators:

$[u,v]=v^{-1}u^{-1}vu.$

######  Lemma 10.10.

Fix a leafwise homogeneous topological Anosov action. Let $\alpha$ and $\beta$
be non-proportional coarse Lyapunov exponents, $u\in N_{\alpha}$ and $v\in
N_{\beta}$. Then for every $x$, there exists a unique collection of elements
$w_{i}:=\rho^{\alpha\beta}_{\gamma_{i}}(u,v,x)\in N_{\gamma_{i}}$, where
$\gamma_{i}$ ranges over all coarse Lyapunov exponents in
$\Sigma(\alpha,\beta)$ listed so that
$(\alpha,\gamma_{1},\gamma_{2},\dots,\gamma_{n},\beta)$ is the canonical
circular ordering, satisfying:

$w_{n}*\dots*w_{2}*w_{1}*[u,v]\cdot x=x.$

Furthermore, the functions $\rho^{\alpha\beta}_{\gamma_{i}}$ are continuous in
all three variables, and satisfy the following equivariance property for any
$g\in\mathbb{R}^{k}\times M$:

(10.1)
$g_{*}\rho^{\alpha\beta}_{\gamma_{i}}(u,v,x)=\rho^{\alpha\beta}_{\gamma_{i}}(g_{*}u,g_{*}v,gx)$

###### Proof.

Consider the points $y=[u,v]\cdot x$, and notice that if $a\in\mathbb{R}^{k}$
satisfies $\alpha(a),\beta(a)<0$, then $d(a^{n}\cdot x,a^{n}\cdot y)\to 0$. In
particular, if $\left\\{a_{1},\dots,a_{m}\right\\}$ are elements such that
$\Delta^{-}(\left\\{a_{1},\dots,a_{m}\right\\})=\left\\{\alpha,\gamma_{1},\dots,\gamma_{n},\beta\right\\}$
(such a choice of $\left\\{a_{i}\right\\}$ exists by Lemma 10.3), then $y\in
W^{s}_{(a_{1},\dots,a_{m})}(x)$. Hence, by (TA-5) there exist elements
$u^{\prime}\in N_{\alpha}$, $v^{\prime}\in N_{\beta}$, $w_{i}\in
N_{\gamma_{i}}$ such that

(10.2) $y=v^{\prime}*w_{n}*\dots*w_{1}*u^{\prime}{{\cdot}}x.$

Choose $a\in\ker\alpha$ such that $\beta(a)<0$. Then since
$\gamma_{i}\in\Sigma(\alpha,\beta)$ for every $i$, $\gamma_{i}(a)<0$. In
particular,
$\lim_{n\to\infty}(a^{n})_{*}(v^{\prime}*w_{n}*\dots*w_{1}*u^{\prime})=u^{\prime}$,
so $\lim_{n\to\infty}d(a^{n}x,a^{n}y)>0$ unless $u^{\prime}=e$. However, since
$(a^{n})_{*}[u,v]=[u,(a^{n})_{*}v]\to[u,e]=e$, $d(a^{n}x,a^{n}y)\to 0$.
Therefore, $u^{\prime}=e$. Similarly, $v^{\prime}=e$ by choosing
$a\in\ker\beta$ such that $\alpha(a)<0$. Since $u^{\prime},v^{\prime}=e$, 10.2
is the desired expression. ∎

######  Definition 10.11.

The functions $\rho^{\alpha\beta}:N_{\alpha}\times
N_{\beta}\to\displaystyle\prod_{i=\\#\Sigma(\alpha,\beta)}^{1}N_{\gamma_{i}}$
and $\rho^{\alpha,\beta}_{\gamma_{i}}:N_{\alpha}\times N_{\beta}\to
N_{\gamma_{i}}$ are called geometric commutators (we do not endow the target
space with an algebraic structure, it is only a topological space). The
element:

(10.3) $\rho^{\alpha,\beta}(u,v,x)*[u,v]$

of $\mathcal{P}$ is called a commutator relation at $x$. It is an element of
the cycle subgroup at $x$.

######  Lemma 10.12.

Let $\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ be a collection of coarse
Lyapunov exponents listed in a circular ordering and $u_{i},v_{i}\in
N_{\gamma_{i}}$. If $u_{1}*\dots*u_{n}\cdot x=v_{1}*\dots*v_{n}\cdot x$ for
every $x\in X$, then $v_{i}=u_{i}$ for $i=1,\dots,n$.

###### Proof.

Suppose that $u_{1}*\dots*u_{r}\cdot x=v_{1}*\dots*v_{r}\cdot x$ for every
$x\in X$, where $u_{i},v_{i}\in N_{\alpha_{i}}$. Then

$u_{1}*\dots*u_{r}*{v_{r}}^{-1}*\dots*{v_{1}}^{-1}$

stabilizes every point of $X$. Picking some $a\in\ker\alpha_{r}$ such that
$\alpha_{i}(a)<0$ for all $i=1,\dots,r-1$ implies that

$(a_{*}u_{1})*\dots*(a_{*}u_{r-1})*u_{r}*{v_{r}}^{-1}*(a_{*}v_{r-1})^{-1}\dots*(a_{*}v_{1})^{-1}$

also stabilizes every point of $X$. Letting $\alpha_{i}(a)\to\infty$, $i<r$
implies that $u_{r}{v_{r}}^{-1}$ stabilizes every point of $X$. Since the
action of $N_{\alpha_{1}}$ is faithful, $u_{r}=v_{r}$. Iterating this
procedure by choosing $a_{i}\in\ker\beta_{i}$ such that $\alpha_{j}(a_{i})<0$
for $j=1,\dots,i-1$ inductively shows $u_{i}=v_{i}$, $i=1,\dots,r-1$. ∎

###  10.2. Groups generated by opposite weights

In this section, we study the interaction of the groups $N_{\alpha}$ and
$N_{-\alpha}$, where $\alpha\in\Delta$ is a weight such that
$-\alpha\in\Delta$. We will show that they fit into a Lie group action, and
establish certain structural features.

######  Lemma 10.13.

If $y\in\overline{\ker\alpha\cdot x}$, then $\overline{\ker\alpha\cdot
y}\subset\overline{\ker\alpha\cdot x}$.

###### Proof.

Suppose that $z\in\overline{\ker\alpha\cdot y}$ and $a\in\ker\alpha$ be such
that $d(ay,z)<\varepsilon$. Since $a:X\to X$ is continuous, we may choose
$\delta>0$ such that if $d(y,y^{\prime})<\delta$, then
$d(ay^{\prime},z)<\varepsilon$. Then choose $b\in\ker\alpha$ such that
$d(bx,y)<\delta$. Then by construction, $d(abx,z)<\varepsilon$ and
$z\in\overline{\ker\alpha\cdot x}$. ∎

The following lemma gives a topological analogue of an ergodic decomposition
for the action of $\ker\alpha$. A topological ergodic decomposition does not
always exist, but using the hyperbolic structures, we can obtain a partition
into closed sets on which $\ker\alpha$ is transitive.

######  Lemma 10.14.

Fix $\alpha\in\Delta$. There is an $\mathbb{R}^{k}$-invariant residual set of
points $x_{0}\in X$ such that $(M\times\ker\alpha)\cdot x_{0}$ is dense and
either

  1. (1)

$\ker\alpha\cdot x_{0}$ is dense, or

  2. (2)

$\mathcal{F}_{\alpha}(m):=m\cdot\overline{\ker\alpha\cdot x_{0}}$, $m\in M$
are a family of closed sets that partition $X$ and each atom is saturated by
$W^{\beta}$-leaves for every $\beta\in\Delta$ (including $\beta=\alpha$). The
indexing of the partition $\left\\{\mathcal{F}_{\alpha}(m):m\in M\right\\}$ by
$m$ depends on $x_{0}$, but the partition itself is independent of the choice
of $x_{0}$ from the residual set.

In case (2), for every $a\in\mathbb{R}^{k}$ and $m\in M$, there exists
$m^{\prime}\in M$ such that
$a\mathcal{F}_{\alpha}(m)=\mathcal{F}_{\alpha}(m^{\prime})$.

###### Proof.

Define $A_{\alpha}=\ker\alpha$ and let $W^{A_{\alpha}}$ be the accessibility
class of a regular element $a\in A_{\alpha}$. That is, $W^{A_{\alpha}}$ is the
set of points reached by finitely many legs in $W^{\beta}$,
$\beta\not=\pm\alpha$. Since periodic points are dense by (TA-4), by Lemma
4.21 of [48], the set of points for which
$W^{A_{\alpha}}(x)\subset\overline{A_{\alpha}\cdot x}$ is residual. Since
$W^{A_{\alpha}}(gx)=gW^{A_{\alpha}}(x)$ and $\overline{A_{\alpha}\cdot
gx}=g\overline{A_{\alpha}x}$, the same is true for every point along the
$\mathbb{R}^{k}\times M$-orbit of such a point $x$. We may therefore choose
$x_{0}$ such that $W^{A_{\alpha}}(x_{0})\subset\overline{A_{\alpha}\cdot
x_{0}}$ and $(M\times A_{\alpha})\cdot x_{0}$ is dense in $X$. Notice that
since the $M$-action normalizes the $N_{\beta}$ actions and $M\times
A_{\alpha}$ actions, the same is true for $m\cdot x_{0}$ for every $m\in M$.

To see that the sets
$\mathcal{F}_{\alpha}(m)=m\cdot\overline{A_{\alpha}x_{0}}=\overline{A_{\alpha}mx_{0}}$
partition the space $X$, observe first that they cover the space since
$(M\times A_{\alpha})\cdot x$ is dense by (TA-8), and $M$ is compact. We now
show that they partition the space. Indeed, assume
$\mathcal{F}_{\alpha}(m_{1})\cap\mathcal{F}_{\alpha}(m_{2})\not=\emptyset$, so
that there exists
$y\in\overline{A_{\alpha}m_{1}x_{0}}\cap\overline{A_{\alpha}m_{2}x_{0}}$. We
will show that $m_{1}x_{0}\in\overline{A_{\alpha}m_{2}x_{0}}$, which implies
that $\overline{A_{\alpha}m_{1}x_{0}}\subset\overline{A_{\alpha}m_{2}x_{0}}$
by Lemma 10.13 (the opposite inclusion follows similarly).

By assumption, $\overline{A_{\alpha}m_{i}x_{0}}$ each contain their respective
$W^{A_{\alpha}}(m_{i}x_{0})$. Since $y\in\overline{A_{\alpha}m_{1}\cdot
x_{0}}\cap\overline{A_{\alpha}m_{2}x_{0}}$, there exist points $x_{k}\in
A_{\alpha}m_{1}\cdot x_{0}$ and $x_{k}^{\prime}\in A_{\alpha}m_{2}\cdot x_{0}$
such that $x_{k},x_{k}^{\prime}\to y$. Choose $k$ large enough so that we may
find a connection from $x_{k}$ to $x_{k}^{\prime}$ in the following way:
$x_{k}$ is connected to a point $z_{k}$ by moving along a small piece of
$\mathbb{R}^{k}\times M$-orbit, a small $\alpha$-leaf and then a small
$-\alpha$ leaf. That is, $z_{k}=vugx_{k}$, where $v\in N_{-\alpha}$, $u\in
N_{\alpha}$ and $g\in\mathbb{R}^{k}\times M$ are all small group elements.
Then $z_{k}$ is connected to $x_{k}^{\prime}$ by the $W^{\beta}$ leaves,
$\beta\not=\pm\alpha$. Such a connection is possible when $x_{k}$ and
$x_{k}^{\prime}$ are close by (TA-7).

Now, by construction, $z_{k}\in W^{A_{\alpha}}(x_{k}^{\prime})$, so since
$W^{A_{\alpha}}(x_{k}^{\prime})\subset\overline{A_{\alpha}x_{k}^{\prime}}=\overline{A_{\alpha}m_{2}x_{0}}$
by the genericity assumption, we may find $b_{n}\in A_{\alpha}$ such that
$b_{n}m_{2}x_{0}\to z_{k}$. But if $x_{k}=a_{k}m_{1}x_{0}$, then
$z_{k}=vuga_{k}m_{1}x_{0}$, where $u\in N_{\alpha}$, $v\in N_{-\alpha}$ and
$g\in\mathbb{R}^{k}\times M$ become smaller with $k$. Since
$a_{k}\in\ker\alpha$, $a_{k}^{-1}z_{k}=vugm_{1}x_{0}$. Therefore, given
$\varepsilon>0$, by choosing $k$ sufficiently large, we guarantee that
$d(a_{k}^{-1}z_{k},m_{1}x_{0})<\varepsilon$. Fixing such a $k$, we may find
$\delta>0$ such that if $d(z,z_{k})<\delta$, then
$d(a_{k}^{-1}z,m_{1}x_{0})<\varepsilon$. Since $b_{n}m_{2}x_{0}\to z_{k}$, we
may choose $n$ so large to obtain that $d(b_{n}m_{2}x_{0},z_{k})<\delta$. Then
by construction, $d(a_{k}^{-1}b_{n}m_{2}x_{0},m_{1}x_{0})<\varepsilon$.
Therefore, since $\varepsilon$ was arbitrarily small, and $a_{k}^{-1}b_{n}\in
A_{\alpha}$, $m_{1}x_{0}\in\overline{A_{\alpha}m_{2}x_{0}}$.

Finally, we claim that every coarse Lyapunov leaf is contained in an atom of
$\mathcal{F}_{\alpha}$. This is true by choice of $x_{0}$ for $W^{\beta}$,
$\beta\not=\pm\alpha$, we now show it for $W^{\pm\alpha}$. Indeed, notice that
by definition, if $a\in\mathbb{R}^{k}$, then $a\cdot
m\cdot\overline{A_{\alpha}x_{0}}=m\cdot\overline{A_{\alpha}(a\cdot x_{0})}$
and that if $y\in W^{\alpha}(x_{0})$, then $d(a^{n}\cdot x_{0},a^{n}\cdot
y)\to 0$ (similarly for $W^{-\alpha}(x_{0})$). If the points were on different
$\mathcal{F}_{\alpha}$-partition elements, they would remain a fixed positive
distance from one another (since the atoms continue to differ by a fixed, non-
identity $m\in M$), a contradiction.

Finally, notice that if we had chosen $x_{1}$ and $x_{0}$ from the residual
set of points which contain $W^{A_{\alpha}}$ to define the orbit closures, the
proof would not change. In particular, the partition depends only on picking
orbit closures which contain these sets, so the partition is into “maximal
$\ker(\alpha)$-orbit closures.” In particular, $a\mathcal{F}_{\alpha}(m)$ must
be another partition element 111 By “maximal orbit closures” property it is
obvious that for any $a\in\mathbb{R}^{k},a\mathcal{F}_{\alpha}(m)$ is
contained in some partition element since $a\mathcal{F}_{\alpha}(m)$ is an
orbit closure of some point, then considering the $-a$ action we get the
equality., proving the last claim of the lemma. ∎

Fix a weight $\alpha\in\Delta$ such that $-\alpha\in\Delta$ as well.

######  Lemma 10.15.

There exists a connected Lie group $G_{\alpha}$ and a continuous group action
$G_{\alpha}\curvearrowright X$ such that

  1. (1)

$\operatorname{Lie}(G_{\alpha})\cong\operatorname{Lie}(N_{\alpha})\oplus\mathfrak{g}_{0}\oplus\operatorname{Lie}(N_{-\alpha})$,
where $\mathfrak{g}_{0}\subset\mathbb{R}^{k}\oplus\operatorname{Lie}(M)$,

  2. (2)

the inclusion of $\operatorname{Lie}(N_{\pm\alpha})$ in
$\operatorname{Lie}(G_{\alpha})$ induces a local isomorphism from
$N_{\pm\alpha}$ onto its image,

  3. (3)

the action of $\exp_{G_{\alpha}}(\operatorname{Lie}(N_{\pm\alpha}))$ coincides
with the existing action of $N_{\pm\alpha}$,

  4. (4)

the subgroups $\exp_{G_{\alpha}}(\operatorname{Lie}(N_{\alpha}))$ and
$\exp_{G_{\alpha}}(\operatorname{Lie}(N_{-\alpha}))$ generate $G_{\alpha}$,
and

  5. (5)

the action of $\exp_{G_{\alpha}}(\mathfrak{g}_{0})$ coincides with the
existing action as a subgroup of $\mathbb{R}^{k}\times M$.

###### Proof.

Consider the stabilizer of a point, $\mathcal{C}_{\alpha}(x)$, under the
action of the free product $N_{\alpha}*N_{-\alpha}$. Then $\ker\alpha$ takes
an $\alpha,-\alpha$-cycle to the same cycle at a new basepoint (since by
Definition 6.1(TA-3), $\ker\alpha$ acts trivially on every leg),
$\mathcal{C}_{\alpha}(x)=a_{*}\mathcal{C}_{\alpha}(x)=\mathcal{C}_{\alpha}(a\cdot
x)$. Choose $x_{0}$ as in Lemma 10.14, and let $\mathcal{F}_{\alpha}$ be the
corresponding partition if it exists. Since the atoms of
$\mathcal{F}_{\alpha}$ are saturated by coarse Lyapunov leaves,
$N_{\alpha}*N_{-\alpha}$-orbits are contained in a single $\ker\alpha$-orbit
closure, so the cycles at any point contain those of $m\cdot x_{0}$ for some
$m\in M$. Therefore, by Theorem 6.8 at each $m\in M$ the group
$G_{\alpha}(m)=(N_{\alpha}*N_{-\alpha})/\mathcal{C}_{\alpha}(m\cdot x_{0})$ is
Lie, and has a canonical Hölder action
$G_{\alpha}(m)\curvearrowright\mathcal{F}_{\alpha}(m)$ into which the group
actions of $N_{\alpha}$ and $N_{-\alpha}$ canonically embed.

We wish to show that the group $G_{\alpha}(m)$ is independent of $m\in M$, and
that there is a global group action $G_{\alpha}\curvearrowright X$. Recall
that if $g\in\mathbb{R}^{k}\times M$, $g$ normalizes the
$N_{\pm\alpha}$-actions. Let $g_{*}$ denote the induced automorphism of
$N_{\alpha}*N_{-\alpha}$. Any globally-defined map preserving the coarse
Lyapunov foliations will take cycles to cycles, so
$\mathcal{C}_{\alpha}(g\cdot n)=g_{*}\mathcal{C}_{\alpha}(n)$. Since $g_{*}$
is an automorphism of $N_{\alpha}*N_{-\alpha}$ taking
$\mathcal{C}_{\alpha}(m)$ to $\mathcal{C}_{\alpha}(g\cdot m)$, $g_{*}$ induces
an isomorphism $g_{*}:G_{\alpha}(m)\to G_{\alpha}(g\cdot m)$.

As a remark, note that this determines the isomorphism class of
$G_{\alpha}(m)$, but that this is insufficient for our purposes. To obtain a
group action on $X$, we need to know the cycles $\mathcal{C}_{\alpha}(m)$ are
constant, which is to say that the generating relations for the groups are the
same. This is equivalent to showing that the maps
$\beta_{m}:\operatorname{Lie}(N_{\alpha})\otimes\operatorname{Lie}(N_{-\alpha})\to\operatorname{Lie}(G_{\alpha})$
defined by
$\beta_{m}(X,Y)=[X,Y]_{\operatorname{Lie}(G_{\alpha}(m))}\in\operatorname{Lie}(G_{\alpha}(m))$
are independent of $m$ in some sense. It is possible that the groups
$G_{\alpha}(m)$ have a graded family of isomorphisms, even though they do not
induce the same actions. In fact, we will show that each
$\operatorname{Lie}(G_{\alpha}(m))$ sits inside a global vector space
independent of $m$ and that the maps $\beta_{m}$ are constant there, proving
constancy of relations and hence of $\mathcal{C}_{\alpha}(m)$. This is made
more precise below. So we now consider the Lie group structure of
$G_{\alpha}(m)$. Notice that $N_{\pm\alpha}$ has a grading that corresponds to
the Oseledets splitting, $\operatorname{Lie}(N_{\alpha})=E^{\alpha}\oplus
E^{c_{2}\alpha}\oplus\dots\oplus E^{c_{\ell_{1}}\alpha}$ and
$\operatorname{Lie}(N_{-\alpha})=E^{-d_{1}\alpha}\oplus
E^{-d_{2}\alpha}\oplus{\cdots\oplus}E^{-d_{\ell_{2}}\alpha}$. Suppose that
${Z}\in E^{c_{i}\alpha}$ and $Y\in E^{-d_{j}\alpha}$. Then
$\exp_{G_{\alpha}(m)}(t[{Z},Y]_{G_{\alpha}(m)})$ is a Hölder curve in $X$, but
a priori only a Hölder curve (recall that by (TA-2), the evaluation maps of
the $N_{\pm\alpha}$ are assumed to be locally bi-Lipschitz, but the global
actions of $N_{\pm\alpha}$, and hence $G_{\alpha}(m)$, are only Hölder). For
each $a\in\mathbb{R}^{k}$, $a_{*}$ is an isomorphism between $G_{\alpha}(m)$
and $G_{\alpha}(m^{\prime})$, where $m^{\prime}$ is such that
$a\mathcal{F}_{\alpha}(m)=\mathcal{F}_{\alpha}(m^{\prime})$. Let us consider
the asymptotic behavior $\exp_{G_{\alpha}(m)}(t[{Z},Y]_{G_{\alpha}(m)})$. If
$x\in\mathcal{F}_{\alpha}(m)$ and $d_{j}>c_{i}$, then as $\alpha(a)\to\infty$:

$\displaystyle d(ax,a\exp_{G_{\alpha}(m)}(t[{Z},Y]_{G_{\alpha}(m)})x)$
$\displaystyle=$ $\displaystyle
d(\exp_{G_{\alpha}(m^{\prime})}(ta_{*}[{Z},Y]_{G_{\alpha}(m)})ax,ax)$
$\displaystyle=$ $\displaystyle
d(\exp_{G_{\alpha}(m^{\prime})}(te^{(c_{i}-d_{j})\alpha(a)}[{Z},Y]_{G_{\alpha}(m^{\prime})})ax,ax)$
$\displaystyle\to$ $\displaystyle 0.$

Importantly, note that we do not yet know the precise rate of convergence
using the distance on $X$, since the group action itself is only Hölder.
However, if $d_{j}>c_{i}$, then the curve
$\exp_{G_{\alpha}(m)}(t[{Z},Y]_{G_{\alpha}(m)})$ is contained in the
$N_{-\alpha}$-orbit by Lemma 10.9 (it is similarly in the $N_{\alpha}$-orbit
for $c_{i}>d_{j}$).

From this we conclude that if $c_{i}\not=d_{j}$,
$[{Z},Y]_{G_{\alpha}(m)}\in\operatorname{Lie}(N_{\pm\alpha})$. Assume that
$c_{i}>d_{j}$, so $[{Z},Y]_{G_{\alpha}(m)}\in\operatorname{Lie}(N_{\alpha})$
(the opposite case is identical). Choose any $b\in\mathbb{R}^{k}$ such that
$\alpha(b)>0$, and let $m\in M$. Then
$b\mathcal{F}_{\alpha}(m)=\mathcal{F}_{\alpha}(gm)$ for some $g\in M$, and
$bg^{-1}$ fixes $\mathcal{F}_{\alpha}(m)$. Choose any norm on
$\operatorname{Lie}(G_{\alpha}(m))$ such that the action of $M$ is isometric.
Then $(bg^{-1})_{*}$ is an automorphism of $G_{\alpha}(m)$ and if
${Z}^{\prime}=g_{*}{Z}$ and $Y^{\prime}=g_{*}Y$. Then, using that the
$M$-action is isometric:

$\displaystyle(bg^{-1})_{*}[{Z},Y]_{G_{\alpha}(m)}$ $\displaystyle=$
$\displaystyle[(bg^{-1})_{*}{Z},(bg^{-1})_{*}Y]_{G_{\alpha}(m)},\mbox{ so}$
$\displaystyle\left|\left|(bg^{-1})_{*}[{Z},Y]_{G_{\alpha}(m)}\right|\right|$
$\displaystyle=$
$\displaystyle\left|\left|[e^{c_{i}\alpha(b)}{Z}^{\prime},e^{-d_{j}\alpha(b)}Y^{\prime}]_{G_{\alpha}(m)}\right|\right|$
$\displaystyle=$ $\displaystyle
e^{(c_{i}-d_{j})\alpha(b)}\left|\left|[{Z}^{\prime},Y^{\prime}]_{G_{\alpha}({gm})}\right|\right|$
$\displaystyle=$ $\displaystyle
e^{(c_{i}-d_{j})\alpha(b)}\left|\left|[{Z},Y]_{G_{\alpha}(m)}\right|\right|.$

Moreover since $(bg^{-1})_{\ast}$ preserves the Oseledets splitting in
$\operatorname{Lie}(N_{\pm\alpha})$, by similar proof we have

$\displaystyle\left|\left|(bg^{-1})^{n}_{*}[{Z},Y]_{G_{\alpha}(m)}\right|\right|=e^{n(c_{i}-d_{j})\alpha(b)}\left|\left|[{Z},Y]_{G_{\alpha}(m)}\right|\right|.$

Since $[{Z},Y]_{G_{\alpha}(m)}\in\operatorname{Lie}(N_{\alpha})$, this implies
that $[{Z},Y]_{G_{\alpha}(m)}$ is in the direct sum of generalized eigenspaces
of $(bg^{-1})_{*}$ with eigenvalues whose moduli are
$e^{(c_{i}-d_{j})\alpha(b)}$ and hence that $[{Z},Y]_{G_{\alpha}(m)}\in
E^{(c_{i}-d_{j})\alpha}$.

In the case when $c_{i}=d_{j}$, then the flow generated by $[{Z},Y]$ commutes
with the $\mathbb{R}^{k}$ action, and therefore its orbits are contained in
the $\mathbb{R}^{k}\times M$-orbits by Lemma 10.8. Define $\varphi(x)$ to be
the element of $A\cdot M$ such that $\exp([{Z},Y])\cdot x=\varphi(x)\cdot x$.
Then $\varphi$ is constant on $\ker\alpha$ orbits, and hence $\varphi(x)$ is
constant on $\mathcal{F}_{\alpha}(m)$. Therefore, the flow generated by
$[{Z},Y]$ is a one-parameter subgroup of $\mathbb{R}^{k}{\times}M$ on
$\mathcal{F}_{\alpha}(m)$. Therefore, $\operatorname{Lie}(G_{\alpha}(m))$ has
a canonical vector space embedding into
$V=\operatorname{Lie}(N_{\alpha})\oplus\operatorname{Lie}(\mathbb{R}^{k}\times
M)\oplus\operatorname{Lie}(N_{-\alpha})$ (note that for now, $V$ is only a
vector space since there is no well-defined Lie bracket defined on all of
$V$).

Finally, we wish to show that if ${Z}\in\operatorname{Lie}(N_{\alpha})$ and
$Y\in\operatorname{Lie}(N_{-\alpha})$, then
$[{Z},Y]_{G_{\alpha}(m)}=[{Z},Y]_{G_{\alpha}(m^{\prime})}$ for any
$m,m^{\prime}\in M$ (where we think of each
$\operatorname{Lie}(G_{\alpha}(m))$ as a vector subspace of $V$). Notice that
by bilinearity of the Lie bracket, we may assume that ${Z}\in E^{c_{i}\alpha}$
and $Y\in E^{-d_{j}\alpha}$. We first prove the claim when
$\mathcal{F}_{\alpha}(m^{\prime})=a\mathcal{F}_{\alpha}(m)$. With these
assumptions, using the fact that $[{Z},Y]\in E^{(c_{i}-d_{j})\alpha(a)}$ if
$c_{i}\not=d_{j}$, or otherwise commutes with the $\mathbb{R}^{k}$-action, we
see that

$\displaystyle[{Z},Y]_{G_{\alpha}(m)}$ $\displaystyle=$
$\displaystyle(a^{-1})_{*}[a_{*}{Z},a_{*}Y]_{G_{\alpha}(m^{\prime})}$
$\displaystyle=$ $\displaystyle
e^{(d_{j}-c_{i})\alpha(a)}[e^{c_{i}\alpha(a)}{Z},e^{-d_{j}\alpha(a)}Y]_{G_{\alpha}(m^{\prime})}$
$\displaystyle=$ $\displaystyle[{Z},Y]_{G_{\alpha}(m^{\prime})}.$

Therefore, the group $G_{\alpha}(m)$ is constant on $\mathbb{R}^{k}$-orbits,
and hence $\mathcal{C}_{\alpha}(x_{0})$, the space of cycles in $N_{\alpha}$
and $N_{-\alpha}$ is constant on a dense set by (TA-1). The sets
$\mathcal{C}_{\alpha}(x)$ are semi-continuously varying in the following
sense: if $\lim_{n\to\infty}x_{n}=x$, and
$\sigma_{n}\in\mathcal{C}_{\alpha}(x_{n})$ is a sequence of cycles converging
to a cycle $\sigma$, then $\sigma\in\mathcal{C}(x)$. By density of the
$\mathbb{R}^{k}$ orbit and this semicontinuity, $\mathcal{C}_{\alpha}(x_{0})$
is contained in $\mathcal{C}_{\alpha}(x)$ for every $x\in X$. Therefore, there
is a Lie group
$G_{\alpha}=(N_{\alpha}*N_{-\alpha})/\mathcal{C}_{\alpha}(x_{0})$ which acts
on the total space $X$, as described. ∎

##  11\. Polynomial forms of geometric commutators

######  Theorem 11.1.

If $\mathbb{R}^{k}\times M\curvearrowright X$ is a leafwise homogeneous
topological Anosov action satisfying the assumptions of Theorem 10.7, then the
functions $\rho^{\alpha,\beta}(u,v,x)$ are independent of $x$.

###  11.1. Reduction to Oseledets subspaces

If $\alpha$ is a coarse Lyapunov exponent, recall from the definition of a
topological Anosov action that $\operatorname{Lie}(N_{\alpha})$ splits as a
direct sum of Oseledets subspaces. That is,
$\operatorname{Lie}(N_{\alpha})=E^{c_{1}^{\alpha}\alpha}\oplus\dots\oplus
E^{c_{\ell(\alpha)}^{\alpha}\alpha}$. Let $e(u)=\exp(u)$ for ease of notation,
and $\log$ denote its inverse (which exists because $N_{\alpha}$ is
nilpotent). Then by (TA-3), if $u\in E^{c_{i}^{\alpha}\alpha}$,
$a\in\mathbb{R}^{k}$,

(11.1) $a_{*}e(u)=e(e^{c_{i}^{\alpha}\alpha(a)}u).$

######  Definition 11.2.

Given Oseledets spaces $E^{c_{i}^{\alpha}\alpha}$ and
$E^{c_{j}^{\beta}\beta}$, let
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{{c_{m}^{\gamma}}\gamma}:E^{c_{i}^{\alpha}\alpha}\times
E^{c_{j}^{\beta}\beta}\times X\to\operatorname{Lie}(N_{\gamma})$ be defined by

$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)=\pi_{m}^{\gamma}(\log\rho^{\alpha,\beta}_{\gamma}(e(u),e(v),x)),$

where $\pi_{m}^{\gamma}:\operatorname{Lie}(N_{\gamma})\to
E^{c_{m}^{\gamma}\gamma}$ is the projection onto the corresponding Oseledets
space induced by the Oseledets splitting.

We now turn to the last technical assumption which appears in Theorem 10.7. If
$c_{m}^{\gamma}\gamma=\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta$
for some $\sigma,\tau>0$, we call $\sigma$ and $\tau$ the Lyapunov
coefficients of $c_{m}^{\gamma}\gamma$ with respect to $c_{i}^{\alpha}\alpha$
and $c_{j}^{\beta}\beta$. We say that an action has integral Lyapunov
coefficients if
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}\equiv
0$ whenever both Lyapunov coefficients are less than 1.

######  Lemma 11.3.

If for every pair of Oseledets subspaces
$E^{c_{i}^{\alpha}\alpha},E^{c_{j}^{\beta}\beta}$ with $\alpha$ and $\beta$
nonproportional, the functions
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$are
independent of $x$, then the functions $\rho^{\alpha,\beta}$ are independent
of $x$ for every $\alpha,\beta\in\Delta$.

###### Proof.

Let $u\in N_{\alpha}$ and $v\in N_{\beta}$. Then we may write
$u=e(u_{1})e(u_{2})\cdots e(u_{\ell(\alpha)})$ and $v=e(v_{1})e(v_{2})\cdots
e(v_{\ell(\beta)})$, where $u_{i}\in E^{c_{i}^{\alpha}\alpha}$ and $v_{j}\in
E^{c_{j}^{\beta}\beta}$. We wish to compute $[u,v]$ using commutator relations
coming from commutators from the exponentials of Oseledets subpaces. Using
their expression in terms of Oseledets subspaces, we may write

$[u,v]=e(u_{1})\cdots e(u_{\ell(\alpha)})\cdot e(v_{1})\cdots
e(v_{\ell(\beta)})\cdot e(-u_{\ell(\alpha)})\cdots e(-u_{1})\cdot
e(-v_{\ell(\beta)})\cdots e(-v_{1}).$

We wish to push the $e(-u_{\ell(\alpha)})$ term past the $e(v_{i})$-terms to
cancel with $e(u_{\ell(\alpha)})$. Since we have constant commutator relations
among the exponentials of Oseledets spaces, we may do so, but accumulate their
commutators along the way. We may choose to put them on the left or right
whenever we commute, we choose to put them on the left. That is, we write
$e(v_{i})e(-u_{\ell(\alpha)})=ge(-u_{\ell(\alpha)})e(v_{i})$, where $g$ is a
geometric commutator of $e(-v_{i})$ and $e(u_{\ell(\alpha)})$ determined by
the $\rho$-function, which is constant. Recall that $g$ is a product of
elements from the coarse Lyapunov groups strictly between $\alpha$ and
$\beta$, which are constant by the assumption of the lemma.

The element $g$ itself may be decomposed as a product of Oseledets subspaces,
which we may push past $e(v_{1})\cdots e(v_{i-1})$ by the same method,
accumulating new terms on the left in the process between the weights of $g$
and $\beta$. These terms are independent of $x$ since we only commute terms
coming from exponentials of Oseledets spaces. Since $g$ will always take
values in weights between $\alpha$ and $\beta$, there is a clear induction on
$\\#\Sigma(\alpha,\beta)$, which terminates since there are only finitely many
such weights. We may express $[u,v]$ as

$[u,v]=e(u_{1})\cdots e(u_{\ell(\alpha)})ge(-u_{\ell(\alpha)})e(v_{1})\cdots
e(v_{\ell(\beta)})\cdot e(-u_{\ell(\alpha)-1})\cdots e(-u_{1})\cdot
e(-v_{\ell(\beta)})\cdots e(-v_{1}),$

where $g$ is a product of exponentials of Oseledets space from coarse Lyapunov
exponents that lie strictly between $\alpha$ and $\beta$, which is still
independent of $x$. Using the same procedure as above, we may push all
$g$-terms to the far left, and cancel the $e(u_{\ell(\alpha)})$-term with its
inverse to obtain the following expression, with $g^{\prime}$ independent of
$x$:

$[u,v]=g^{\prime}e(u_{1})\cdots e(u_{\ell(\alpha)-1})\cdot e(v_{1})\cdots
e(v_{\ell(\beta)})\cdot e(-u_{\ell(\alpha)-1})\cdots e(-u_{1})\cdot
e(-v_{\ell(\beta)})\cdots e(-v_{1}).$

We now repeat this process until we have canceled each $e(u_{i})$-term which
then further allows for the cancelling of all $e(v_{j})$-terms, leaving only a
product exponentials of Oseledets spaces from $\Sigma(\alpha,\beta)$, which is
independent of $x$.

We have now reduced $[u,v]$ to a product of exponentials of Oseledets spaces
of $\gamma$, where $\gamma\in\Sigma(\alpha,\beta)$. To deduce that
$\rho^{\alpha,\beta}(u,v,x)$ is independent of $x$, we write
$\Sigma_{\Omega}(\alpha,\beta)$ in a circular ordering. Then push all of the
exponential terms from each the Oseledets subspace of the first coarse
exponent to the left. Since we have constant relations among the exponentials
of the Oseledets spaces, this is possible, accumulating their commutators on
the right. The corresponding reduction yields an element, written in circular
ordering, which is independent of $x$. Uniqueness follows from Lemma 10.12. ∎

###  11.2. Setting up the inductions

We prove Theorem 11.1 using Lemma 11.3 by showing each
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
is constant. We use use three inductions. The outermost induction is on
$\\#\Sigma(\alpha,\beta)$, we call this Induction I. In each step of the
induction, we will show for Lyapunov exponents $c_{i}^{\alpha}\alpha$ and
$c_{j}^{\beta}\beta$,

(11.2) if
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}\not\equiv
0$, then
$c_{m}^{\gamma}\gamma=\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta$
for some $\sigma,\tau\in\mathbb{Z}_{+}$, and (11.3) if
$c_{m}^{\gamma}\gamma=\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta$,
then
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$
is a polynomial which is $\sigma$-homogeneous in $u$, $\tau$-homogeneous in
$v$ and independent of $x$.

We first state a key consequence of the induction. Let
$\mathcal{P}_{|\alpha,\beta|}$ denote the group freely generated by the groups
$N_{\gamma}$,
$\gamma\in\Sigma(\alpha,\beta)\cup\left\\{\alpha,\beta\right\\}$.

######  Lemma 11.4.

If (11.2) and (11.3) hold for all linearly independent $\alpha,\beta$ such
that $\\#\Sigma(\alpha,\beta)\leq n$, then for any such $\alpha,\beta$,

  * •

the action of $\mathcal{P}_{|\alpha,\beta|}$ factors through the action of a
nilpotent Lie group $N_{|\alpha,\beta|}$,

  * •

$\operatorname{Lie}(N_{|\alpha,\beta|})=\bigoplus_{\gamma\in\Sigma(\alpha,\beta)\cup\left\\{\alpha,\beta\right\\}}\operatorname{Lie}(N_{\gamma})$,
and

  * •

the family of automorphisms $a_{*}$ defined in (TA-3) descend to an
automorphism of $N_{|\alpha,\beta|}$.

###### Proof.

Write
$\Sigma(\alpha,\beta)\cup\left\\{\alpha,\beta\right\\}=\left\\{\alpha=\gamma_{1},\gamma_{2},\dots,\gamma_{r}=\beta\right\\}$
in the induced circular ordering. Let $\mathcal{G}$ denote the factor of the
group $\mathcal{P}_{\alpha,\beta}$ modulo the commutator relations (10.3). By
(11.3) and Lemma 11.3, they are independent of $x$. We first claim that every
$\rho\in\mathcal{G}$ can be written as

(11.4) $u_{1}*\dots*u_{r},$

where each $u_{i}\in N_{\gamma_{i}}$ is unique. Indeed, any
$\rho\in\mathcal{G}$ can be written as $\rho=v_{1}*\dots*v_{k}$ (where each
$v_{i}\in N_{\beta_{n_{i}}}$).

We may begin by pushing all of the terms from the $\beta_{1}$ component to the
left. We do this by looking at the first term to appear with $\beta_{1}$. Each
time we pass it through, we may accumulate some
$[u^{\beta_{1}},v^{\beta_{j}}]$ which may be rewritten as
$\rho(u^{\beta_{1}},v^{\beta_{j}})$, having no $\beta_{1}$ terms, since we
have quotiented by the commutator relations (10.3). So we have shown that in
$\mathcal{G}$, $\rho$ is equal to $u_{1}*\rho^{\prime}$, where $\rho^{\prime}$
consists only of terms without $\beta_{1}$, and $u_{1}\in N_{\beta_{1}}$.

We now proceed inductively. We may in the same way push all $\beta_{2}$ terms
to the left. Notice now that each time we pass through, the “commutator”
$\rho(u^{\beta_{2}},v^{\beta_{j}})$, $j\geq 3$ has no $\beta_{1}$ or
$\beta_{2}$ terms. Iterating this process yields the desired presentation of
$\rho$.

Thus, every element of $\mathcal{G}$ has a unique presentation of the form
(11.4), where the uniqueness follows from Lemma 10.12. The map which assigns
an element $\rho$ to such a presentation gives a an injective map from
$\mathcal{G}$ to $\prod N_{\beta_{i}}$ (but the map may not be a homomorphism
of groups). By Lemma 6.4, it will be continuous once its lift to
$\mathcal{P}_{\alpha,\beta}$ is continuous. In each combinatorial cell
$C_{\overline{\beta}}$, the map is given by composition of the group
multiplications in each $N_{\beta_{i}}$ and the functions
$\rho^{\alpha,\beta}(\cdot,\cdot)$ evaluated on cell coordinates, which are
continuous. Therefore, the lift is continuous, so the map from $\mathcal{G}$
is continuous.

Therefore, there is an injective continuous map from $\mathcal{G}$ to a
finite-dimensional space, and $\mathcal{G}$ is a Lie group by Theorem 6.9. Fix
$a$ which contracts every $\beta_{i}$. The fact that $\mathcal{G}$ is
nilpotent follows from the fact that it has a contracting automorphism. ∎

Each step of the outer induction on $\\#\Sigma(\alpha,\beta)$ will be proved
using two further inductions. We introduce a partial order on
$\left\\{(i,j):1\leq i\leq\ell(\alpha),1\leq j\leq\ell(\beta)\right\\}$ by
saying that $(i_{1},j_{1})\preceq(i_{2},j_{2})$ if and only if $i_{1}\leq
i_{2}$ and $j_{1}\leq j_{2}$. The second induction will utilize this partial
order: we will show (11.2) and (11.3) for a pair $c_{i}^{\alpha}\alpha$ and
$c_{j}^{\beta}\beta$ assuming that we have concluded it for all choices of
$c_{m}^{\gamma}\gamma$ and all $c_{i^{\prime}}^{\alpha}\alpha$,
$c_{j^{\prime}}^{\beta}\beta$ such that $(i,j)\preceq(i^{\prime},j^{\prime})$
and $(i,j)\not=(i^{\prime},j^{\prime})$. The base of this induction will then
be $(\ell(\alpha),\ell(\beta))$, the unique maximal element. It is clear from
the structure of the partial order that such an induction will exhaust all
choices of $(i,j)$. We call this induction Induction II.

Given $c_{i}^{\alpha}\alpha$ and $c_{j}^{\beta}\beta$, let

$[c_{i}^{\alpha}\alpha,c_{j}^{\alpha}\beta]=\left\\{c_{m}^{\gamma}\gamma:\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}\not\equiv
0\right\\}\mbox{ and
}\Omega_{l}=\left\\{\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta:\sigma+\tau=l,\;\sigma,\tau>0\right\\}\cap\left\\{c_{m}^{\gamma}\gamma:\gamma\in\Delta\right\\}.$

Then there are finitely many values $l_{0}<l_{1}<\dots<l_{m}$ such that

$\Sigma(\alpha,\beta)=\bigcup_{p=0}^{m}\Omega_{l_{p}}.$

Given a subset $S$ of Lyapunov exponents, and a weight
$c_{i}^{\alpha}\alpha\in\Delta$, let

$[c_{i}^{\alpha}\alpha,S]=\left\\{c_{m}^{\gamma}\gamma:\rho^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}\not\equiv
0\mbox{ for some }c_{j}^{\beta}\beta\in S\right\\}.$

Let $\mathcal{P}_{(\alpha,\beta)}$, $\mathcal{P}_{|\alpha,\beta)}$ and
$\mathcal{P}_{(\alpha,\beta|}$ denote the groups freely generated by the
weights of $\Sigma(\alpha,\beta)$,
$\Sigma(\alpha,\beta)\cup\left\\{\alpha\right\\}$ and
$\Sigma(\alpha,\beta)\cup\left\\{\beta\right\\}$, respectively. By the
induction hypothesis, the action of each group factors through Lie groups
which we denote by $N_{(\alpha,\beta)}$, $N_{|\alpha,\beta)}$ and
$N_{(\alpha,\beta|}$, respectively. Let
$F_{l}=\bigoplus_{c_{m}^{\gamma}\gamma\in\Omega_{l}}E^{c_{m}^{\gamma}\gamma}$,
so that $\operatorname{Lie}(N_{(\alpha,\beta)})=\bigoplus_{a=1}^{m}F_{l_{a}}$.

It now suffices to show Claims 11.2 and 11.3 for each weight
$\chi\in\Omega_{l_{p}}$. We will do this using a final induction on $p$,
starting from $p=0$. We call this induction Induction III. We summarize each
induction below, noting that the proof of (11.2) and (11.3) runs in a
lexicographical ordering: for each step of Induction I, we do every step of
Induction II, and for each of Induction II, we do every step of Induction III:

  * •

Induction I: $\\#\Sigma(\alpha,\beta)$, base case $\\#\Sigma(\alpha,\beta)=0$.

  * •

Induction II: Partial order on $(c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta)$,
base case maximal element, induction moves downward.

  * •

Induction III: $c_{m}^{\gamma}\gamma\in\Omega_{l}$, induction on $l$, base
case $l=l_{0}$ smallest coefficients.

###  11.3. Proving the inductive steps

Fix $l$, and given $g\in N_{(\alpha,\beta)}$, write $\log
g=\check{g}+\bar{g}+\hat{g}$, where $\check{g}$ is the component of $\log g$
from the weight spaces of $\Omega_{l^{\prime}}$, $l^{\prime}<l$, $\bar{g}$ is
the component from the weight spaces of $\Omega_{l}$ and $\hat{g}$ is the
component from the weight spaces of $\Omega_{l^{\prime}}$, $l^{\prime}>l$.

By Induction I and Lemma 11.4, there are groups $N_{(\alpha,\beta)}$,
$N_{|\alpha,\beta)}$ and $N_{(\alpha,\beta|}$ generated by the weights
$\Sigma(\alpha,\beta)$, $\left\\{\alpha\right\\}\cup\Sigma(\alpha,\beta)$ and
$\left\\{\beta\right\\}\cup\Sigma(\alpha,\beta)$. The following important
lemma uses these group structures to describe the action of ${N_{\alpha}}$ on
$N_{(\alpha,\beta)}$.

######  Lemma 11.5.

If $u\in E^{c_{i}^{\alpha}\alpha}$, then the conjugation action of $\exp(u)$
on $N_{|\alpha,\beta)}$ preserves $N_{(\alpha,\beta)}$. Furthermore,
$\operatorname{ad}(u)$ a nilpotent automorphism such that
$\operatorname{ad}(u)(F_{l})\subset F_{l+1}$.

###### Proof.

That $\operatorname{ad}(u)$ is unipotent follows from the fact that
$\operatorname{ad}(u)(F_{l})\subset F_{l+1}$, which we now show. Assume
$c_{m}^{\gamma}\gamma\in\Omega_{l}$. The Lie group $N_{|\alpha,\beta)}$ also
carries an automorphism $a_{*}$ which expands $E^{c_{i}^{\alpha}\alpha}$ by
$e^{c_{i}^{\alpha}\alpha(a)}$ and $E^{c_{m}^{\gamma}\gamma}$ by
$e^{c_{m}^{\gamma}\gamma(a)}$. Therefore,
$[E^{c_{m}^{\gamma}\gamma},E^{c_{i}^{\alpha}\alpha}]$ consists of vectors
which are expanded by $e^{c_{m}^{\gamma}\gamma(a)+c_{i}^{\alpha}\alpha(a)}$.
Since $a$ is arbitrary we conclude that
$[E^{c_{m}^{\gamma}\gamma},E^{c_{i}^{\alpha},\alpha}]\subset
E^{c_{m}^{\gamma}\gamma+c_{i}^{\alpha}\alpha}$. Since
$c_{m}^{\gamma}\gamma\in\Omega_{l}$, the result follows. ∎

A completely symmetric version holds for $u\in E^{c_{j}^{\beta}\beta}$. The
following are immediate consequences of Lemma 11.5, and the Baker-Campbell-
Hausdorff formula.

######  Corollary 11.6.

If $u\in E^{c_{i}^{\alpha}\alpha}\cup E^{c_{j}^{\beta}\beta}$, and
$g=\exp(\check{g}+\bar{g}+\hat{g})\in N_{(\alpha,\beta)}$, let
$v=\exp(u)*g*\exp(-u)$. Then:

  1. (1)

$v\in N_{(\alpha,\beta)}$,

  2. (2)

$\check{v}$ is a polynomial of $u$ and $\check{g}$,

  3. (3)

$\bar{v}$ takes the following form:

$\bar{v}=p(u,\check{g})+\bar{g}$

for some polynomial $p$ such that $p(0,\cdot)=p(\cdot,0)=0$, and

  4. (4)

$\hat{v}$ is a polynomial in $u$ and $\log g$.

######  Corollary 11.7.

If $g_{1},g_{2}\in N_{(\alpha,\beta)}$ and we write $g_{3}=g_{1}g_{2}$, then

$\check{g}_{3}=p_{1}(\check{g_{1}},\check{g_{2}})\qquad\bar{g}_{3}=\bar{g}_{1}+\bar{g}_{2}+p_{2}(\check{g}_{1},\check{g}_{2})\qquad\hat{g}_{3}=p_{3}(g_{1},g_{2}),$

for some polynomials $p_{1}$, $p_{2}$ and $p_{3}$.

Notice that $\rho^{\alpha,\beta}(u,v,x)$ is a formal product of elements from
the groups $N_{\gamma}$, $\gamma\in\Sigma(\alpha,\beta)$, written in a
circular ordering. Therefore, it represents a unique element of
$N_{(\alpha,\beta)}$, which we abusively denote with the same notation. Fix
$c_{m}^{\gamma}\gamma\in\Omega_{l}$, and define a function
$r^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$ to
be the $E^{c_{m}^{\gamma}\gamma}$-component of
$\log\rho^{\alpha,\beta}(e(u),e(v),x)$.

######  Corollary 11.8.

$r^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)=\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)+p(u,v)$
for some polynomial $p:E^{c_{i}^{\alpha}\alpha}\oplus
E^{c_{j}^{\beta}\beta}\to E^{c_{m}^{\gamma}\gamma}$ independent of $x$.

###### Proof.

The definitions of
$r^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$ and
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
are quite similar, the only difference being that
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
uses the $c_{m}^{\gamma}\gamma$-component of
$\log\rho^{\alpha,\beta}_{\gamma}$, while
$r^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$ regards
$\rho^{\alpha,\beta}$ as an element of $N_{(\alpha,\beta)}$, then takes the
$\log$ and the $c_{m}^{\gamma}\gamma$-component. Therefore, we wish to compare
the standard exponential coordinate system on $N_{(\alpha,\beta)}$ and the
coordinate system given by $(v_{1},\dots,v_{n})\mapsto e(v_{1})\dots
e(v_{n})$, where $v_{i}\in N_{\gamma_{i}}$ and the $\gamma_{i}$ are listed in
a circular ordering. By Corollary 11.7, the $c_{m}^{\gamma}\gamma$-components
will differ only by polynomials that depend on the terms of the commutator
coming from $\Omega_{l^{\prime}}$, $l^{\prime}<l$ (ie, the $\check{\cdot}$
-terms). Since by Induction III, such terms are polynomials,
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
and $r^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
differ by a polynomial in $u$ and $v$ which is independent of $x$. ∎

Figure 1 gives an example of the structures above. We give a description of
the features available for one step of the induction for this particular
example. We assume that we are at the stage of the induction to analyze
$\hat{\rho}^{c_{2}^{\alpha}\alpha,c_{2}^{\beta}\beta}_{c_{j}^{\gamma_{m}}\gamma_{m}}$.
Then $\Omega_{l_{0}}=\left\\{c_{1}^{\gamma_{1}}\gamma_{1}\right\\}$,
$\Omega_{l_{1}}=\left\\{c_{1}^{\gamma_{2}}\gamma_{2}\right\\}$ and
$\Omega_{l_{2}}=\left\\{c_{2}^{\gamma_{1}}\gamma_{1},c_{2}^{\gamma_{2}}\gamma_{2}\right\\}$,
since they are the intersections of lines parallel to the one passsing through
$c_{2}^{\beta}\beta$ and $c_{2}^{\alpha}\alpha$. At the first stage of the
innermost induction, we would analyze only the function
$\hat{\rho}^{c_{2}^{\alpha}\alpha,c_{2}^{\beta}\beta}_{c_{1}^{\gamma_{1}}\gamma_{1}}$.
The crucial feature for the base step is that, by Lemma 11.5, no terms in the
Oseledets space $c_{1}^{\gamma_{1}}\gamma_{1}$ can appear by commuting the
$c_{i}^{\gamma_{j}}\gamma_{j}$ with another weight in the figure. The second
induction is necessary, due to the fact that $E^{c_{2}^{\alpha}\alpha}$ may
only be a vector subspace of $\operatorname{Lie}(N_{\alpha})$ and not be a
subalgebra, and the algebraic properties of this subspace will be crucial in
understanding the dependence of
$\hat{\rho}^{c_{2}^{\alpha}\alpha,c_{2}^{\beta}\beta}_{c_{1}^{\gamma_{1}}\gamma_{1}}(u,v,x)$
on $u$. Luckily, some algebraic features remain. If $u_{1},u_{2}\in
E^{c_{2}^{\alpha}\alpha}$, we may write $e(u_{1}+u_{2})=e(u_{1})e(u_{2})\cdot
g$ for some $g\in N_{\alpha}$. In fact, such a $g$ must lie in
$\exp(E^{c_{3}^{\alpha}\alpha}\oplus E^{c_{4}^{\alpha}\alpha})$. By the second
induction on the pairs $c_{i}^{\alpha}\alpha$ and $c_{j}^{\beta}\beta$, we
know that this additional term $g$ will have polynomial relations with
$c_{2}^{\beta}\beta$ and has polynomial relations with each
$c_{j}^{\gamma_{m}}\gamma_{m}$ by the first induction on
$\\#\Sigma(\alpha,\beta)$. This allows the analysis to go through.

In the next step of the induction on the $\Omega_{l_{i}}$,
$c_{1}^{\gamma_{2}}\gamma_{2}$ terms may appear when commuting the
$c_{1}^{\gamma_{1}}\gamma_{1}$ terms with a multiple of $\alpha$ or $\beta$
(which will be needed in Lemma 11.9), but for this example, this is the only
Lyapunov exponent strictly between $\alpha$ and $\beta$ with this property (by
considering Figure 1 and Lemma 11.5). This can and will appear when analyzing
how the function
$\hat{\rho}^{c_{2}^{\alpha}\alpha,c_{2}^{\beta},\beta}_{c_{1}^{\gamma_{2}}\gamma_{2}}(u,v,x)$
depends on $u$. Such terms will contribute polynomials by the induction
hypothesis, leading to the final polynomial form.

$c_{1}^{\alpha}\alpha$$c_{2}^{\alpha}\alpha$$c_{4}^{\alpha}\alpha$$c_{3}^{\alpha}\alpha$$c_{1}^{\beta}\beta$$c_{2}^{\beta}\beta$$c_{1}^{\gamma_{1}}\gamma_{1}$$c_{2}^{\gamma_{1}}\gamma_{1}$$c_{1}^{\gamma_{2}}\gamma_{2}$$c_{2}^{\gamma_{2}}\gamma_{2}$$\beta$$\alpha$$\gamma_{1}$$\gamma_{2}$
Figure 1. Lyapunov exponents in $\Sigma(\alpha,\beta)$

We now return to the formal proof, assuming the induction hypotheses. Assume
Claims 11.2 and 11.3 hold for $c_{m}^{\gamma}\gamma\in\Omega_{l_{q}}$, $q<p$.

Fix $v\in E^{c_{j}^{\beta}\beta}$ and let
$\varphi(u,x)=\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$.
Notice that when $a\in\ker\beta$ and
$c_{m}^{\gamma}\gamma=\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta$,
(10.1), (TA-3) and the definition of
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
implies that

(11.5) $\varphi(u,x)=e^{-\sigma
c_{i}^{\alpha}\alpha(a)}\varphi(e^{c_{i}^{\alpha}\alpha(a)}u,a\cdot x).$

We are now ready to establish the key lemma which gives a cocycle-like
property to the function $\varphi$. While the proof requires checking some
complicated details, the following lemma follows from two simple ideas:
splitting a commutator into a sum of two commutators requires a conjugation
and reordering, and with careful bookkeeping, the reordering and conjugation
can be shown to contribute polynomial terms only. By the Baker-Campbell-
Hausdorff formula, for homogeneous systems where the functions $\varphi$ are
compositions of multiplication in a nilpotent Lie group, such polynomials will
be nonvanishing unless $\sigma=\tau=1$, and the cocycle equation without them
will not hold. We assume that the induction hypotheses hold.

######  Lemma 11.9.

$\varphi(u_{1}+u_{2},x)=\varphi(u_{1},x)+\varphi(u_{2},e(u_{1})x)+p(u_{1},u_{2})$
for some polynomial $p:E^{c_{i}^{\alpha}\alpha}\times
E^{c_{i}^{\alpha}\alpha}\to E^{c_{m}^{\gamma}\gamma}$ such that
$p(0,\cdot)\equiv 0$ and $p(\cdot,0)\equiv 0$.

###### Proof.

We assume that $c_{m}^{\gamma}\gamma\in\Omega_{l_{p}}$. Recall that
$\varphi(u,x)={\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}}(u,v,x)$
is the $c_{m}^{\gamma}\gamma$-component of the unique path
$\rho^{\alpha,\beta}_{\gamma}(u,v,x)$, written in circular ordering of the
weights in $\Sigma(\alpha,\beta)$, which connects $[e(u),e(v)]\cdot x$ and
$x$. Given $u_{1},u_{2}\in E^{c_{i}\alpha}$, there exists
$q(u_{1},u_{2})\in\bigoplus_{i^{\prime}>i}E^{c_{i^{\prime}}^{\alpha}\alpha}$
such that $e(u_{1}+u_{2})=e(q(u_{1},u_{2}))e(u_{2})e(u_{1})$. Since
$N_{\alpha}$ is nilpotent, $q$ is a polynomial in $u_{1}$ and $u_{2}$. Notice
that using only the free product relations, we get that:

$\displaystyle[e(u_{1}+u_{2}),e(v)]$ $\displaystyle=$ $\displaystyle
e(-v)*e(-u_{1}-u_{2})*e(v)*e(u_{1}+u_{2})$ $\displaystyle=$ $\displaystyle
e(-v)*e(-u_{1}-u_{2})*e(v)*e(q(u_{1},u_{2}))*e(u_{2})*e(u_{1})$
$\displaystyle=$ $\displaystyle
e(-v)*e(-u_{1}-u_{2})*e(v)*e(q(u_{1},u_{2}))*e(u_{2})*\big{(}e(-v)*e(u_{1})*e(v)\big{)}$
$\displaystyle\qquad*\;\big{(}e(-v)*e(-u_{1})*e(v))*e(u_{1})$ $\displaystyle=$
$\displaystyle
e(-v)*e(-u_{1})*e(-u_{2})*e(-q(u_{1},u_{2}))*e(v)*e(q(u_{1},u_{2}))*e(u_{2})$
$\displaystyle\qquad*\;\big{(}e(-v)*e(u_{1})*e(v)\big{)}*[e(u_{1}),e(v)]$
$\displaystyle=$ $\displaystyle
e(-v)*e(-u_{1})*e(-u_{2})*e(-q(u_{1},u_{2}))*e(v)*e(q(u_{1},u_{2}))*e(-v)*e(u_{2})*e(v)$
$\displaystyle\qquad*\;[e(u_{2}),e(v)]*\big{(}e(-v)*e(u_{1})*e(v)\big{)}*[e(u_{1}),e(v)]$
$\displaystyle=$
$\displaystyle\big{(}e(-v)*e(-u_{1})*e(-u_{2})\big{)}*[e(-v),e(q(u_{1},u_{2}))]*\big{(}e(u_{2})*e(u_{1})*e(v)\big{)}$
$\displaystyle\qquad*\;\big{(}e(-v)*e(-u_{1})*e(v)\big{)}*[e(u_{2}),e(v)]*\big{(}e(-v)*e(u_{1})*e(v)\big{)}*[e(u_{1}),e(v)]$
$\displaystyle=$
$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\big{(}e(-v)*e(-u_{1})*e(-u_{2})\big{)}*[e(-v),e(q(u_{1},u_{2}))]*\big{(}e(u_{2})*e(u_{1})*e(v)\big{)}}$
$\displaystyle\qquad*\;{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\big{(}e(-v)*e(-u_{1})*e(v)\big{)}*[e(u_{2}),e(v)]*\big{(}e(-v)*e(u_{1})*e(v)\big{)}}*{\color[rgb]{0.5,0.5,0}\definecolor[named]{pgfstrokecolor}{rgb}{0.5,0.5,0}\pgfsys@color@cmyk@stroke{0}{0}{1}{.5}\pgfsys@color@cmyk@fill{0}{0}{1}{.5}[e(u_{1}),e(v)]}$

The last equality is simply the second-to-last expression rewritten with
color-coding. First, consider the red term. Since $q$ takes values in
$\bigoplus_{i^{\prime}>i}E^{c_{i^{\prime}}^{\alpha}\alpha}$, we know the
commutators of $q$ with $e(-v)$ are polynomial and independent of their
basepoint by Induction II. Therefore we may rewrite the red term as
$\big{(}e(-v)*e(-u_{1})*e(-u_{2})\big{)}*e(\tau_{0}(u_{1},u_{2},v))*\big{(}e(u_{2})*e(u_{1})*e(v)\big{)}$,
where $\tau_{0}\in\operatorname{Lie}(N_{(\alpha,\beta)})$ is independent of
$x$, depending polynomially on $u_{1}$, $u_{2}$ and $v$. Then by Induction I,
we know how each of the conjugating terms act on the term $\tau_{0}$, which
must be polynomially. Therefore, the entire first red term is independent of
$x$ and can be replaced by some $e(\tau(u_{1},u_{2},v))$ for some polynomial
$\tau$ taking values in $\operatorname{Lie}(N_{(\alpha,\beta)})$.

We now turn to the blue terms. Let $y=e(u_{1})\cdot x$, so that
$h_{2}(y):=\rho^{\alpha,\beta}(e(u_{2}),e(v),y)\in N_{(\alpha,\beta)}$ acts on
$y$ in the same way as $[e(u_{2}),e(v)]$. By induction, we may apply the
conjugation of $h_{2}(y)$ by $e(-v)*e(u_{1})*e(v)$ as 3 independent ones,
which have well-understood forms by Corollary 11.6. Indeed, applying parts (2)
and (3) of the corollary three times shows that the blue terms act on $y$ in
the same way that $g(y)=\exp(\check{g}+\bar{g}(y)+\hat{g}(y))\in
N_{(\alpha,\beta)}$ does, where $\check{g}$ is polynomial in $u_{2}$ and $v$
(by applying part (2) and Induction I), $\bar{g}(y)=\bar{h}_{2}(y)+p(u_{2},v)$
for some polynomial $p$ (by applying part (3)) and
$\hat{g}(y)\in\bigoplus_{l^{\prime}>l_{p}}\bigoplus_{c_{m}^{\gamma}\gamma\in\Omega_{l}}E^{c_{m}^{\gamma}\gamma}$
is, for now, uncontrolled.

The final green term is straightforward, letting
$h_{1}(x)=\rho^{\alpha,\beta}(e(u_{1}),e(v),x)$ be the element of
$N_{(\alpha,\beta)}$ acting on $x$ in the same way as $[e(u_{1}),e(v)]$. Then
$\check{h}_{1}$ is a polynomial in $u_{1}$ and $v$ independent of $x$ by
Induction III. Putting the conclusions together yields that
$[e(u_{1}+u_{2}),e(v)]$ acts on $x$ in the same way as:

$h(x)=e(\tau(u_{1},u_{2},v))*g(y)*h_{1}(x),$

so by Corollary 11.7, $\check{h}$ is a polynomial in $u_{1}$, $u_{2}$ and $v$
which is independent of $x$, and $\bar{h}(x)=\bar{h}_{1}(e(u_{1})\cdot
x)+\bar{h}_{2}(x)+p(u_{1},u_{2},v)$ for some polynomial $p$ in $u_{1}$,
$u_{2}$ and $v$ (depending only on the group structure of $N_{\alpha,\beta}$
and is hence independent of $x$). Therefore, the $\Omega_{l_{p}}$ terms have
exactly the prescribed form. ∎

######  Lemma 11.10.

If $V$ and $W$ are vector spaces and $f:V\to W$ is a continuous function such
that $f(0)=0$ and

(11.6) $f(v_{1}+v_{2})=f(v_{1})+f(v_{2})+p(v_{1},v_{2})$

for some polynomial $p$, then $f$ is a polynomial.

###### Proof.

First, observe that since $f(0+v)=f(0)+f(v)+p(0,v)$, it follows that
$p(0,v)=0$. Therefore, $p(v,w)$ has no constant terms and $p$ is symmetric by
(11.6). Furthermore, every term of $p$ must have degree at least one in both
$v$ and $w$ since it is symmetric. If $q(v)=p(v,v)$, then every term of $q$
has degree at least two. In particular, since each nonzero term of $q$ is
multiplied by at least $2^{n}\cdot 2^{-2n}$, $\sum 2^{n}q(2^{-n}v)$ is
summable and the sum converges to a polynomial of the same degree.

From (11.6), it follows that
$f(v)=2f\left(\frac{1}{2}v\right)+q\left(\frac{1}{2}v\right)$. Inductively, it
follows that
$f(v)=2^{n}f\left(2^{-n}v\right)+\sum_{i=1}^{n}2^{i-1}q\left(2^{-i}v\right)$.
Since the sum on the right hand side converges to a polynomial, it follows
that $\lim_{n\to\infty}2^{n}f\left(2^{-n}v\right)$ converges to a vector
uniformly bounded as $v$ varies in a compact set. Therefore, the map
$D:v\mapsto\lim_{n\to\infty}2^{n}f\left(2^{-n}v\right)$ is well-defined and
satisfies

$D(v+w)=\lim_{n\to\infty}2^{n}f\left(2^{-n}(v+w)\right)=\lim_{n\to\infty}2^{n}\left(f\left(2^{-n}v\right)+f\left(2^{-n}w\right)+p(2^{-n}v,2^{-n}w)\right)\\\
=D(v)+D(w).$

Furthermore, since
$D(v)=f(v)-\sum_{i=1}^{\infty}2^{i-1}q\left(2^{-i}v\right)$, it follows that
$D$ is continuous and hence linear. Therefore,
$f(v)=D(v)+\sum_{i=1}^{\infty}2^{i-1}q\left(2^{-i}v\right)$, and $f$ is a
polynomial. ∎

Recall that if we write
$c_{m}^{\gamma}\gamma=\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta$,
we say that $\sigma$ and $\tau$ are the Lyapunov coefficients. Since we have
assumed integral Lyapunov coefficients (Definition 11.2), we may assume that
either $\sigma\geq 1$ or $\tau\geq 1$. We without loss of generality assume
that $\sigma\geq 1$.

######  Corollary 11.11.

The function $\varphi(u,x)$ is a polynomial in $u$, whose coefficients are
functions of $x$ which are constant along each the sets
$\mathcal{F}_{\beta}(m)$ defined in Lemma 10.14.

###### Proof.

By Lemma B.1, it suffices to show the following claim:

Claim 11.12.1. For every $u,v\in E^{c_{i}^{\alpha}\alpha}$ such that
$\left|\left|u\right|\right|=1$, $\varphi(tu+v,x)$ is a polynomial in $t$
whose coefficients are functions of $x$ which are constant along the sets
$\mathcal{F}_{\beta}(m)$.

Claim 11.12.1 will follow from the following weaker claim:

Claim 11.12.2.  For every $u\in E^{c_{i}^{\alpha}\alpha}$ such that
$\left|\left|u\right|\right|=1$, $\varphi(tu,x)$ is a polynomial in $t$ whose
coefficients are functions of $x$ which are constant along the sets
$\mathcal{F}_{\beta}(m)$.

Let us deduce Claim 11.12.1 from Claim 11.12.2. By Lemma 11.9,

$\varphi(tu+v,x)=\varphi(tu,x)+\varphi(v,e(tu)\cdot x)+p(tu,v).$

If we can show Claim 11.12.2, then it follows that $\varphi(tu,x)$ is a
polynomial in $t$ whose coefficients are functions of $x$ which are constant
along the sets $\mathcal{F}_{\beta}(m)$. By Lemma 10.14 the atoms of
$\mathcal{F}_{\beta}$ are saturated by the leaves of $W^{\alpha}$. Since
$\varphi(sv,y)$ is a polynomial in $s$ whose coefficients are functions of $y$
which are constant along atoms of $\mathcal{F}_{\beta}$, it follows that
$\varphi(v,e(tu)\cdot x)$ is independent of $t$ and is constant as $x$ moves
within the sets $\mathcal{F}_{\beta}(m)$. We have shown that $\varphi(tu+v,x)$
is the sum of a polynomial in $t$ whose coefficients are functions of $x$
which are constant along the sets $\mathcal{F}_{\beta}(m)$, a function of $x$
which is constant along the sets $\mathcal{F}_{\beta}(m)$ and a polynomial in
$t$. Claim 11.12.1 follows.

So we aim to prove Claim 11.12.2. Fix $u\in E^{c_{i}^{\alpha}\alpha}$ with
$\left|\left|u\right|\right|=1$, and for notational convenience, let
$g(t,x)=\varphi(tu,x)$. We claim that for every $x\in X$ and Lebesgue almost
every $t_{1}\in\mathbb{R}$,
$\displaystyle\left.\frac{d}{dt}\right|_{t=t_{1}}g(t,x)$ exists. To prove the
claim, it suffices to show that $g$ is locally Lipschitz. By Lemma 11.9 and
(11.5),

$\displaystyle\left|\left|g(t,x)-g(t_{1},x)\right|\right|$ $\displaystyle=$
$\displaystyle\left|\left|g(t-t_{1},e(tu)\cdot
x)+p(tu,(t-t_{1})u)\right|\right|$ $\displaystyle\leq$
$\displaystyle\left|\left|t-t_{1}\right|\right|^{\sigma}\left|\left|g\left(1,a\cdot
e(tu)\cdot
x\right)\right|\right|+{\left|\left|p(t_{1}u,(t-t_{1})u)\right|\right|}$

for a suitable choice of $a\in\ker\beta$. Recall that $\sigma\geq 1$ by the
integrality assumption and the choice made directly before the statement of
Corollary 11.11. Hence, since $g(1,\cdot)$ is bounded, $p$ is a polynomial and
$\sigma\geq 1$, $g(t,x)$ is Lipschitz in any compact neighborhood of $t_{1}$.
Therefore, for almost every $t_{1}\in E^{c_{i}^{\alpha}\alpha}$, $g$ is
differentiable in $t$ at $t_{1}$. Therefore, since $g$ is differentiable in
$t$ at $t=0$ on a dense set of each orbit of the one-parameter subgroup
generated by $u$, $g$ is differentiable in $t$ at $t=0$ on a dense subset of
$X$.

By (11.5), the set of points for which $g$ is differentiable in $t$ is
invariant under $\mathbb{R}^{k}$. Therefore, if $f(x)$ denotes the derivative
of $g$ in $t$ at $t=0$, $f$ exists on a dense subset of $X$. We claim that
$\left|\left|f(x)\right|\right|\leq B$ for some $B\in\mathbb{R}$ whenever it
exists. Indeed,

$\left|f(x)\right|=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\left|g(\varepsilon,x)\right|=\lim_{\varepsilon\to
0}\frac{1}{\varepsilon}\varepsilon^{\sigma}\left|g(1,a_{\varepsilon}\cdot
x)\right|\leq\sup_{y\in X}\left|g(1,y)\right|$

where $a_{\varepsilon}\in\ker\beta$ is chosen appropriately (using (11.5)),
since $\sigma\geq 1$. Notice that if $a\in\ker\beta$, then again by (11.5) and
the chain rule,

$f(a\cdot x)=e^{(\sigma-1)c_{i}^{\alpha}\alpha(a)}f(x).$

Therefore, either $\sigma=1$ or $f\equiv 0$, since otherwise one may apply an
element $a\in\ker\beta$ with $\alpha(a)$ arbitrarily large to contradict the
boundedness of $f$. Since $\sigma=1$ or $f\equiv 0$, $f$ is constant along
$\ker\beta$ orbits.

We claim that $f$ is also constant along $W^{\alpha}$ leaves whenever it
exists. Assume it is not identically 0, otherwise the claim follows
immediately. In this case $\sigma=1$, and $g$ must be a cocycle over the
$e(tu)$-action by comparing Lemma 11.9 and (11.5). By (TA-9), there exists an
SRB-like measure $\mu$ which is invariant under the $\ker\beta$-action and has
absolutely continuous disintegrations along $N_{\alpha}$-leaves. By standard
Hopf argument, it follows that each ergodic component for the ergodic
decomposition of $\mu$ with respect to the $\ker\beta$-action is also
absolutely continuous along $N_{\alpha}$-leaves.

Hence, for $\mu$-almost every point $x$, $f$ is constant at Lebesgue-almost
every point of $W^{\alpha}$ (since it is a $\ker\beta$-invariant function, and
defined at Lebesgue almost every point of every $W^{\alpha}$ leaf). Since the
$e(tu)$-orbit foliation is a smooth subfoliation of $N_{\alpha}$, it follows
that it is absolutely continuous on almost every such orbit. Since $f$ is the
derivative of $g$, and $g$ is a cocycle over the $e(tu)$-flow, it follows that

$g(t,x)=\int_{0}^{t}f(e(ru)\cdot x)\,dr=v_{x}t$

where $v_{x}\in E^{c_{m}^{\gamma}\gamma}$ is the commmon value of $f$ at
almost every point of $W^{\alpha}(x)$. Then $g$ is linear on a dense set of
$W^{\alpha}$ leaves, and has derivative invariant under $\ker\beta$. Since $g$
is continuous, it follows that $g(t,x)=v_{x}t$, where $v_{x}$ is a derivative
which depends only on the atom of $\mathcal{F}_{\beta}$. This concludes the
case when $\sigma=1$.

We return to the general case (either $\sigma=1$ or $f\equiv 0$). Let $v_{x}$
denote the common value for $f$ along $W^{\alpha}(x)$ and $\ker\beta\cdot x$.
Fix $t_{1}\in\mathbb{R}$ such that $f(e(t_{1}u)\cdot x)$ exists (recall that
the collection of such $t_{1}$ has full Lebesgue measure). Then by Lemma 11.9

(11.7)
$\left.\frac{d}{dt}\right|_{t=t_{1}}g(t,x)=\left.\dfrac{d}{dt}\right|_{t=0}\varphi((t+t_{1})u,x)\\\
=\left.\dfrac{d}{dt}\right|_{t=0}\Big{(}\varphi(t_{1}u,x)+\varphi(tu,e(t_{1}u)\cdot
x)+p(t_{1}u,tu)\Big{)}=f(e(t_{1}u)\cdot x)=v_{x}+q(t_{1})$

where $q$ is some polynomial independent of $x$. Since we know the initial
condition $g(0,x)=0$, one may integrate $v_{x}+q$ to a get a polynomial form
for $g$ at each $x$. Furthermore, since $v_{x}$ does not vary as $x$ moves
along $\ker\beta$, the coefficients of the polynomial are constant along the
atoms of $\mathcal{F}_{\beta}$. This shows Claim 11.12.2, and finishes the
proof. ∎

###### Proof of Claims 11.2 and 11.3.

Corollary 11.11 implies that for fixed $v$, the function
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$
is a polynomial in $u$, whose coefficients are functions of the atoms of
$\mathcal{F}_{\beta}$. It is not difficult to see that an analogous version of
Lemma 11.9 holds when fixing $u$ and varying $v$. Indeed, if we define
$\psi(v,x)=\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$,
then a nearly identical proof shows that

(11.8) $\psi(v_{1}+v_{2},x)=\psi(v_{1},x)+\psi(v_{2},e(v_{1})*e(u)\cdot
x)+p(u,v_{1},v_{2})$

for some polynomial $p$. Furthermore, since we have already shown that
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$
is a polynomial in $u$, it follows that the following equation for $\psi$
analogous to (11.5) for $\varphi$ holds for all $a\in\mathbb{R}^{k}$, not just
$a\in\ker\alpha$:

(11.9) $\psi(v,x)=e^{-\tau
c_{j}^{\beta}\beta(a)}\psi(e^{c_{i}^{\beta}\beta(a)}v,a\cdot x).$

Thus, $\psi$ is invariant under $\ker\beta$, and hence constant on each atom
of $\mathcal{F}_{\beta}$.

Since the leaves of $W^{\alpha}$ and $W^{\beta}$ are contained in the atoms of
$\mathcal{F}_{\beta}$ by Lemma 10.14, the dependence of $\psi$ on $x$ does not
affect (11.8). Therefore, on each atom of $\mathcal{F}_{\beta}$, the map
$\psi$ satisfies the assumption of Lemma 11.10 and on each atom of
$\mathcal{F}_{\beta}$, the function
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$
is a polynomial in $u$ and $v$, with coefficients depending only on the atom.

We must show that the polynomials are independent of $x$. Notice that
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}(u,v,x)$
is a family of polynomials whose coefficients depend on $x$, so (10.1) implies
that the polynomial is $\sigma$-homogeneous in $u$ and $\tau$-homogeneous in
$v$, since otherwise the coefficients would grow to $\infty$ by applying
contracting or expanding elements of $\mathbb{R}^{k}$ (so, in particular,
$\sigma,\tau\in\mathbb{Z}$ and (11.2) holds). Therefore, the polynomial is
unchanged as $x$ moves along its $\mathbb{R}^{k}$-orbit by (10.1). Since there
is a dense $\mathbb{R}^{k}$-orbit and
$\hat{\rho}^{c_{i}^{\alpha}\alpha,c_{j}^{\beta}\beta}_{c_{m}^{\gamma}\gamma}$
is continuous in all variables, it follows that it is independent of $x$.
Hence (11.3) holds. ∎

##  12\. Partial homogeneity implies homogeneity

The goal of this section is to produce homogeneous structures related to the
partial homogeneous structures coming from the $\mathbb{R}^{k}$\- and
$N_{\alpha}$-actions. The arguments and approach expand those of [48, Section
14] in several ways. While the overall scheme is similar, several new
obstacles appear due to the presence of the compact group $M$ as well the
coarse Lyapunov foliations being parameterized by general nilpotent Lie groups
(rather than copies of $\mathbb{R}$). In particular, we build group actions to
describe homogeneous spaces and use Corollary 6.11 to produce Lie structures
on them.

We carefully piece together such homogeneous structures to build one on $X$.
In particular, we use specific, computable relations between the flows of
negatively proportional weights provided by the classification in Lemma 10.15.
We will see that such relations yield a canonical presentation of paths, but
only for a dense set of paths containing an open neighborhood of the identity
in $\hat{\mathcal{P}}_{\left\\{\alpha,-\alpha\right\\}}$.

Our approach is to find canonical presentations for words in
$\hat{\mathcal{P}}$ using only commutator relations and symplectic relations
which we assume are constant and well-defined. By fixing a regular element, we
will be able to use such relations to rearrange the terms in an open set of
words to write them using only stable legs, then only unstable legs, then the
action (Proposition 12.9). This will imply that the quotient group of
$\hat{\mathcal{P}}$ by the commutator relations and symplectic relations is
locally compact, which allows us to use the structure of locally compact
groups (Corollary 6.11). The core of the approach is Lemma 12.8, which gives
the ability to commute stable and unstable paths.

In the end we will have shown that if an ideal factor has constant
$\rho$-functions, it is conjugate to a homogeneous action. In particular, this
holds for any maximal factor, which sets up the induction on factoring out by
a chain of ideals in Section 11.

######  Definition 12.1.

We say that a leafwise homogeneous topological Anosov action has constant
pairwise cycle structure if

  1. (CPCS-1)

for each pair of nonproportional $\alpha,\beta\in\Delta$,
$\gamma\in\Sigma(\alpha,\beta)$ and fixed $u\in N_{\alpha}$ and $v\in
N_{\beta}$, $\rho^{\alpha,\beta}_{\gamma}(u,v,y)$ (see Definition 10.11) is
independent of $y\in X$, and

  2. (CPCS-2)

for each $\alpha\in\Delta$ such that $-\alpha\in\Delta$, the action of
$\mathcal{P}_{\left\\{\alpha,-\alpha\right\\}}$ factors through a Lie group
action on $X$.

In applications, the conditions above are deduced from the genuinely higher-
rank assumptions (TA-8) (indeed, they are consequences of Theorem 11.1 and
Lemma 10.15, respectively). However, we do not need the genuinely higher-rank
assumptions to conclude the main goal of this section:

######  Theorem 12.2.

If $\mathbb{R}^{k}\times M\curvearrowright X$ is a leafwise homogeneous,
topological Anosov action with constant pairwise cycle structure, then the
action is topologically conjugate to a translation action on a homogeneous
space $G/\Gamma$, with $\Gamma$ discrete.

###  12.1. Stable-unstable-neutral presentations

We let $\rho^{\alpha,\beta}_{\gamma}(u,v)$ denote the common value of
$\rho^{\alpha,\beta}_{\gamma}(u,v,x)$ for $\alpha,\beta\in\Delta$,
$\gamma\in\Sigma(\alpha,\beta)$. This is guaranteed to be independent of $x$
since the action has constant pairwise cycle structure. Let
$\mathcal{P}\;(=\mathcal{P}_{\Delta})$ be the group freely generated by the
groups $N_{\alpha}$. Let $\mathcal{C}^{\prime}$ be the smallest closed normal
subgroup containing all cycles of the form

  * •

$[u,v]*\rho^{\alpha,\beta}(u,v)$ for $u\in N_{\alpha}$ and $v\in N_{\beta}$ as
described in Definition 10.11 and

  * •

any element of $\mathcal{P}_{\left\\{\alpha,-\alpha\right\\}}$ which factors
through the identity of the Lie group action provided by (CPCS-2) (ie,
$\ker(\mathcal{P}_{\left\\{\alpha,-\alpha\right\\}}\to G_{\alpha})$, where
$G_{\alpha}$ is as in Lemma 10.15).

Since such cycles are cycles at every point by assumption,
$\mathcal{C}^{\prime}\subset\mathcal{C}(x):=\operatorname{Stab}_{\mathcal{P}}(x)$
and $\mathcal{C}^{\prime}$ is normal. Consider the quotient group
$\mathcal{G}=\mathcal{P}/\mathcal{C}^{\prime}$.

Fix a regular element $a_{0}\in\mathbb{R}^{k}$. The goal of this subsection is
to show that any $\rho\in\mathcal{G}$ can be reduced (via the relations in
$\mathcal{C}^{\prime}$) to some $\rho_{+}*\rho_{-}*\rho_{0}$, with $\rho_{+}$
having only terms from $N_{\chi}$ with $\chi\in\Delta^{+}(a_{0})$, $\rho_{-}$
having only terms from $N_{\chi}$ with $\chi\in\Delta^{-}(a_{0})$, and
$\rho_{0}$ being a product of elements of $\mathbb{R}^{k}\times M$ generated
by symplectic pairs (see Lemma 10.15). Rather, we will show this for the group
obtained by taking the semidirect product of the $\mathbb{R}^{k}\times M$ with
$\mathcal{G}$ see Proposition 12.9. We begin by identifying well-behaved
subgroups of $\mathcal{G}$. Given a subset $\Xi\subset\Delta$, let
$\mathcal{G}_{\Xi}$ denote the subgroup of $\mathcal{G}$ generated by the
subgroups $N_{\chi}$, as $\chi$ ranges over $\Xi$. We say that $\Xi$ is stable
if $\Xi\subset\Delta^{-}(a)$ for some $a\in\mathbb{R}^{k}$. We say that it is
closed if for any $\alpha,\beta\in\Xi$, $\Sigma(\alpha,\beta)\subset\Xi$.

######  Lemma 12.3.

If $\Xi$ is stable, closed collection of weights, then $\mathcal{G}_{\Xi}$ is
a nilpotent Lie group.

###### Proof.

The lemma follows as in the proof of Lemma 11.4. Indeed, the proof works
verbatim after restricting the weights $\chi\in\Xi$ to a generic
$\mathbb{R}^{2}\subset\mathbb{R}^{k}$. Then one uses the circular ordering to
find unique presentations of $\mathcal{G}_{\Xi}$. See also [48], Section 5.2
and Lemma 17.5. ∎

Let $\chi\in\Delta$ be a weight such that $-\chi\in\Delta$, and
$\beta\in\Delta$ be any linearly independent weight. Let
$\Xi=\left\\{t\beta+s\chi:t\geq 0,s\in\mathbb{R}\right\\}\cap\Delta$, and
$\Xi^{\prime}=\left\\{t\beta+s\chi:t>0,s\in\mathbb{R}\right\\}\cap\Delta=\Xi\setminus\left\\{\chi,-\chi\right\\}$.

######  Proposition 12.4.

If $\Xi$ is as above and $\rho\in\mathcal{G}_{\Xi}$ is any element, then
$\rho=\rho_{\chi}*\rho_{\Xi^{\prime}}$, where
$\rho_{\chi}\in\mathcal{G}_{\left\\{\chi,-\chi\right\\}}$, and
$\rho_{\Xi^{\prime}}\in\mathcal{G}_{\Xi^{\prime}}$. Furthermore, such a
decomposition is unique.

###### Proof.

The proof technique is the same as that of Lemma 11.4. Using constancy of
commutator relations, we may push any elements of $N_{\pm\chi}$ to the left,
accumulating elements of $\mathcal{G}_{\Xi^{\prime}}$ as the commutator on the
right.

To see uniquness, suppose that
$\rho_{\chi}*\rho_{\Xi^{\prime}}=\rho_{\chi}^{\prime}*\rho_{\Xi^{\prime}}^{\prime}$.
Then
$(\rho_{\chi}^{\prime})^{-1}*\rho_{\chi}=\rho_{\Xi^{\prime}}^{\prime}*\rho_{\Xi^{\prime}}^{-1}$.
But $\mathcal{G}_{\Xi^{\prime}}$ is a a subgroup of $\mathcal{G}_{\Xi}$ and it
is clear that
$\mathcal{G}_{\Xi^{\prime}}\cap\mathcal{G}_{\left\\{\chi,-\chi\right\\}}=\left\\{e\right\\}$.
Therefore, $\rho_{\chi}^{\prime}=\rho_{\chi}$ and
$\rho_{\Xi^{\prime}}=\rho_{\Xi^{\prime}}^{\prime}$, and the decomposition is
unique. ∎

######  Corollary 12.5.

If $\Xi$ is as above, $\mathcal{G}_{\Xi}$ is a Lie group. Furthermore,
$\mathcal{G}_{\Xi}$ has the semidirect product structure
$\mathcal{G}_{\left\\{\chi,-\chi\right\\}}\ltimes\mathcal{G}_{\Xi^{\prime}}$,
with $\mathcal{G}_{\Xi^{\prime}}$ a nilpotent group.

###### Proof.

Notice that in the proof of Proposition 12.4, we get a unique expression by
moving the elements of $\mathcal{G}_{\left\\{\chi,-\chi\right\\}}$ to the
left, and doing so accumulates only the $\mathcal{G}_{\Xi^{\prime}}$ terms.
Therefore, the decomposition gives $\mathcal{G}_{\Xi}$ the structure of a
semidirect product of $\mathcal{G}_{\left\\{\chi,-\chi\right\\}}$ and
$\mathcal{G}_{\Xi^{\prime}}$. These groups are Lie by (CPCS-2) and Lemma 12.3,
respectively. The action of $\mathcal{G}_{\left\\{\chi,-\chi\right\\}}$ on
$\mathcal{G}_{\Xi^{\prime}}$ is continuous since the action of its generating
subgroups corresponding to $\chi$ and $-\chi$ are given by commutators, which
are continuous. Therefore, $\mathcal{G}_{\Xi}$ is the semidirect product of
the Lie group $\mathcal{G}_{\left\\{\chi,-\chi\right\\}}$ with the Lie group
$\mathcal{G}_{\Xi^{\prime}}$, with a continuous representation, and is hence a
Lie group. ∎

The crucial tool in showing that $\mathcal{P}/\mathcal{C}^{\prime}$ is Lie is
to show that it is locally Euclidean. To that end, the crucial result is Lemma
12.8. Fix a regular element $a_{0}\in\mathbb{R}^{k}$, then define
$\mathcal{G}_{+}=\mathcal{G}_{\Delta^{+}(a_{0})}$ and
$\mathcal{G}_{-}=\mathcal{G}_{\Delta^{-}(a_{0})}$. Note that
$\mathcal{G}_{\pm}$ are nilpotent Lie groups by Lemma 12.3. Let
$\mathcal{D}_{\chi}$ be the subgroup of $\mathcal{G}_{\\{\chi,-\chi\\}}$ which
coincides with the $\mathbb{R}^{k}\times M$-action (see Lemma 10.15). Let
$\mathcal{D}\subset\mathcal{G}$ be the group freely generated by all such
$\mathcal{D}_{\chi}$. Note that the action of each element of $\mathcal{D}$
coincides with the action of some element of the $\mathbb{R}^{k}\times M$
action by construction. However, the action of $\mathcal{D}$ is not obviously
faithful, and may fail to be, as is the case for the Weyl chamber flow on
$SL(3,\mathbb{R})$ where there are 3 symplectic pairs of weights, each
generating one-parameter subgroups of
$\operatorname{Diag}\cong\mathbb{R}^{2}$.

######  Lemma 12.6.

Suppose the $\mathbb{R}^{k}$ orbit of $x_{0}$ is dense. Then if
$d\in\mathcal{D}$ is a cycle at $x_{0}$, then $d$ is a cycle everywhere.

###### Proof.

The action of the generators of $\mathcal{D}$ coincides with the
$\mathbb{R}^{k}\times M$ action, therefore the action of $\mathcal{D}$
commutes with the $\mathbb{R}^{k}$-action. Then if $d$ is a cycle at $x_{0}$,
$d$ is a cycle at any point in $\mathbb{R}^{k}\cdot x_{0}$, hence everywhere
as the $\mathbb{R}^{k}$ orbit of $x_{0}$ is dense. ∎

Denote by $\mathcal{C}_{D}$ the group of cycles in $\mathcal{D}$ at a point
$x_{0}$ with a dense $\mathbb{R}^{k}$-orbit (such a point exists by (TA-1)),
and set $G=\mathcal{G}/\mathcal{C}_{D}$ and $D=\mathcal{D}/\mathcal{C}_{D}$.
Notice that $G$ and $D$ are topological groups since $\mathcal{C}_{D}$ is a
closed normal subgroup by Lemma 12.6. Furthermore, let $G_{\pm}$ denote the
projections of the groups $\mathcal{G}_{\pm}$ to $G$. The following lemma is
immediate from the fact that the action of $D$ coincides with that of the
$\mathbb{R}^{k}\times M$ action and Corollary 12.5:

######  Lemma 12.7.

Let $\rho^{\pm}\in G_{\pm}$ (resp. $G_{\pm}(\bar{x})$) and $\rho^{0}\in D$.
Then

$\rho^{0}\rho^{\pm}(\rho^{0})^{-1}\in G_{\pm}\mbox{ (resp.
}G_{\pm}(\bar{x})).$

The following is the basic commutation argument. It is an adaptation of Lemma
14.11 of [48], with changes to account for the nilpotent groups being
multidimensional.

######  Lemma 12.8.

For an open set of elements $\rho^{+}\in G_{+}$, $\rho^{-}\in G_{-}$
containing $\left\\{e\right\\}\times\left\\{e\right\\}$ there exist
$(\rho^{+})^{\prime}\in G_{+}$, $(\rho^{-})^{\prime}\in G_{-}$ and
$\rho^{0}\in D$ such that

$\rho^{+}*\rho^{-}=(\rho^{-})^{\prime}*(\rho^{+})^{\prime}*\rho^{0}.$

Furthermore, $(\rho^{+})^{\prime}$, $(\rho^{-})^{\prime}$ and $\rho^{0}$
depend continuously on $\rho^{+}$ and $\rho^{-}$.

###### Proof.

Order the weights of $\Delta^{+}(a_{0})$ and $\Delta^{-}(a_{0})$ using a fixed
circular ordering as
$\Delta^{+}(a_{0})=\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ and
$\Delta^{-}(a_{0})=\left\\{\beta_{1},\dots,\beta_{m}\right\\}$. Since
$\Delta^{+}(a_{0})$ is a stable subset, $G_{+}$ is a nilpotent group by Lemma
12.3. Therefore, we may write $\rho_{+}=u_{n}*\dots*u_{1}$ for some $u_{i}\in
N_{\alpha_{i}}$. We will inductively show that we may write the product
$\rho^{+}*\rho^{-}$ as
$u_{n}*\dots*u_{k}*(\rho^{-})^{\prime}*v_{k-1}*\dots*v_{1}*\rho^{0}$ for some
$v_{i}\in N_{\alpha_{i}}$, $(\rho^{-})^{\prime}\in\mathcal{G}_{-}$ and
$\rho^{0}\in D$ (all of which depend on $k$, the index of the induction). Our
given expression is the base case $k=1$.

Suppose we have this for $k$. If $-\alpha_{k}\in\Delta$, then it must be in
$\Delta_{-}$. Let $l(k)$ denote the index for which $\beta_{l(k)}=-\alpha_{k}$
if $-\alpha_{k}$ is a weight. Otherwise, since there is no weight negatively
proportional to $\alpha_{k}$, we set $\beta_{l(k)}=-\alpha_{k}$ with $l(k)$ a
half integer so that $-\alpha_{k}$ appears between $\beta_{l(k)-1/2}$ and
$\beta_{l(k)+1/2}$. Then decompose $\Delta$ into six (possibly empty) subsets:
$\left\\{\alpha_{k}\right\\}$, $\left\\{-\alpha_{k}\right\\}$,
$\Delta_{1}=\left\\{\alpha_{l}:l<k\right\\}$,
$\Delta_{2}=\left\\{\alpha_{l}:l>k\right\\}$,
$\Delta_{3}=\left\\{\beta_{l}:l<l(k)\right\\}$ and
$\Delta_{4}=\left\\{\beta_{l}:l>l(k)\right\\}$. See Figure 2.

$\alpha_{k}$$-\alpha_{k}$$\left\\{\chi:\chi(a_{0})=0\right\\}$$\Delta_{1}$$\Delta_{2}$$\Delta_{3}$$\Delta_{4}$
Figure 2. Decomposing $(\mathbb{R}^{k})^{*}$ into quadrants

We let $G_{\Delta_{i}}$ denote the subgroup of $G$ generated by the
$\Delta_{i}$. Notice that
$\Delta_{-}=\Delta_{3}\cup\left\\{-\alpha_{k}\right\\}\cup\Delta_{4}$ (with
$\left\\{-\alpha_{k}\right\\}$ omitted if there is no weight of this form) is
stable, so again, since $G_{-}$ is nilpotent, $(\rho^{-})^{\prime}$ may be
expressed uniquely as $q_{3}*w*q_{4}$ with $q_{3}\in G_{\Delta_{3}}$ and
$q_{4}\in G_{\Delta_{4}}$ and $w\in N_{-\alpha_{k}}$ (if $-\alpha_{k}$ is not
a weight, we omit this term). Now,
$\left\\{\alpha_{k}\right\\}\cup\Delta_{2}\cup\Delta_{3}$ is a stable set
whose associated group is nilpotent. So
$u_{k}*q_{3}=q_{2}*(q_{3})^{\prime}*v_{k}$ for some $q_{2}\in G_{\Delta_{2}}$,
$(q_{3})^{\prime}\in G_{\Delta_{3}}$ and $v_{k}\in N_{\alpha_{k}}$. Notice
that by iterating some $a\in\mathbb{R}^{k}$ for which $\alpha_{k}(a)=0$, and
$\beta(a)<0$ for all $\beta\in\Delta_{2}\cup\Delta_{3}$, we actually know that
$v_{k}=u_{k}$. Thus, we have put our expression in the form:

$\displaystyle
u_{n}*\dots*u_{k}*(\rho^{-})^{\prime}*v_{k-1}*\dots*v_{1}*\rho^{0}$
$\displaystyle=$ $\displaystyle
u_{n}*\dots*u_{k}*(q_{3}*w*q_{4})*v_{k-1}*\dots*v_{1}*\rho^{0}$
$\displaystyle=$ $\displaystyle
u_{n}*\dots*(u_{k}*q_{3})*w*q_{4}*v_{k-1}*\dots*v_{1}*\rho^{0}$
$\displaystyle=$ $\displaystyle
u_{n}*\dots*u_{k+1}*(q_{2}*(q_{3})^{\prime}*v_{k})*w*q_{4}$
$\displaystyle\qquad*\,v_{k-1}*\dots*v_{1}*\rho^{0}$

Now, there are two cases: $-\alpha_{k}\not\in\Delta$ in which case $v_{k}=e$.
If $-\alpha_{k}\in\Delta$, then as long as $v_{k}$ and $w$ are both
sufficiently small, $v_{k}*w=w^{\prime}*v_{k}^{\prime}*g$ for some
$w^{\prime}\in N_{-\alpha_{k}}$, $v_{k}^{\prime}\in N_{\alpha_{k}}$ and $g\in
D$ (since by Lemma 10.15, the corresponding subalgebras form a splitting of
$G_{\alpha}$). Furthermore, notice that $q_{2}$ and the terms appearing before
$q_{2}$ all belong to $\Delta_{2}$, so we may combine them to reduce the
expression to:

$u_{n}^{\prime}*\dots*u_{k+1}^{\prime}*(q_{3})^{\prime}*w^{\prime}*v_{k}^{\prime}*g*q_{4}*v_{k-1}*\dots*v_{1}*\rho^{0}$

for some collection of $u_{i}^{\prime}\in N_{\alpha_{i}}$, and $g\in D$. But
by Lemma 12.7, $g$ may be pushed to the right preserving the form of the
expression and being absorbed into $\rho^{0}$. We abusively do not change
these terms and drop $g$ from the expression.

Now, we do the final commutation by commuting $v_{k}^{\prime}$ and $q_{4}$.
Notice that $\left\\{\alpha_{k}\right\\}\cup\Delta_{1}\cup\Delta_{4}$ is a
stable subset. Therefore, we may write $v_{k}^{\prime}*q_{4}$ as
$(q_{4})^{\prime}*v_{k}^{\prime\prime}*q_{1}$ with $q_{1}\in
G_{\Delta_{1}},(q_{4})^{\prime}\in G_{\Delta_{4}}$ and
$v_{k}^{\prime\prime}\in N_{\alpha_{k}}$. Inserting this into the previous
expression, we see that the $q_{1}$ term can be absorbed into the remaining
product of the $v_{i}$ terms. This yields the desired form

$u_{n}^{\prime}*\dots*u_{k+1}^{\prime}*(q_{3})^{\prime}*w^{\prime}*(q_{4})^{\prime}*v_{k}^{\prime\prime}*(q_{1}*v_{k-1}*\dots*v_{1})*\rho^{0},$

with the new $\rho^{-}$ equal to
$(q_{3})^{\prime}*w^{\prime}*(q_{4})^{\prime}$. ∎

Recall that the action of $D$ coincides with a subgroup the
$\mathbb{R}^{k}\times M$ action, let $f:D\to\mathbb{R}^{k}\times M$ be the
homomorphism which associates an element of $D$ with the corresponding element
of the $\mathbb{R}^{k}\times M$ action. Then let $\hat{G}$ be the quotient of
$\hat{\mathcal{P}}:=(\mathbb{R}^{k}\times M)\ltimes\mathcal{P}$ (see
Definition 6.6) by the smallest closed normal subgroup containing all elements
from $\ker(\mathcal{P}\to G)$ (ie, the relations already constructed in $G$)
and elements of the form $f(d){\rho_{d}}^{-1}$, where $d\in D\subset G$ and
$\rho_{d}$ is some representative of $d$ in $\mathcal{P}$.

Recall that $\mathcal{P}$ is the free product of nilpotent Lie groups, and has
a canonical CW-complex structure as described in Section 6.2. The cell
structure can be seen by considering subcomplexes corresponding to sequences
of weights $\bar{\chi}=(\chi_{1},\dots,\chi_{n})$ and letting
$C_{\bar{\chi}}=\left\\{u_{1}*\dots*u_{n}:u_{i}\in N_{\chi_{i}}\right\\}\cong
N_{\chi_{1}}\times\dots\times N_{\chi_{n}}$. Then a neighborhood of the
identity is a union of neighborhoods in each cell $C_{\bar{\chi}}$ containing
0.

Let $\pi:\hat{\mathcal{P}}\to\hat{G}$ denote the canonical projection, and
note that $\ker\pi$ is exactly the group generated by (conjugates of)
commutator cycles, nontrivial symplectic cycles, cycles in $\mathcal{C}_{D}$,
and identifications of elements of $\mathbb{R}^{k}\times M$ with their
generated versions in $\mathcal{P}$ (ie, elements of the form
$f(d)\rho_{d}^{-1}$). Furthermore, notice that putting $\Delta^{+}(a)$ and
$\Delta^{-}(a)$ in a circular ordering makes $G_{+}\times
G_{-}\times(\mathbb{R}^{k}\times M)$ a combinatorial cell in
$\hat{\mathcal{P}}$.

######  Proposition 12.9.

There exists an open neighborhood $U$ of $e\in\hat{\mathcal{P}}$ and a
continuous, open map $\Phi:U\to G_{+}\times G_{-}\times(\mathbb{R}^{k}\times
M)$ such that $\pi\circ\Phi=\pi$.

###### Proof.

We describe the map $\Phi$, whose domain will become clear from the
definition. Let
$\Delta^{+}(a_{0})=\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ and
$\Delta^{-}(a_{0})=\left\\{\beta_{1},\dots,\beta_{m}\right\\}$ be the weights
as described in the proof of Lemma 12.8. Given a word
$\rho=u_{1}*\dots*u_{n}$, $u_{i}\in N_{\chi_{i}}$, $\chi_{i}\in\Delta$ for
every $i$, we begin by taking all occurrences of $\alpha_{n}$ in $\rho$ and
pushing them to the left, starting with the leftmost term. When we commute it
past another $\alpha_{i}$, we accumulate only other $\alpha_{j}$, $i+1\leq
j\leq n-1$, in $\rho^{\alpha_{i},\alpha_{n}}$, which we may canonically
present in increasing order on the right of the commutation. A similar
statement holds for the commutation of $\alpha_{n}$ with $\beta_{i}$. We
iterate this procedure as in the proof of Lemma 12.8 to obtain the desired
presentation. Since the commutation operations involved are determined by the
combinatorial type, the resulting presentation is continuous from the cell
$C_{\bar{\chi}}$. Furthermore, if one of the terms happens to be e, the
procedure yields the same result whether it is considered there or not. Thus,
it is a well-defined continuous map from a neighborhood of the identity in
$\hat{\mathcal{P}}$ to $G_{+}\times G_{-}\times(\mathbb{R}^{k}\times M)$. It
is continuous from $\hat{\mathcal{P}}$ because it is continuous from each
$C_{\bar{\chi}}$.

Notice that in the application of Lemma 12.8, we require that all terms are
sufficiently small. Thus, in each combinatorial pattern, since the algorithm
is guaranteed to have a finite number of steps and swaps appearing, and each
term appearing will depend continuously on the initial values of the terms, we
know that for each $\bar{\chi}$, some neighborhood of 0 will be in the domain
of $\Phi$, by the neighborhood structure described above.

Notice that the reduction of a word $u$ to a word of the form
$u_{+}*u_{-}*a\in G_{+}\times G_{-}\times(\mathbb{R}^{k}\times M)$ uses only
relations defining $\hat{G}$. Therefore, if after the reductions, the same
form is obtained, the original words must represent the same element of
$\hat{G}$. That is, $\pi\circ\Phi=\pi$. ∎

######  Corollary 12.10.

The group $\hat{G}$ is a Lie group.

###### Proof.

Choose $U$ is as in Lemma 12.9, and let $K\subset\Phi(U)$ be a compact
neighborhood of $(e,e,e)\in G_{+}\times G_{-}\times(\mathbb{R}^{k}\times M)$.
Note that such a neighborhood exists since $G_{\pm}$ and $\mathbb{R}^{k}\times
M$ are all Lie groups using Lemma 12.3. Then $\Phi^{-1}(K)$ is a neighborhood
of $e\in\hat{\mathcal{P}}$, and $\pi(\Phi^{-1}(K))=\pi(K)$ is a neighborhood
of $e\in\hat{G}$. Therfore, $\hat{G}$ is locally compact. Furthermore, since
$\hat{G}$ is the factor of the locally path-connected group
$\hat{\mathcal{P}}$, $\hat{G}$ is locally path-connected. Hence $\hat{G}$ is a
projective limit of Lie groups by Corollary 6.11. Hence $\hat{G}$ has an
associated sequence $G_{n}$ of connected Lie groups, factor maps
$q_{n}:\hat{G}\to G_{n}$, and projections $p_{n}:G_{n}\to G_{n-1}$ such that
$\ker p_{n}$ is compact, $q_{n}=p_{n+1}\circ q_{n+1}$, and
$\bigcap_{n=1}^{\infty}\ker q_{n}=\left\\{e\right\\}$ (see the diagram in
Corollary 6.11).

We first claim that there exists $N,d\in\mathbb{N}$ such that $\dim(G_{n})=d$
for all $n\geq N$. Indeed, note that since each $p_{n}$ is surjective,
$\dim(G_{n})\geq\dim(G_{n-1})$, so it suffices to show that $\dim(G_{n})\leq
d$ for all $n\in\mathbb{N}$ and some $d\in\mathbb{N}$. We have that
$q_{n}(G_{+})$, $q_{n}(G_{-})$ and $q_{n}(\mathbb{R}^{k}\times M)$ are Lie
subgroups of $G_{n}$, and since $q_{n}\circ\pi\circ\Phi:G_{+}\times
G_{-}\times(\mathbb{R}^{k}\times M)\to G_{n}$ is an open map, we conclude that
the map $(g_{1},g_{2},a)\mapsto g_{1}\cdot g_{2}\cdot a$ from
$q_{n}(G_{+})\times q_{n}(G_{-})\times(\mathbb{R}^{k}\times M)$ is an open
map. It follows that
$\dim(G_{n})\leq\dim(q_{n}(G_{+}))+\dim(q_{n}(G_{-}))+\dim(q_{n}(\mathbb{R}^{k}\times
M))\leq\dim(G_{+})+\dim(G_{-})+\dim(\mathbb{R}^{k}\times M)$, which is
independent of $n$.

Since $\dim(G_{n})=\dim(G_{n+1})$ for all $n\geq N$, $p_{n}$ is a local
isomorphism for all $n\geq N+1$. It follows that the algebras
$\operatorname{Lie}(G_{n})$ are all isomorphic, and there exists a unique
simply connected group $\tilde{G}$ such that
$\operatorname{Lie}(\tilde{G})\cong\operatorname{Lie}(G_{n})$ for sufficiently
large $n$. We may therefore construct local isomorphisms $f_{n}:\tilde{G}\to
G_{n}$ inductively by defining $f_{n+1}$ to be the unique Lie group
homomorphism with derivative $(dp_{n})^{-1}\circ df_{n}$. We therefore obtain
the following commutative diagram:

${\hat{G}}$${\cdots}$${G_{n+1}}$${G_{n}}$${G_{n-1}}$${G_{n-2}}$${\cdots}$${G_{0}}$${\tilde{G}}$$\scriptstyle{q_{n+1}}$$\scriptstyle{q_{n}}$$\scriptstyle{q_{n-1}}$$\scriptstyle{q_{n-2}}$$\scriptstyle{q_{0}}$$\scriptstyle{p_{n+1}}$$\scriptstyle{p_{n}}$$\scriptstyle{p_{n-1}}$$\scriptstyle{p_{n-2}}$$\scriptstyle{f_{n}}$$\scriptstyle{f_{n+1}}$$\scriptstyle{f_{n-1}}$$\scriptstyle{f_{n-2}}$$\scriptstyle{f_{0}}$

By the universal property of inverse limits, there exists a unique
homomorphism $F:\tilde{G}\to\hat{G}$ such that $q_{n}\circ F=f_{n}$. We claim
that the image of $F$ is exactly the path component of $\hat{G}$. Indeed, if
$\gamma:[0,1]\to\hat{G}$ is any path such that $\gamma(0)=e$, then
$\gamma_{n}=q_{n}\circ\gamma$ is a path in $G_{n}$, and
$p_{n}\circ\gamma_{n}=\gamma_{n-1}$. Since $f_{n}$ is a local isomorphism,
there exists a unique $\tilde{\gamma}_{n}:[0,1]\to\tilde{G}$ such that
$f_{n}\circ\tilde{\gamma}_{n}=\gamma_{n}$. Since $p_{n}\circ f_{n}=f_{n-1}$,
the maps $\tilde{\gamma}_{n}$ all coincide, let
$\tilde{\gamma}:[0,1]\to\tilde{G}$ denote the corresponding lift. Then by
construction, $F\circ\tilde{\gamma}=\gamma$, and the endpoint of $\gamma$ can
be reached in the image of $F$.

Finally, since $\hat{G}$ is path connected, the path identity component is
exactly $\hat{G}$, so $F$ is onto. Therefore, $\hat{G}$ is the image of a Lie
group, and therefore Lie. ∎

###  12.2. Description of the homogeneous spaces

By Corollary 12.10, $X$ is the homogeneous space of a Lie group $\hat{G}$,
which is generated by subgroups which are images of $N_{\alpha}$,
$\alpha\in\Delta$ and $\mathbb{R}^{k}\times M$. Furthermore, the group
$\hat{G}$ is a factor of the group $\hat{\mathcal{P}}=(\mathbb{R}^{k}\times
M)\ltimes\mathcal{P}$, where $\mathcal{P}$ is the free product of the groups
$N_{\alpha}$, and let $\mathcal{C}$ denote the kernel of
$\hat{\mathcal{P}}\to\hat{G}$. Therefore, $\mathcal{C}$ is the normal closure
of the group generated by commutator relations $\rho^{\alpha,\beta}(u,v,x)$
(which do not depend $x$), symplectic relations, and identifications of the
diagonal elements of $G_{\alpha}$ with the $\mathbb{R}^{k}$-action.

Let $S(x)=\operatorname{Stab}_{\hat{G}}(x)^{\circ}$, and notice that $S_{1}$
is the closed subgroup of a Lie group and therefore Lie.

######  Lemma 12.11.

$\dim(S(x))$ is independent of $x\in X$.

###### Proof.

Since $\hat{G}$ acts transitively on $X$ by (TA-7), it follows that $X$ is the
homogeneous space of the Lie group $\hat{G}$. The result is now immediate
since if $g\in\hat{G}$, $S(x)$ and $S(g\cdot x)$ are subgroups conjugated by
$g$. ∎

Let $a_{0}\in\mathbb{R}^{k}$ be a generic element (ie, an element such that
$\alpha(a_{0})\not=0$ for all $\alpha\in\Delta$ and
$c_{i}\alpha(a_{0})\not=c_{j}\beta(a_{0})$ for all functionals
$c_{i}\alpha\in[\alpha]$, $c_{j}\beta\in[\beta]$.

######  Lemma 12.12.

If $x$ has a dense $\mathbb{R}^{k}$-orbit and is $a_{0}^{\pm 1}$-recurrent,
and $S(x)\not=\left\\{e\right\\}$, then $\operatorname{Lie}(S(x))$ contains an
element of an Oseledets subspace of $\operatorname{Lie}(N_{\alpha})$ for some
$\alpha\in\Delta$.

###### Proof.

Suppose that $S(x)\not=\left\\{e\right\\}$, and consider
$\operatorname{Lie}(S(x))\subset\operatorname{Lie}(G)$. Notice that
$x\mapsto\operatorname{Lie}(S(x))$ is semi-continuous in the following sense:
if $x_{n}\to x$ and $\mathbb{R}v_{n}\subset\operatorname{Lie}(S(x_{n}))$, with
$\left|\left|v_{n}\right|\right|=1$ and $v_{n}\to v$, then
$\mathbb{R}v\subset\operatorname{Lie}(S(x))$.

Now, simply notice that if $S(x)\not=\left\\{e\right\\}$ and ${a_{0}}^{n}\cdot
x\to x$, then $S(x)$ contains its fastest Oseledets space (either forward or
backward), or is contained in $(\mathbb{R}^{k}\times M)$. $S(x)$ is transverse
to $\mathbb{R}^{k}\times M$ since the action of $\mathbb{R}^{k}\times M$ is
locally free by the definition of Lyapunov homogeneous topological Anosov
action. Therefore, $S(x)$ contains an Oseledets space if it is nontrivial. ∎

######  Corollary 12.13.

$S(x)=\left\\{e\right\\}$ for all $x\in X$.

###### Proof.

If $S(x)\not=\left\\{e\right\\}$ for some point, it is nontrivial at every
point by Lemma 12.11. Since $a_{0}^{\pm 1}$-recurrence and dense
$\mathbb{R}^{k}$-orbit are both residual properties, we may find some $x_{1}$
such that $x_{1}$ has a dense $\mathbb{R}^{k}$-orbit and for which
$\operatorname{Lie}(S(x_{1}))$ contains an element of an Oseledets space by
Lemma 12.12. But since $a_{*}S(x_{1})=S(a\cdot x_{1})$ and Oseledets spaces
are invariant under $a_{*}$, it follows that $\operatorname{Lie}(S(x))$
contains that element for all $x$. This contradicts (TA-2), since locally bi-
Lipschitz implies locally free. ∎

###### Proof of Theorem 12.2.

We have just shown that $S(x)=\left\\{e\right\\}$,
$\operatorname{Stab}_{G}(x)$ is discrete. The group $\hat{G}$ is Lie by
Corollary 12.10 and contains the $\mathbb{R}^{k}\times M$-action as a
subgroup. The result follows. ∎

###  12.3. Proof of Theorem 10.7

By Theorem 11.1 and Lemma 10.15, any action satisfying the assumptions of
Theorem 10.7 has pairwise constant cycle structure (recall Definition 12.1).
Therefore, Theorem 10.7 is a consequence of Theorem 12.2.

##  13\. Smooth Anosov Actions

In this section, we verify that smooth abelian actions of
$\mathbb{R}^{k}\times M$ satsifying assumptions (FA-1) and (FA-2) must be
leafwise-homogeneous topological Anosov actions. Much of the work establishing
this was done in Section 9 (culminating in Theorem 9.15). The main things left
to check are the integral Lyapunov coefficients property (recall the paragraph
following Definition 11.2), as well as verifying that the conjugating map is
smooth.

######  Proposition 13.1.

If $\mathbb{R}^{k}\times M\curvearrowright X$ is a $C^{\infty}$ Anosov action
satisfying conditions (FA-1) and (FA-2) of the introduction, then the compact
extension of the action provided by Theorem 9.15 is a genuinely higher-rank
leafwise homogeneous topological Anosov action with integral Lyapunov
coefficients and SRB measures.

###### Proof.

Properties (TA-1) and (TA-8) follow immediately from (FA-1). The group actions
of Property (TA-2) come from Theorem 9.15(3), and the normalization property
(TA-3) follows from (4) and (5) of that Theorem.

Property (TA-4) follows from the Anosov closing lemma for actions of
$\mathbb{R}^{k}\times M$. The proof of Lemma 4.15 in [48] is written for
$\mathbb{R}^{k}$-actions, but works verbatim for $\mathbb{R}^{k}\times M$.
Then, as in Lemma 4.17 of that paper, one may deduce density of periodic
orbits when there is an invariant measure of full support. Properties (TA-5)
and (TA-6) follow from Lemma 5.11 of [48]. Property (TA-7) is a direct
consequence of the usual local product structure for Anosov actions. Property
(TA-9) follows from the existence of SRB measures for Anosov actions and the
smoothness of holonomies along fast foliations in an unstable. This is shown
in [48, Proposition 4.22] for $\mathbb{R}^{k}$-actions, the proof works
verbatim for $\mathbb{R}^{k}\times M$. Since the proof of integral Lyapunov
coefficients is more involved, we prove it in Lemma 13.3. ∎

Recall that a common stable foliation is a foliation $\mathcal{W}$ whose
leaves are given by $\bigcap_{k=1}^{n}W^{s}_{a_{k}}(x)$ for some collection
$a_{1},\dots,a_{k}$ of Anosov elements of $\mathbb{R}^{k}$. Common stable
foliations are Hölder foliations with smooth leaves (see [48, Lemma 4.5]).

######  Lemma 13.2.

Consider a common stable foliation $\mathcal{W}$, which is the sum of coarse
Lyapunov distributions $E_{\alpha_{1}},\dots,E_{\alpha_{n}}$. Let
$\left\\{\beta_{1},\dots,\beta_{N}\right\\}$ be an arbitrary listing of
Lyapunov functionals such that
$\bigcup_{i=1}^{n}[\alpha_{i}]=\left\\{\beta_{1},\dots,\beta_{N}\right\\}$.
Then the map

$\phi_{x}:\bigoplus_{i=1}^{N}E^{\beta_{i}}\to X$

defined by

$\phi_{x}(u_{1},\dots,u_{N})=\exp(u_{1})\exp(u_{2})\dots\exp(u_{N})\cdot x$

is surjective from a neighborhood of 0 in $\bigoplus_{i=1}^{N}E^{\beta_{i}}$
onto a neighborhood of $x$ in $\mathcal{W}(x)$.

This lemma follows from [46, Lemma 3.2], which uses ideas from [35],
particularly Corollary 4.5. We do not offer a complete proof, but summarize
the strategy: note that if the actions of $N_{\alpha}$ were smooth, this would
be true from the inverse function theorem. One may then use smooth
approximations of the one-parameter subgroups and show that in the limit the
original action is recovered, and the onto property persists by topological
degree arguments.

######  Lemma 13.3.

If $\mathbb{R}^{k}\times M\curvearrowright X$ is a $C^{2}$ Anosov action
satisfying (FA-1) and (FA-2) of the introduction, then it has integral
Lyapunov coefficients.

Before proceeding with the proof, we recall notations developed in Section 10.
Each coarse Lyapunov exponent is an equivalence class of functionals
$[\alpha]$ defined up to positive scalar multiple, and we choose $\alpha$ as
the smallest representative. Given $\alpha,\beta\in\Delta$,
$\Sigma(\alpha,\beta)$ is the set of coarse Lyapunov exponents which can be
written as positive linear combinations of $\alpha$ and $\beta$. Let
$\Omega(\alpha,\beta)=\cup_{\gamma\in\Sigma(\alpha,\beta)}[\gamma]$. That is,
$\Omega(\alpha,\beta)$ is a set of Lyapunov functionals between $\alpha$ and
$\beta$.

###### Proof.

Fix Lyapunov functionals $c_{i}^{\alpha}\alpha\in[\alpha]$ and
$c_{j}^{\beta}\beta\in[\beta]$. Then divide $\Omega(\alpha,\beta)=A\cup B$,
where
$A=\left\\{\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta:\sigma\geq
1\right\\}\cap\Delta$ and
$B=\left\\{\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta:\sigma<1\right\\}\cap\Delta$.
Let

$B^{\prime}=\left\\{\sigma\,c_{i}^{\alpha}\alpha+\tau\,c_{j}^{\beta}\beta:\sigma<1,\tau\geq
1\right\\}\cap\Delta\subset B.$

Give the coarse Lyapunov exponents in $\Sigma(\alpha,\beta)$ a circular
ordering $\left\\{\alpha=\chi_{1},\dots,\chi_{n}=\beta\right\\}$. We will show
that if $c_{p}^{\chi_{k}}\chi_{k},c_{q}^{\chi_{\ell}}\chi_{\ell}\in A\cup
B^{\prime}$, then
$[c_{p}^{\chi_{k}}\chi_{k},c_{q}^{\chi_{\ell}}\chi_{\ell}]\subset A\cup
B^{\prime}$ by induction on $\left|k-\ell\right|$. Then the result follows
since $\alpha=\chi_{1}$ and $\beta=\chi_{n}$.

The base case is trivial: if $\left|k-\ell\right|=1$, then $\chi_{k}$ and
$\chi_{\ell}$ are adjacent in the circular ordering, and therefore
$N_{\chi_{k}}$ and $N_{\chi_{\ell}}$ commute. We now try to commute
$c_{p}^{\chi_{k}}\chi_{k},c_{q}^{\chi_{\ell}}\chi_{\ell}\in A\cup B^{\prime}$,
$\left|k-\ell\right|>1$. We break into cases based on whether each weight lies
in $A$ or $B^{\prime}$. The easiest occurs when both belong to $A$ or both
belong to $B^{\prime}$, so we cover these first.

Let $\mathcal{W}$ be the foliation whose leaves are tangent to
$\bigoplus_{k=1}^{n}E^{\chi_{k}}$, which is a common stable manifold. Notice
that choosing $a\in\ker\beta$, and perturbing by a very small amount will
yield an element $a^{\prime}$ close to $a$ for which $\left(\bigoplus_{\chi\in
A}E^{\chi}\right)\oplus\left(\bigoplus_{\chi\in B}E^{\chi}\right)$ is a
dominated splitting of $T\mathcal{W}$. Therefore,
$\bigoplus_{c_{p}^{\chi_{k}}\chi_{k}\in A}E^{c_{p}^{\chi_{k}}\chi_{k}}$ is
tangent to a foliation $\mathcal{W}^{A}$.

Assuming we have fixed $c_{i}^{\alpha}\alpha$ and $c_{j}^{\beta}\beta$, if
$\chi\in\Sigma(\alpha,\beta)$, let $\hat{N}_{\chi}$ be the nilpotent group
tangent to the subalgebra

$\hat{\mathfrak{n}}_{\chi}=\bigoplus_{c_{p}^{\chi}\chi\in
A}E^{c_{p}^{\chi}\chi}.$

This is a (possibly trivial) subalgebra since each $a\in\mathbb{R}^{k}$ acts
by an automorphism of $N_{\chi}$, so by standard Lie theory,
$[E^{c_{p}^{\chi}\chi},E^{c_{q}^{\chi}\chi}]\subset
E^{(c_{p}^{\chi}+c_{q}^{\chi})\chi}$. Notice that each leaf $\mathcal{W}^{A}$
has local $C^{0}$ surjections given by the maps from
$\hat{N}_{\chi_{1}}\times\hat{N}_{\chi_{2}}\times\dots\hat{N}_{\chi_{n}}\to\mathcal{W}^{A}(x)$
by

(13.1) $(u_{1},\dots,u_{n})\mapsto u_{1}\cdot u_{2}\cdot\dots\cdot u_{n}\cdot
x.$

The proof of this claim uses Lemma 13.2. See also Lemma 5.11 of [48].
Essentially it follows from transversality of the distributions, the
difficulty being that they are only Hölder. Indeed, these show that (13.1)
gives coordinates in a neighborhood near 0 and $x$, then use the expanding
dynamics to show they are global coordinates. In particular, if both
$c_{p}^{\chi_{k}}\chi_{k}$ and $c_{q}^{\chi_{\ell}}\chi_{\ell}\in A$,
$[c_{p}^{\chi_{k}}\chi_{k},c_{q}^{\chi_{\ell}}\chi_{\ell}]\subset A$.
Similarly, if both $c_{p}^{\chi_{k}}\chi_{k}$ and
$c_{q}^{\chi_{\ell}}\chi_{\ell}$ belong to $B^{\prime}$, then they both belong
to $C=\left\\{u\alpha+v\beta:v\geq 1\right\\}\cap\Delta$. By identical
arguments, choosing a perturbation of $b\in\ker\alpha$, gives
$[B^{\prime},B^{\prime}]\subset[C,C]\subset C\subset A\cup B^{\prime}$.

We now consider the case when $c_{p}^{\chi_{k}}\chi_{k}\in B^{\prime}$ and
$c_{q}^{\chi_{\ell}}\chi_{\ell}\in A$ with $\left|k-\ell\right|>1$ (the case
when $c_{p}^{\chi_{k}}\chi_{k}\in A$ and $c_{q}^{\chi_{\ell}}\chi_{\ell}\in
B^{\prime}$ follows from a symmetric argument). Let $x\in X$, $u\in
E^{c_{p}^{\chi_{k}}\chi_{k}}\subset\operatorname{Lie}(N_{\chi_{k}})$ and $v\in
E^{c_{q}^{\chi_{\ell}}\chi_{\ell}}\subset\operatorname{Lie}(N_{\chi_{\ell}})$.
We construct points related to a geometric commutator in the following way

$\begin{array}[]{ccc}y=e(u)\cdot x&&x^{\prime}=e(v)\cdot x\\\
y^{\prime}=e(v)\cdot y&&w=e(u)\cdot x^{\prime}\end{array}$

Notice that $y^{\prime}=[e(-v),e(-u)]\cdot w$ (see Figure 3).

Let $\Sigma_{k\ell}:=\left\\{\chi_{k+1},\dots,\chi_{\ell-1}\right\\}$ be the
set of coarse exponents strictly between $\chi_{k}$ and $\chi_{\ell}$. Each
coarse exponent of $\Sigma_{k\ell}$ splits into Lyapunov exponents, let
$\Omega_{k\ell}$ denote the set of Lyapunov exponents proportional to a coarse
exponent of $\Sigma_{k\ell}$. Now write $\Omega_{k\ell}$ as

$\Omega_{k\ell}=\left\\{\gamma_{1},\dots,\gamma_{m_{1}},\delta_{1},\dots,\delta_{m_{2}},\epsilon_{1},\dots,\epsilon_{m_{3}}\right\\},$

where $\left\\{\gamma_{\bullet}\right\\}$, $\left\\{\delta_{\bullet}\right\\}$
and $\left\\{\epsilon_{\bullet}\right\\}$ are the exponents of
$A\cap\Omega_{k\ell}$, $(B\setminus B^{\prime})\cap\Omega_{k\ell}$ and
$B^{\prime}\cap\Omega_{k\ell}$, respectively, with each subset listed in an
order to be clarified later.

We assume $y$ and $w$ are sufficiently close, to be determined later (if we
show it for sufficiently small $u,v$, we may use the dynamics of
$\mathbb{R}^{k}\curvearrowright X$ and (9.4) to conclude it for arbitrary
$u,v$).

Notice that the distribution $\bigoplus_{s=k+1}^{\ell-1}E^{\chi_{s}}$ is
uniquely integrable to a foliation $\mathcal{W}^{k\ell}$ with $C^{2}$ leaves
since it is the intersection of stable manifolds for the action. Since
$y^{\prime}=[e(-v),e(-u)]\cdot w$, $y^{\prime}\in\mathcal{W}^{k\ell}(w)$ by
Lemma 10.10. Therefore, by Lemma 13.2 applied to the splitting
$T\mathcal{W}^{k\ell}$ into the bundles $E^{\gamma_{s}}$, $E^{\delta_{s}}$ and
$E^{\epsilon_{s}}$, there exists a path moving from $w$ to $y^{\prime}$ which
first moves along curves of the form $\exp(w)$, where $w\in E^{\gamma_{s}}$,
to arrive at a point $p$. Then similarly along exponential images of $w\in
E^{\delta_{s}}$ to arrive at a point $q$ from $p$. Finally, we move along the
exponential images of vectors in $E^{\epsilon_{s}}$ to connect $q$ to
$y^{\prime}$. In this way, $p$ is obtained from $w$ after moving along curves
tangent to $B^{\prime}$, and $q$ is obtained from $p$ by moving along curves
tangent to $B\setminus B^{\prime}$. Then $q$ is also connected to $y^{\prime}$
via curves tangent to $A$. See Figure 3.

$x$$e(u)$$y$$e(v)$$x^{\prime}$$e(v)$$e(u)$$w$$p$$B^{\prime}$$q$$y^{\prime}$$B\setminus
B^{\prime}$$A$ Figure 3. A geometric commutator

Recall that $\left|\alpha,\beta\right|$ denotes the set of coarse exponents
between $\alpha$ and $\beta$, including $\alpha$ and $\beta$, and let
$W^{\left|\alpha,\beta\right|}$ denote the foliation whose leaves are tangent
to the sum of these distributions (since it is a common stable manifold, it is
a Hölder foliation with $C^{2}$ leaves). Choose any pair of $C^{2}$ discs
$D_{1}\ni x,y$, $D_{2}\ni x^{\prime},q$ of dimension $\sum_{\omega\in
B}\dim(E^{\omega})$ transverse to $\mathcal{W}^{A}$ inside of
$W^{\left|\alpha,\beta\right|}(x)$. This is possible since $x$ and $y$ are
connected via $c_{p}^{\chi_{k}}\chi_{k}\in B^{\prime}$ and $x^{\prime}$ and
$q$ are connected via only curves tangent to coarse Lyapunov foliations
corresponding to exponents in $B$. Therefore, $x^{\prime}$ and $q$ are the
images of $x$ and $y$ under the $\mathcal{W}^{A}$ holonomy from $D_{1}$ to
$D_{2}$. By [37, Section 8.3, Lemma 8.3.1], there exist bi-Lipschitz
coordinates for which the leaves of the foliation $\mathcal{W}^{A}$ are
parallel Euclidean hyperspaces. In particular, the holonomy along
$\mathcal{W}^{A}$ is uniformly Lipschitz independent of the choice of $D_{1}$
and $D_{2}$, given sufficiently good transversality conditions, so
$d(x,y)/d(x^{\prime},q)$ is bounded above and below by a constant.

We claim that $p=q$ (ie, that no weights of $(B\setminus
B^{\prime})\cap\Omega_{kl}$ appear). Roughly, the reason is that such weights
contract too slowly. Indeed, pick an element $a^{\prime}\in\ker\alpha$ such
that $\beta(a^{\prime})=-1$. We may perturb $a^{\prime}$ to an element $a$
which is regular and such that $\alpha(a)<0$, and such that if $\delta\in
B\setminus B^{\prime}$, $\chi_{k}(a)<\delta(a)<0$. This is possible because if
$\delta=\sigma\,\alpha+\tau\,\beta\in B\setminus B^{\prime}$, then $\tau<1$,
so $\delta(a^{\prime})=-\tau>-1=\beta(a^{\prime})\geq
c_{p}^{\chi_{k}}\chi_{k}(a^{\prime})$, since when $c_{p}^{\chi_{k}}\chi_{k}\in
B^{\prime}$, the $\beta$ coefficient is at least 1. This is clearly an open
condition for each $\delta$, so we may choose $a$ as indicated. We may also
assume, by rescaling $a$ as necessary, that $\beta(a)=-1$.

Since $y=\exp(u)x$, we can estimate distance between iterates of $x$ and $y$
using the intertwining property (TA-3). Recall that since
$c_{p}^{\chi_{k}}\chi_{k}\in B^{\prime}$,
$c_{p}^{\chi_{k}}\chi_{k}(a)<\beta(a)=-1$. Therefore
$d_{W^{\chi_{k}}}(a^{t}\cdot x,a^{t}\cdot
y)=e^{tc_{p}^{\chi_{k}}\chi_{k}(a)}d_{W^{\chi_{k}}}(x,y)<e^{-t}d_{W^{\chi_{k}}}(x,y)$
using the Hölder metric along the leaves of $W^{\chi_{k}}$. Now, suppose
$p\not=q$. Recall that $p$ and $q$ are connected by legs in $B\setminus
B^{\prime}=\left\\{\delta_{1},\dots,\delta_{m}\right\\}$, so that there exist
$p=x_{0},x_{1},\dots,x_{m}=q$ such that $x_{s}$ $x_{s-1}$ are connected by a
short curve everywhere tangent to $E^{\delta_{s}}$. Since the distributions
$E^{\delta_{s}}$ are transverse, if $p\not=q$, there exists some $s$ for which
$x_{s}\not=x_{s-1}$. Without loss of generality, we assume that
$\left\\{\delta_{s}\right\\}$ are ordered such that
$0>\delta_{1}(a)>\delta_{2}(a)>\dots>\delta_{m}(a)>\chi_{i}(a)$. Then let
$s_{0}$ be the minimal $s$ for which $x_{s}\not=x_{s-1}$, and
$c_{1}=\delta_{s_{0}}(a)$, $c_{2}=\delta_{s_{0}+1}(a)$. Notice that
$0>c_{1}>c_{2}>-1$. By minimality, we get $x_{s_{0}-1}=p$.

Let $d$ denote the Riemannian distance on the manifold. Since for any
$\chi\in\Delta$, the distance along each $W^{\chi}$ leaf is locally Lipschitz
equivalent to the distance on the manifold, there exists $L>0$ such that for
all $\chi\in\Delta$ and sufficiently close points $z\in W^{\chi}(z^{\prime})$,
we have $L^{-1}d_{W^{\chi}}(z,z^{\prime})\leq d(z,z^{\prime})\leq
Ld_{W^{\chi}}(z,z^{\prime})$. Then after applying the triangle inequality, for
sufficiently large $t$, we get:

$\displaystyle d(a^{t}\cdot x^{\prime},a^{t}\cdot q)$ $\displaystyle\geq$
$\displaystyle d(a^{t}\cdot x_{s_{0}},a^{t}\cdot p)-d(a^{t}\cdot
x^{\prime},a^{t}\cdot p)-d(a^{t}\cdot x_{s_{0}},a^{t}q)$ $\displaystyle\geq$
$\displaystyle L^{-1}d_{W^{[\delta_{s_{0}}]}}(a^{t}\cdot x_{s_{0}},a^{t}\cdot
p)-d(a^{t}\cdot x^{\prime},a^{t}\cdot p)-d(a^{t}\cdot x_{s_{0}},a^{t}q)$
$\displaystyle\geq$ $\displaystyle
L^{-2}e^{c_{1}t}d(x_{s_{0}},p)-L^{2}Ce^{c_{2}t}\geq C^{\prime}e^{c_{1}t}$

since by construction, apply the triangle inequality to all legs connecting
$a^{t}\cdot x_{s_{0}}$ and $a^{t}\cdot q$ and $a^{t}\cdot x^{\prime}$ and
$a^{t}\cdot p$, which contract faster than $e^{c_{2}t}$ and $e^{-t}$,
respectively, since $c_{1}>c_{2}>-1$. We may construct new disks $D_{1,t}$ and
$D_{2,t}$ with sufficient transversality conditions to $\mathcal{W}^{A}$
connecting $a^{t}\cdot x$ and $a^{t}\cdot y$, and $a^{t}\cdot x^{\prime}$ and
$a^{t}\cdot q$ (note that we may not simply iterate the disks $D_{1}$ and
$D_{2}$ forward, since $\mathcal{W}^{A}$ is not the fast foliation for $a$,
and the transversality may degenerate). Therefore, since each of the
foliations along weights of $B$ are uniformly transverse to those of $A$, and
the holonomies are Lipschitz with a uniform Lipschitz constant on sufficiently
transverse discs, we arrive at a contradiction to the fact that $d(a^{t}\cdot
x,a^{t}\cdot y)<C^{\prime\prime}e^{-t}$, so $p=q$.

Therefore, the connection between $w$ and $y^{\prime}$ only involves weights
of $A\cup B^{\prime}$. By the induction hypothesis, the commutator of two
weights in $(A\cup B^{\prime})\cap\Omega_{k\ell}$ produces only weights in
$(A\cup B^{\prime})\cap\Omega_{k\ell}$. Hence, for any $x$,

$e(v)*e(u)\cdot x=\rho_{1}*e(u)*e(v)\cdot x=e(u)*e(v)*\rho_{2}\cdot x$

for some paths $\rho_{1}$ and $\rho_{2}$ which only involve $e(w)$ for some
$w$ in the weight spaces $A\cup B^{\prime}$, but may depend on $x$, $u$ and
$v$. Using such relations, for any word in the weights $A\cup B^{\prime}$, we
may put it in a desired circular ordering without weights in $B\setminus
B^{\prime}$. This proves the inductive step, and hence the lemma, since
$B\setminus
B^{\prime}=\left\\{\sigma\,\alpha+\tau\,\beta:\sigma,\tau<1\right\\}\cap\Delta$.
∎

###  13.1. Regularity of conjugacies and proof of Theorem 2.6

In this section, we work in two regularities: $C^{2}$ and $C^{\infty}$.

######  Theorem 13.4.

If $\mathbb{R}^{k}\times M\curvearrowright X$ is a totally Anosov
$C^{r}$-action on a $C^{\infty}$-manifold $X$ for $r=\infty$ or $2$ satisfying
assumptions (FA-1) and (FA-2), then there exists a (bi)homogeneous space
$K\backslash H/\Gamma$ and a $C^{r^{\prime}}$ diffeomomorphism $h:X\to
K\backslash H/\Gamma$ such that, setting $\tilde{a}=h\circ a\circ h^{-1}$ for
any $a\in\mathbb{R}^{k}\times M$, defines a bi-homogeneous action, where
$r^{\prime}=\infty$ or $(1,\theta)$ for any $\theta\in(0,1)$, respectively.
Moreover, any $C^{0}$ conjugacy of the $\mathbb{R}^{k}\times M$-action is
$C^{r^{\prime}}$.

###### Proof.

By Proposition 13.1 and Theorem 10.7, it follows that there is a
$C^{0}$-conjugacy between the canonical lift $\tilde{X}$ produced in Theorem
9.15 and a translation action on a homogeneous space $H/\Gamma$. The $C^{0}$
conjugacy between $\tilde{X}$ and $H/\Gamma$ induces a conjugacy $h$ between
$X$ and $K\backslash H/\Gamma$. So it suffices to show that any $C^{0}$
conjugacy between $X$ and $K\backslash H/\Gamma$ is $C^{r^{\prime}}$.

The coarse Lyapunov manifolds are parameterized by nilpotent group actions,
which are subgroups of the full group $H$ in the homogeneous model (and hence
act smoothly on the space $H/\Gamma$). They are conjugated to the actions
$N_{\alpha}\curvearrowright X$ produced in Theorem 9.15, which act by
$C^{r^{\prime}}$ diffeomorphisms on their orbits by Lemma 9.12 or 9.10.

By standard Lie theoretic arguments (see, e.g., [42], Section 5.1), since each
element is a diffeomorphism, the group action is $C^{1}$ in both cases. In the
$C^{\infty}$ setting, it also follows immediately that the group action is
$C^{\infty}$, and therefore provides $C^{\infty}$-coordinates. Therefore, the
conjugacy is $C^{\infty}$ restricted to each leaf of a coarse Lyapunov
foliations.

In the $C^{2}$-case, we still have that $h$ is $C^{1}$ along leaves of
$W^{\alpha}$. Notice that if $\left|\left|\cdot\right|\right|_{\alpha}$ is the
Hölder metric on $TW^{\alpha}$ constructed using (FA-2) in Section 9.1, then
it is invariant under the isometries in Proposition 9.2. Then
$h_{*}\left|\left|\cdot\right|\right|_{\alpha}$ is a norm which is invariant
under right translation on each leaf of $W^{\alpha}$ in $H/\Gamma$ (notice
that while right translation is not defined on all of $H/\Gamma$, it is well-
defined on each $W^{\alpha}$-leaf). In particular,
$h_{*}\left|\left|\cdot\right|\right|_{\alpha}$ is $C^{\infty}$ on each
$W^{\alpha}$-leaf of $H/\Gamma$. Therefore, $h:W^{\alpha}(x)\to
W^{\alpha}(h(x))$ is an isometry between a Hölder metric on the leaf in
$\tilde{X}$ and $C^{\infty}$ metric on the leaf in $H/\Gamma$, and is
therefore $C^{r^{\prime}}$ by Theorem 5.4.

The conjugacy $h$ is clearly $C^{r^{\prime}}$ along the $\mathbb{R}^{k}\times
K\times M$-orbits. The smoothness of $h$ will now follow from iterated
applications of Theorem 5.3. Fix a regular element $a$. We show that $h$ is
$C^{r^{\prime}}$ along the foliations $W^{s}_{a}$, $W^{u}_{a}$ and
$\mathbb{R}^{k}\times M\times K$. Then by Theorem 5.3, since
$\mathbb{R}^{k}\times M\times K$ and $W^{s}_{a}$ foliate the center-stable
manifold as transverse subfoliations, it will follow that $h$ is
$C^{r^{\prime}}$ along the center-stable foliation. Then again, since the
conjugacy is $C^{r^{\prime}}$ along the center-stable foliation and the
unstable foliation, another application Theorem 5.3 shows that $h$ is
$C^{r^{\prime}}$.

So we must show that $h$ is $C^{r^{\prime}}$ along the foliations $W^{s}_{a}$
and $W^{u}_{a}$. We show it for $W^{s}_{a}$, the proof for $W^{u}_{a}$ follows
by considering $-a$. List the coarse exponents
$\Delta^{-}(a)=\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ is a circular
ordering, so that there are foliations $\mathcal{W}_{i}$ such that
$T\mathcal{W}_{i}=\bigoplus_{j=1}^{i}TW^{\alpha_{j}}$. These foliations are
obtained as intersections of stable manifolds from different Weyl chambers. We
claim that $h$ is $C^{r^{\prime}}$ along $\mathcal{W}_{i}$ by induction on
$i$.

We have already established the base case of $i=1$. Assume that $h$ is
$C^{r^{\prime}}$ along $\mathcal{W}_{i}$. Then by construction,
$\mathcal{W}_{i+1}$ is a foliation whose leaves have two transverse
subfoliations: $\mathcal{W}_{i}$ and $W^{\alpha_{i+1}}$. Since we know that
$h$ is $C^{r^{\prime}}$ along each by induction and since $W^{\alpha_{i+1}}$
is a coarse Lyapunov foliation, it follows that $h$ is $C^{r^{\prime}}$ along
$\mathcal{W}_{i+1}$ (again by Theorem 5.3). This proves the inductive step.
Since $W^{s}_{a}=\mathcal{W}_{n}$, it follows that $h$ is $C^{r^{\prime}}$
along $W^{s}_{a}$ and therefore, $h$ is $C^{r^{\prime}}$. ∎

##  Appendix A Normal forms for contracting foliations

We recall some aspects of normal forms theory, following [27, 26] which
contains optimal results in the uniformly contracting setting. If $f:X\to X$
is a $C^{\infty}$ diffeomorphism of a Riemannian manifold with norm
$\left|\left|\cdot\right|\right|$ preserving a continuous foliation
$\mathcal{W}$ with $C^{\infty}$ leaves, and
$\chi=(\chi_{1},\dots,\chi_{\ell})$ is an $\ell$-tuple of negative numbers
such that $\chi_{1}<\dots<\chi_{\ell}<0$, we say that $f$ has
$(\chi,\varepsilon)$-spectrum on $\mathcal{W}$ if there is a splitting
$T\mathcal{W}=\mathcal{E}^{1}\oplus\dots\mathcal{E}^{\ell}$ into invariant
subbundles such that for every $v\in\mathcal{E}^{i}$,

$e^{\chi_{i}-\varepsilon}\leq\left|\left|df(v)\right|\right|/\left|\left|v\right|\right|\leq
e^{\chi_{i}+\varepsilon}.$

By Remark 4.2 of [26], this is sufficient to obtain the usual narrow band
condition on the Mather spectrum if $\varepsilon$ is sufficiently small. Write
a vector $v\in T_{x}\mathcal{W}$ in coordinates as $v=(v_{1},\dots,v_{\ell})$,
where $v_{i}\in\mathcal{E}^{i}_{x}$. A polynomial
$q:\mathcal{E}_{x}\to\mathcal{E}_{y}$ is said to be
$(s_{1},\dots,s_{\ell})$-homogeneous if

$q(\lambda_{1}v_{1},\dots,\lambda_{\ell}v_{\ell})=\lambda_{1}^{s_{1}}\dots\lambda_{\ell}^{s_{\ell}}q(v_{1},\dots,v_{\ell})).$

Then with a fixed $\ell$-tuple $(\chi_{1},\dots,\chi_{\ell})$, say that a
polynomial $p:\mathcal{E}_{x}\to\mathcal{E}_{y}$ is of subresonance type (with
respect to $\chi$) if

$p(v_{1},\dots,v_{\ell})=(p_{1}(v_{1},\dots,v_{\ell}),\dots,p_{\ell}(v_{1},\dots,v_{\ell})),$

and for $i=1,\dots,\ell$, $p_{i}$ is a a sum of
$(s_{1},\dots,s_{\ell})$-homogeneous polynomials such that $\chi_{i}\leq\sum
s_{j}\chi_{j}$. It is of resonance type (with respect to $\chi$) if each
$p_{i}$ is a sum of $(s_{1},\dots,s_{\ell})$-homogeneous polynomials such that
$\sum s_{j}\chi_{j}=\chi_{i}$.

The following is a consequence of Theorem 4.6 of [26].

######  Theorem A.1.

Let $f:X\to X$ be a $C^{\infty}$ diffeomorphism of a $C^{\infty}$ manifold $X$
preserving a continuous foliation $\mathcal{W}$ with $C^{\infty}$ leaves. If
$\chi=(\chi_{1},\dots,\chi_{\ell})$ is an $\ell$-tuple as above, then there
exists a constant $\varepsilon=\varepsilon(\chi)>0$ with the following
property: if there exists a smooth Riemannian metric for which
$df|_{T\mathcal{W}}$ has $(\chi,\varepsilon)$-spectrum, then there exists a
family of $C^{\infty}$ diffeomorphisms
$\mathcal{H}_{x}:T_{x}\mathcal{W}\to\mathcal{W}(x)$ such that

  1. (NF-1)

for every $x\in X$, $\mathcal{H}_{f(x)}^{-1}\circ f\circ\mathcal{H}_{x}$ is a
subresonance polynomial,

  2. (NF-2)

for every $x\in X$, $d\mathcal{H}_{x}=\mathrm{Id}$,

  3. (NF-3)

if $\mathcal{G}_{x}$ is any other such family, then
$\mathcal{G}_{x}=\mathcal{H}_{x}\circ p_{x}$, for some family of subresonance
polynomials $p_{x}$,

  4. (NF-4)

if $y\in\mathcal{W}(x)$, then $\mathcal{H}_{y}=\mathcal{H}_{x}\circ q_{x,y}$
for a composition of a translation with some subresonance polynomial
$q_{x,y}$,

  5. (NF-5)

if $g:X\to X$ is a $C^{\infty}$ diffeomorphism which commutes with $f$, then
$\mathcal{H}_{g(x)}^{-1}\circ g\circ\mathcal{H}_{x}$ is a subresonance
polynomial.

##  Appendix B Polynomial functions along transverse foliations

The following was shown by Margulis ([40, Lemma 4] or [41, Lemma 17]), who
proved it for rational functions in the measurable setting assuming Lebesgue
almost everywhere properties (cf. also [57, Theorem 3.4.4] and its proof.) We
provide a proof for polynomials in the continuous case which is much simpler
and more straightforward.

######  Lemma B.1.

Let $V$ be a vector space, and $f:V\to\mathbb{R}$ be a continuous function
such that $t\mapsto f(v+tw)$ is a polynomial in $t$ for every $v,w\in V$. Then
$f$ is a polynomial.

###### Proof.

We prove that for any collection of linearly independent elements
$v_{1},\dots,v_{n}$, the map

$(t_{1},\dots,t_{n})\mapsto f(w+\sum t_{i}v_{i})$

is a polynomial in $(t_{1},\dots,t_{n})$ by induction on $n$. Notice that the
base case of $n=1$ is the assumption of the lemma, and the case of $n=\dim(V)$
proves the conclusion.

The degree of the polynomials must be uniformly bounded by some constant $N$
since the map $f$ is continuous. Assume we have shown it for subspaces of
dimension $n-1$, and consider a collection $v_{1},\dots,v_{n}$. Then by
assumption, fixing $t_{2},\dots,t_{n}$ and letting $t_{1}$ vary yields a
continuous family of polynomials in $t_{1}$, so

(B.1)
$f(t_{1}v_{1}+\dots+t_{n}v_{n})=a_{0}(t_{2},\dots,t_{n})+a_{1}(t_{2},\dots,t_{n})t_{1}+\dots+a_{N}(t_{2},\dots,t_{n}){t_{1}}^{N}$

for some collection of continuous functions
$a_{i}:\mathbb{R}^{n-1}\to\mathbb{R}$. By induction, the functions

$p_{k}(t_{2},\dots,t_{n}):=f(kv_{1}+t_{2}v_{2}+\dots+t_{n}v_{n})$

are polynomials. Applying (B.1) to the definition of the polynomials $p_{k}$
yields the following system of equations:

$\begin{pmatrix}1&1&1&\dots&1\\\ 2&4&8&\dots&2^{N}\\\ 3&9&27&\dots&3^{N}\\\
\vdots&&&\ddots&\\\
N+1&(N+1)^{2}&(N+1)^{3}&\dots&(N+1)^{N+1}\end{pmatrix}\begin{pmatrix}a_{0}\\\
a_{2}\\\ a_{3}\\\ \vdots\\\ a_{N}\end{pmatrix}=\begin{pmatrix}p_{1}\\\