# On Partially Hyperbolic diffeomorphisms in dimension three
via a notion of autonomous dynamics

Souheib Allout and Kambiz Moghaddamfar  Souheib Allout
Ruhr-Universitaet Bochum, Germany<EMAIL_ADDRESS>Kambiz Moghaddamfar
UMPA, ENS de Lyon, France, and
Sharif university of technology, Tehran, Iran<EMAIL_ADDRESS>
###### Abstract.

We introduce a notion of autonomous dynamical systems and apply it to prove
rigidity of partially hyperbolic diffeomorphisms on closed compact three-
manifolds under some smoothness hypothesis of their associated framing.

###### Key words and phrases:

Autonomous, partially hyperbolic, three-dimensional Lie groups, stable,
unstable and central spaces, affine diffeomorphisms.

###### Contents

  1. 1 Introduction and statement of main results
    1. 1.1 Definition
    2. 1.2 Examples
      1. 1.2.1 Toral affine automorphisms:
      2. 1.2.2 The flat case:
      3. 1.2.3 The algebraic case:
    3. 1.3 Partially hyperbolic diffeomorphisms
      1. 1.3.1 A rigidity hypothesis:
      2. 1.3.2 Smooth vs topological classification, related results:
    4. 1.4 Autonomous diffeomorphisms in dimension two
  2. 2 General constructions
    1. 2.1 Some properties
    2. 2.2 Lie groups
    3. 2.3 Integration of infinitesimal actions of Lie algebras
  3. 3 Proof of Theorem 1.2 in the non-parabolic case
    1. 3.1 The hyperbolic case:
    2. 3.2 The elliptic case
  4. 4 Proof of Theorem 1.2 in the parabolic case
  5. 5 Proof of Theorem 1.1
    1. 5.1 $C^{\infty}-$Regularity:
    2. 5.2 Algebraic classification
    3. 5.3 The $C^{1}-$regularity
      1. 5.3.1 Non-constant structure coefficients:
      2. 5.3.2 Case of constant structure coefficients:
  6. 6 Counterexamples to Theorem 1.1 in the $C^{0}-$regularity case

##  1\. Introduction and statement of main results

The study of partially hyperbolic diffeomorphisms in dimension three, both in
the flexible and rigid settings, has been widely explored and it is still an
active area of research. Rigidity of these diffeomorphisms, satisfying some
additional requirements, has been investigated in both smooth and topological
frameworks. In this paper, which is motivated by [9] (and also [23]), we
provide smooth ”rigid” classifications of partially hyperbolic diffeomorphisms
satisfying some additional hypothesis. We show then how to extend this to the
$C^{1}-$regularity case. For this, let us start with some definitions:

###  1.1. Definition

Let $M$ be a compact smooth manifold of dimension $n$ endowed with a
parallelization (framing) $\mathcal{F}$ of its tangent bundle, $i.e.$ a system
of vector fields defining a basis of each tangent space and let $\varphi$ be a
$C^{1}$ diffeomorphism of $M$. The derivative cocycle is the map:

$x\in M\to C_{\varphi}(x)\in{\sf{GL}}(n,\mathbb{R})$

where $C_{\varphi}(x)$ is the matrix of the derivative $D_{x}\varphi:T_{x}M\to
T_{\varphi(x)}M$, when these linear spaces are endowed with the bases
$\mathcal{F}(x)$ and $\mathcal{F}(\varphi(x))$ respectively.

We say that $\varphi$ is autonomous with respect to $\mathcal{F}$ if the
cocycle map $C_{\varphi}$ is constant. The same definition applies to a
$C^{1}$ action of a group $G$ to mean that any element of it is autonomous
with respect to $\mathcal{F}$.

Now, we say that a $G-$action on $M$ is autonomous if it is autonomous with
respect to some parallelization $\mathcal{F}$ on $M$. In this case we have a
representation $G\to\sf{GL}(n,\mathbb{R})$ associating to each $g\in G$ the
cocycle matrix of $g$ (acting on $M$). This is essentially inspired from [30]
and [31] where the terminology autonomous was introduced. Our contribution
here is to prove results towards the classification of such autonomous systems
in some cases. As one may expect, the regularity of the framing is relevant
and it is natural to ask when is possible to have constant matrix? $i.e$ when
is it possible for a diffeomorphism to be autonomous?

###  1.2. Examples

Here we give some examples of autonomous actions that will be explained in
more details in the next section:

####  1.2.1. Toral affine automorphisms:

Consider the space $\mathbb{T}^{n}=\mathbb{R}^{n}/\mathbb{Z}^{n}$. The group
${\sf{GL}}(n,\mathbb{Z})\ltimes\mathbb{T}^{n}$ acts naturally on
$\mathbb{T}^{n}$ by affine automorphisms (where ${\sf{GL}}(n,\mathbb{Z})$ is
the group of integer matrices with determinant $\pm 1$). This action is
autonomous with respect to the canonical framing on $\mathbb{T}^{n}$. The
cocycle matrix of each element
$A\in{\sf{GL}}(n,\mathbb{Z})\ltimes\mathbb{T}^{n}$ coincides with matrix of
the linear part of $A$.

####  1.2.2. The flat case:

Let $\mathcal{F}$ be a framing given by linearly independent vector fields
$X_{1},X_{2},\ldots,X_{n}$ of a compact manifold of dimension $n$. We say that
$\mathcal{F}$ is flat if $[X_{i},X_{j}]=0$ for every
$i,j\in\\{1,2,\ldots,n\\}$. In this case the flows
$\phi_{1}^{t},\phi_{2}^{t},\ldots,\phi_{n}^{t}$ (of the vector fields
$X_{1},X_{2},\ldots,X_{n}$ respectively) commute and generate a transitive
locally free action of $\mathbb{R}^{n}$ on $M$. Hence
$M\approx\mathbb{R}^{n}/\Gamma$ where $\Gamma$ is a discrete subgroup with
compact quotient. This implies that $M\approx\mathbb{T}^{n}$ and we have:

######  Proposition 1.1.

Let $M$ be a compact manifold of dimension $n$ with a flat framing
$\mathcal{F}$. Assume that $\varphi$ is a diffeomorphism of $M$ which is
autonomous with respect to $\mathcal{F}$. Then $\varphi$ is conjugate to an
affine automorphism of the $n-$dimensional torus $\mathbb{T}^{n}$.

Proof: the proof is presented in the next section.

$\square$

####  1.2.3. The algebraic case:

Let $G$ be a Lie group and consider $M=G/\Gamma$ where $\Gamma$ is a discrete
subgroup acting by right translations on $G$. Take a basis $\mathcal{B}$ of
the Lie algebra $\sf{Lie}(G)$ and right translate it to get a framing of $G$.
The projection of this framing defines a framing $\mathcal{F}$ of $G/\Gamma$.
Then the left $G-$action on $G/\Gamma$ is autonomous. In fact, for $g\in G$,
its derivative cocycle when acting on $G/\Gamma$ is just the matrix of
$\sf{Ad}(g)$ acting on $\sf{Lie}(G)$ with respect to the basis $\mathcal{B}$.

Let now $A:G\to G$ be an automorphism of $G$ such that $A(\Gamma)=\Gamma$ and
$g\in G$. Then $L_{g}\circ A$ induces a diffeomorphism on $G/\Gamma$ which is
autonomous with respect to $\mathcal{F}$ and having cocycle matrix
$M_{\sf{Ad(g)}}M_{A}$ where $M_{\sf{Ad}(g)}$ and $M_{A}$ are the cocycle
matrices of $\sf{Ad}(g)$ and $A$ respectively. We call a composition like
$L_{g}\circ A$ an affine automorphism. We will give more details about all
this.

###  1.3. Partially hyperbolic diffeomorphisms

Let $M$ be a compact smooth manifold of dimension $n$ and let $f:M\to M$ be a
diffeomorphism of $M$. The $f$ is said to partially hyperbolic if the tangent
bundle $TM$ splits continuously into the direct sum into three $f-$invariant
sub-bundles $E_{s}$, $E_{c}$, and $E_{u}$ such that $TM=E_{s}\oplus
E_{c}\oplus E_{u}$ and there exists a Riemannian metric on $M$ such that for
all unitary vectors $v_{s},v_{c},v_{u}$ in $E_{s},E_{c}$, and $E_{u}$
respectively, we have:

  * •

$\left\|Df(v_{s})\right\|<1,\ \text{and}\ \left\|Df(v_{u})\right\|>1$

  * •

$\left\|Df(v_{s})\right\|<\left\|Df(v_{c})\right\|<\left\|Df(v_{u})\right\|$

####  1.3.1. A rigidity hypothesis:

In the paper [9], which is recently presented to classify partially hyperbolic
diffeomorphisms in dimension three, the authors considered partially
hyperbolic diffeomorphisms $\varphi:M\to M$ satisfying a rigidity hypothesis,
$i.e$, having constant Lyapunov exponents. It turns out that this is
equivalent to the fact that the diffeomorphism $\varphi$ is autonomous with
respect to a framing $\mathcal{F}$ given by three vector fields $X_{s}$,
$X_{c}$, and $X_{u}$ such that $\varphi_{*}(X_{s})=\lambda_{s}X_{s}$,
$\varphi_{*}(X_{c})=\lambda_{c}X_{c}$, and
$\varphi_{*}(X_{u})=\lambda_{u}X_{u}$ where $\left|\lambda_{s}\right|<1$,
$\left|\lambda_{u}\right|>1$, and
$\left|\lambda_{s}\right|<\left|\lambda_{c}\right|<\left|\lambda_{u}\right|$.

In [9] they provide a classification with the assumption that the vector
fields $X_{s},X_{c},$ and $X_{u}$ are smooth, and that the diffeomorphism
$\varphi$ is either transitive or real analytic. We give here a more explicit
classification assuming only $C^{1}$ regularity without the transitivity (or
analyticity) assumption.

Let $M$ be a compact connected three manifold and $\varphi:M\to M$ be a
partially hyperbolic diffeomorphism satisfying the rigidity hypothesis $i.e$
$\varphi$ is autonomous with respect to a framing $\mathcal{F}$ given by three
vector fields $X_{s},X_{c}$, and $X_{u}$ such that the cocycle matrix of
$\varphi$ equals $\begin{pmatrix}\lambda_{s}&0&0\\\ 0&\lambda_{c}&0\\\
0&0&\lambda_{u}\end{pmatrix}$ where $\left|\lambda_{s}\right|<1$,
$\left|\lambda_{u}\right|>1$, and
$\left|\lambda_{s}\right|<\left|\lambda_{c}\right|<\left|\lambda_{u}\right|$.
In this case we have:

######  Theorem 1.1.

Assume that the vector fields $X_{s},X_{c}$, and $X_{u}$ are $C^{1}$. Then, up
to finite power, $\varphi$ is $C^{1}-$conjugate to:

  * •

A suspension of an Anosov linear automorphism $A$ of $\mathbb{T}^{2}$ by a
diffeomorphism that commutes with $A$.

  * •

Or, to an affine automorphism $L_{g}\circ A:G/\Gamma\to G/\Gamma$ where
$A\in\sf{Aut}(G)$ satisfying $A(\Gamma)=\Gamma$, $L_{g}$ is a left
translation, and $\Gamma\subset G$ is a cocompact lattice of the simply
connected three dimensional Lie group $G$. In addition, $G$ is one of the four
Lie groups: $\mathbb{R}^{3}$, the Heisenberg group (denoted $\sf{Heis}_{3}$),
$\sf{Lor}(1,1)={\sf{SO}}(1,1)\ltimes\mathbb{R}^{2}$, or
$\widetilde{\sf{SL}}(2,\mathbb{R})$.

See 5.6 for more details on the description of the first case, and 5.2 for the
second case. We will present first an elementary proof (relying somehow on
Sard’s theorem) assuming $C^{\infty}$ regularity of the framing, showing that
the conjugacy is smooth. The point is to emphasize that the $C^{1}$ regularity
is by no means trivial which requires a different proof. After that, in the
last section, we show how to construct counterexamples to Theorem 1.1 in the
$C^{0}-$regularity case (see 6.2).

####  1.3.2. Smooth vs topological classification, related results:

$\bullet$ One observation is that, in our case where $\lambda_{c}=1$ and
$\lambda_{s}$, $\lambda_{u}$ have the same sign, the diffeomorphism
$\varphi:M\to M$ preserves a Lorentzian metric on $M$ given by the quadratic
form $g=dx_{s}dx_{u}+dx_{c}$ with respect the framing generated by the vector
fields $X_{s},X_{c},X_{u}$. This implies, in particular, that the isometry
group of $(M,g)$ is non-compact. In the paper [11] C. Frances gives a
topological (and geometrical) classification of closed Lorentzian three-
manifolds with non-compact isometry group. He showed that such a manifold must
be homeomorphic to one of the following, assuming at least $C^{9}$ regularity:

  * •

A quotient $\widetilde{{\sf{SL}}}(2,\mathbb{R})/\Gamma$ where
$\Gamma\subset\widetilde{{\sf{SL}}}(2,\mathbb{R})$ is a cocompact lattice.

  * •

The torus $\mathbb{T}^{3}$, or, a suspension of $\mathbb{T}^{2}$ by a
parabolic or a hyperbolic linear automorphism.

Theorem 1.1 classifies (assuming only $C^{1}$ regularity) not only the
manifold $M$, but even the dynamics for being algebraic. In particular,
comparing to C. Frances’ results, the Lorentzian metric $g$ on $M$
(constructed above) is in fact locally homogeneous.
$\bullet$ Another remark is that in our situation the case
$\sf{Heis}_{3}/\Gamma$ corresponds to a suspension of $\mathbb{T}^{2}$ by a
parabolic element, and the case $\sf{Lor}(1,1)/\Gamma$ corresponds to a
suspension of $\mathbb{T}^{2}$ by a hyperbolic element, in the results of C.
Frances.

$\bullet$ In [23] Mion-Mouton proves a rigidity of partially hyperbolic
diffeomorphisms in another direction. He classifies the three-dimensional
partially hyperbolic diffeomorphisms whose stable, unstable, and central
distributions $E_{s},E_{u},$ and $E_{c}$ are smooth, such that $E_{s}\oplus
E_{u}$ is a contact distribution, and whose non-wandering set equals the whole
manifold. He shows that up to a finite quotient or a finite power, they are
smoothly conjugate either to the time-one map of an algebraic contact Anosov
flow, or to an affine partially hyperbolic automorphism of a nil-manifold.

$\bullet$ In [2] and [3] the authors construct new families of partially
hyperbolic diffeomorphisms in dimension three using topological (surgery)
methods. This shows the flexible side of this theory. Whereas, for example in
[5] and [6] the authors provide some rigid topological classifications, for
instance in [6] they classify assuming transitivity and the existence of
$1-$dimensional neutral central direction. In this paper, we give,
particularly, smooth classifications.

$\bullet$ Lastly, let us say that the rigidity hypothesis (constancy of
Lyapunov exponents), which is also assumed in [9], seems at first to be
restrictive, but it is natural enough because this hypothesis is, in
particular, true for algebraic systems. In fact, our framework can also apply
to handle higher dimension situations. In opposite, as shown in [23], a merely
smoothness hypothesis of distributions, is far from leading to prove rigidity.

###  1.4. Autonomous diffeomorphisms in dimension two

Let $S$ be a compact connected surface and $\mathcal{F}$ a smooth framing of
$S$. Then $S$ is necessarily diffeomorphic to the torus $\mathbb{T}^{2}$. Let
$\varphi:S\to S$ be an orientation preserving diffeomorphism which is
autonomous with respect to $\mathcal{F}$ and having cocycle matrix
$M_{\varphi}$. Then:

######  Theorem 1.2.

We have the following classification:

  * •

If the cocycle matrix $M_{\varphi}$ is hyperbolic, then $\varphi$ is smoothly
conjugate to a linear Anosov automorphism of $\mathbb{T}^{2}$.

  * •

If it is elliptic, then $\varphi$ is smoothly conjugate to an isometry of a
flat torus.

  * •

If it is parabolic (with eigenvalues equal to $1$), then either $\varphi$ is
smoothly conjugate to an affine parabolic automorphism or, a finite power of
$\varphi$ is smoothly conjugate to a linear parabolic automorphism of
$\mathbb{T}^{2}$.

The hyperbolic and the elliptic cases are rather straightforward. The
parabolic case is the challenging one. More facts and details are in the proof
section, in particular, we explain the two possibilities (linear, affine) in
the parabolic case.

### Questions

One may derive some questions: in Theorem 1.1, what happens if the vector
fields $X_{s}$, $X_{c}$, and $X_{u}$ are only Lipschitz?
In Theorem 1.2, we can assume that the surface $S$ is not closed. In this
case, if $S$ has finite volume (given by the framing $\mathcal{F}$) and the
diffeomorphism $\varphi$ has hyperbolic cocycle matrix, then the framing
$\mathcal{F}$ is flat defining a translation structure on $S$. Suppose that
$S$ has finite topology (of finite type) then, is it isomorphic, as a surface
with a translation structure, to a translation surface with a finite number of
points removed (including the singularities)?

### Acknowledgments

We would like to thank Professor Ghani Zeghib for proposing this project to us
and for his guidance. We would like also to thank Pierre Mounoud for reading
the paper and for his helpful remarks and corrections.

##  2\. General constructions

Let us start by proving proposition 1.1: Assume $\mathcal{F}$ is given by $n$
vector fields $X_{1},X_{2},\ldots,X_{n}$ (where $n$ is the dimension of $M$).
We have that $[X_{i},X_{j}]=0$ for every $i,j\in\\{1,2,\ldots,n\\}$. This
implies that the flows $\phi_{1}^{t},\phi_{2}^{t},\ldots,\phi_{n}^{t}$
generated by these vector fields define all together an action of
$\mathbb{R}^{n}$ on $M$ which is locally free because the vector fields define
a basis at each point and it is transitive because the orbits in this case are
open and $M$ is connected (note that the flows are globally defined because
$M$ is compact). Hence $M=\mathbb{R}^{n}/\Gamma$ where $\Gamma$ is a discrete
subgroup and the framing $\mathcal{F}$ is induced by projecting a parallel
framing $\mathcal{P}$ of $\mathbb{R}^{n}$. The diffeomorphism $\varphi$ has a
lifting $\widetilde{\varphi}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ satisfying
$\varphi\circ\pi=\pi\circ\widetilde{\varphi}$ where
$\pi:\mathbb{R}^{n}\to\mathbb{R}^{n}/\Gamma$ is the natural projection. Hence
$\widetilde{\varphi}$ is autonomous with respect to $\mathcal{P}$ with cocycle
matrix $M_{\varphi}=M_{\widetilde{\varphi}}$. This implies that
$\widetilde{\varphi}$ is affine.

###  2.1. Some properties

Let $M$ be a compact manifold of dimension $n$ with framing $\mathcal{F}$ and
$\varphi$ be an autonomous diffeomorphism with respect to $\mathcal{F}$ having
cocycle matrix $M_{\varphi}$. Let $T$ be a tensor defined on $M$ and consider
the map $\widetilde{T}:M\to\mathcal{T}$ where

$\mathcal{T}=\\{\text{space of tensors on}\ \mathbb{R}^{n}\ \text{having the
same type as}\ T\\}$

such that $\mathcal{T}$ associates to $z\in M$ the components of $T(z)$ with
respect to the framing $\mathcal{F}(z)$. We know that $\varphi$ acts naturally
on $T$ and $M_{\varphi}$ acts on $\mathcal{T}$, say via a representation
$\rho_{\mathcal{T}}:{\sf{GL}}(n,\mathbb{R})\to{\sf{GL}}(\mathcal{T})$. The map
$\widetilde{T}$ is equivariant, that is:

$\widetilde{T}\circ\varphi=\rho_{\mathcal{T}}(M_{\varphi})\widetilde{T}$

Let $g_{0}$ be the usual inner product in $\mathbb{R}^{n}$ (with respect to
the canonical basis) and consider the tensor $T_{0}$ on $M$ such that the map
$\widetilde{T}_{0}:M\to\sf{Sym}^{+}$ is constant and equals the identity
matrix (the matrix of $g_{0}$ in the canonical basis). This defines a
Riemannian metric on $M$ for which the basis $\mathcal{F}(z)$ is orthonormal
for every $z\in M$. Consider now the associated Riemannian volume form
$\omega_{0}$ on $M$. We have that
$\varphi_{*}\omega_{0}={\sf{det}}(M_{\varphi})\omega_{0}$. On the other side
we have:

$\int_{M}\varphi_{*}\omega_{0}=\int_{M}{\sf{det}}(M_{\varphi})\omega_{0}=\pm\int_{M}\omega_{0}$

This implies that ${\sf{det}}(M_{\varphi})=\pm 1$. In particular, if $\varphi$
preserves the orientation (given by $\mathcal{F}$) then
$M_{\varphi}\in{\sf{SL}}(n,\mathbb{R})$.

Consider a basis $\mathcal{B}$ of $\mathbb{R}^{n}$ and let $A_{\mathcal{B}}$
be the matrix that sends the canonical basis to the basis $\mathcal{B}$.
Define the framing $\mathcal{F}_{\mathcal{B}}$ such that $A_{\mathcal{B}}$ is
the matrix that sends the basis $\mathcal{F}(x)$ of $T_{x}M$ to the basis
$\mathcal{F}_{\mathcal{B}}(x)$ for every $x\in M$. Then $\varphi$ is
autonomous with respect to $\mathcal{F}_{\mathcal{B}}$ with cocycle matrix
$A_{\mathcal{B}}\circ M_{\varphi}\circ A_{\mathcal{B}}^{-1}$.

###  2.2. Lie groups

Let $G$ be a simply connected Lie group of dimension $n$ with Lie algebra
denoted by $\sf{Lie}(G)$, and let $\Gamma\subset G$ be a discrete subgroup.
The quotient space $G/\Gamma$ is a homogeneous manifold of dimension $n$ whose
points are the classes $x\Gamma$, for $x\in G$. The group $G$ acts naturally
on the left on the quotient $G/\Gamma$ as follows:

$G\times G/\Gamma\to G/\Gamma$ $(g,x\Gamma)\to gx\Gamma$

One thinks of $G/\Gamma$ as the quotient space induced by the action on the
right of $\Gamma$ on $G$, that is:

$\Gamma\times G\to G,\ \ (\gamma,g)\to g\gamma^{-1}$

Now, let $(g^{t})$ be a one-parameter subgroup of $G$, and consider the flow
generated by the action on the left of $(g^{t})$ on $G$, $i.e$:

$\mathbb{R}\times G\to G,\ \ (t,x)\to g^{t}x$

Let $x\in G$ and define the curve $\gamma_{x}(t)$ on $G$ as follows:
$\gamma_{x}(t)=g^{t}x$ for every $t\in\mathbb{R}$. Hence
$\gamma_{x}(t)=R_{x}(g^{t})=R_{x}(\gamma_{1}(t))$ where $1\in G$ is the
identity element and $R_{x}$ is the right multiplication by $x$. It is clear
now that:

$\gamma_{x}^{\prime}(0)=D_{1}R_{x}(\gamma_{1}^{\prime}(0))$

Thus $\gamma_{1}^{\prime}(0)\in T_{1}G=\sf{Lie}(G)$ and the flow defined by
the multiplication by $(g^{t})$ is obtained by integrating the right invariant
vector field $X(x)=D_{1}R_{x}(\gamma_{1}^{\prime}(0))$ for every $x\in G$.
Conversely, if $X$ is a right invariant vector field on $G$, and $(g^{t})$ is
the one-parameter group for which $\left.\frac{\partial}{\partial
t}\right|_{t=0}g^{t}=\gamma_{1}^{\prime}(0)=X(1)$ where $\gamma_{1}(t)=g^{t}$,
then the flow generated by $X$ is nothing but the left action of $(g^{t})$ on
$G$.

One may ask the following question: what are all the diffeomorphisms
$\widetilde{\varphi}:G\to G$ preserving the set of all the right invariant
vector fields on $G$? That is, if $X$ is a right invariant vector field, then
the vector field $Y=\widetilde{\varphi}_{*}(X)$ is also right invariant?

Since any right invariant vector field $X$ corresponds to a left action on $G$
by a one-parameter subgroup $(g^{t})$, then the question is in fact equivalent
to finding all the diffeomorphisms $\widetilde{\varphi}$ that send a left
action by a one-parameter subgroup to a left action by another one-parameter
subgroup. More precisely, for every one-parameter subgroup $(g_{1}^{t})$ there
exists another one $(g_{2}^{t})$ such that:

$\widetilde{\varphi}\circ
L_{g_{1}^{t}}\circ\widetilde{\varphi}^{-1}=L_{g_{2}^{t}}$

where $L_{g}$ denotes the left multiplication by $g$. In particular,
$\widetilde{\varphi}$ preserves (by conjugacy) the set of left
multiplications. Before answering the question, let us consider the following
examples of such diffeomorphisms:

  * •

Automorphisms: If $\widetilde{\varphi}:G\to G$ is a Lie group automorphism,
then:

$\widetilde{\varphi}\circ
L_{g}\circ\widetilde{\varphi}^{-1}(x)=\widetilde{\varphi}(g\widetilde{\varphi}^{-1}(x))=\widetilde{\varphi}(g)\widetilde{\varphi}(\widetilde{\varphi}^{-1}(x))=L_{\widetilde{\varphi}(g)}(x)$

  * •

Right translations: Assume $\widetilde{\varphi}=R_{h}$, then:

$\widetilde{\varphi}\circ
L_{g}\circ\widetilde{\varphi}^{-1}(x)=R_{h}(gxh^{-1})=gx=L_{g}(x)$

  * •

Left translations: If $\widetilde{\varphi}=L_{h}$, then:

$\widetilde{\varphi}\circ L_{g}\circ\widetilde{\varphi}^{-1}=L_{h}\circ
L_{g}\circ L_{h}^{-1}=L_{hgh^{-1}}$

######  Proposition 2.1.

Let $\widetilde{\varphi}:G\to G$ be a diffeomorphism preserving (by conjugacy)
the set of all the left translations. Then, $\widetilde{\varphi}=A\circ R_{h}$
where $A\in\sf{Aut}(G)$.

Proof: consider the map $A=\widetilde{\varphi}\circ R_{h^{-1}}$ where
$h^{-1}=\widetilde{\varphi}^{-1}(1)$. Then, it is enough to show that
$A\in\sf{Aut}(G)$. We have that $A$ preserves the set of all the left
multiplications because both $\widetilde{\varphi}$ and $R_{h^{-1}}$ do. Let
$f:G\to G$ defined as follows: $A\circ L_{g}\circ A^{-1}=L_{f(g)}$. Then:

$A\circ L_{g}\circ L_{h}\circ A^{-1}=(A\circ L_{g}\circ A^{-1})\circ(A\circ
L_{h}\circ A^{-1})=L_{f(g)}\circ L_{f(h)}=L_{f(g)f(h)}$

On the other hand we have:

$A\circ L_{g}\circ L_{h}\circ A^{-1}=A\circ L_{gh}\circ A^{-1}=L_{f(gh)}$

Hence, $f(gh)=f(g)f(h)$, $i.e$ $f$ is a homomorphism. On the other side:

$A\circ L_{g}\circ A^{-1}(1)=L_{f(g)}(1)$

This implies that $A(g)=f(g)$. So $A\in\sf{Aut}(G)$.

$\square$

######  Remark 2.1.

We know that left multiplications preserve (by conjugacy) left
multiplications. So, if $\widetilde{\varphi}=L_{g}$ then
$\widetilde{\varphi}=A\circ R_{g}$ where $A\in\sf{Aut}(G)$. In fact, $A$ is
the inner automorphism $L_{g}R_{g^{-1}}$.

Let now $\mathcal{F}$ be a framing of $G$ obtained by right translating a
basis $\mathcal{B}$ of $\sf{Lie}(G)=T_{1}G$. Hence, $\mathcal{F}$ defined by
the $n$ right invariant vector fields determined by the $n$ vectors of
$\mathcal{B}$ where $n=\dim(G)$. Let $A\in\sf{Aut}(G)$. Since $A$ preserves
the set of all the right invariant vector fields, then $A$ is autonomous with
respect to $\mathcal{F}$ with cocycle matrix $M_{A}$ equals the matrix of
$D_{1}A:\sf{Lie}(G)\to\sf{Lie}(G)$ where $\sf{Lie}(G)$ is identified to
$\mathbb{R}^{n}$ using the basis $\mathcal{B}$.

Consider now $g\in G$ and let $\sf{Ad}(g)$ denotes the inner automorphism
$L_{g}R_{g^{-1}}\in\sf{Aut}(G)$. We know that $\sf{Ad}(g)$ is autonomous with
respect to the right invariant framing $\mathcal{F}$. But we have that
$R_{g^{-1}}$ preserves $\mathcal{F}$. This implies that $L_{g}:G\to G$ is
autonomous with respect to $\mathcal{F}$ and with cocycle matrix
$M_{L_{g}}=M_{\sf{Ad}(g)}$.

In fact, if $\widetilde{\varphi}:G\to G$ is a diffeomorphism which is
autonomous with respect to a right invariant framing $\mathcal{F}$, then
$\widetilde{\varphi}$ preserves the set of all the right invariant vector
fields on $G$, which implies that $\widetilde{\varphi}$ preserves (by
conjugacy), the set of all the left multiplications. Thus, using the previous
proposition, we conclude that:

######  Corollary 2.2.

Let $\widetilde{\varphi}:G\to G$ be an autonomous diffeomorphism with respect
to a right invariant framing $\mathcal{F}$. Then $\widetilde{\varphi}=A\circ
R_{h}$ where $A\in\sf{Aut}(G)$.

Let us investigate now the case of autonomous diffeomorphisms of the quotient
space $G/\Gamma$. But before this, it is useful to remind some general facts.
Consider a diffeomorphism $\varphi:G/\Gamma\to G/\Gamma$ and let
$\varphi(\Gamma)=g\Gamma$ (that is, the point $g\Gamma\in G/\Gamma$ is the
image of the point $\Gamma$ by the diffeomorphism $\varphi$). Define the new
diffeomorphism $\psi=g^{-1}\varphi:G/\Gamma\to G/\Gamma$ (recall that $G$ acts
on the left on $G/\Gamma$). Hence, $\psi$ fixes the point $\Gamma\in
G/\Gamma$.

Consider now the action of $\psi$ on the space of all the curves
$\gamma:[0,1]\to G/\Gamma$ such that $\gamma(0)=\Gamma\in G/\Gamma$. Let
$\pi:G\to G/\Gamma$ be the natural projection $i.e$ the universal cover of
$G/\Gamma$. Then for any curve $\gamma:[0,1]\to G/\Gamma$ such that
$\gamma(0)=\Gamma$ there is a unique lifting $\widetilde{\gamma}:[0,1]\to G$
of $\gamma$ and satisfies $\widetilde{\gamma}(0)=1\in G$. Thus, in this way,
$G$ is identified with the space of all the homotopy classes of the curves on
$G/\Gamma$ starting from the point $\Gamma$. Let $\gamma$ be such a curve,
then it corresponds to a unique point in $G$, that is, the point
$\widetilde{\gamma}(1)$ where $\widetilde{\gamma}$ is the unique lifting of
$\gamma$ satisfying $\widetilde{\gamma}(0)=1$. Define the map
$\widetilde{\psi}:G\to G$ as follows:

$\widetilde{\psi}(x)=\widetilde{\psi\circ\gamma}(1)$

where $\widetilde{\psi\circ\gamma}$ is the lifting of $\psi\circ\gamma$. The
map $\widetilde{\psi}$ preserves $\Gamma$, and it is clear that
$\widetilde{\psi}$ is bijective and covers the diffeomorphism $\psi$, $i.e$
$\pi\circ\widetilde{\psi}=\psi\circ\pi$. Hence, $\widetilde{\psi}$ is a
diffeomorphism, and we obtain that $\varphi=g\psi$ is covered by the
diffeomorphism $\widetilde{\varphi}=g\widetilde{\psi}:G\to G$.

Coming back to our situation, let us assume that $\mathcal{F}$ is a framing of
$G/\Gamma$ induced by projecting a right invariant framing
$\widetilde{\mathcal{F}}$ of $G$ using the natural map $\pi:G\to G/\Gamma$
(recall that this is well defined since the quotient space $G/\Gamma$ is
obtained by the action of $\Gamma$ on the right). Let $\varphi:G/\Gamma\to
G/\Gamma$ be an autonomous diffeomorphism with respect to the framing
$\mathcal{F}$ and let $\widetilde{\varphi}=g\widetilde{\psi}:G\to G$ be the
diffeomorphism that covers $\varphi$ as constructed above. Since
$\pi\circ\widetilde{\varphi}=\varphi\circ\pi$ then $\widetilde{\varphi}$ is
autonomous with respect to $\widetilde{\mathcal{F}}$ with cocycle matrix
$M_{\widetilde{\varphi}}=M_{\varphi}$ (because the map $\pi$ transfers,
locally, what happens in $G$ to what happens in $G/\Gamma$). This implies that
$g^{-1}\widetilde{\varphi}=\widetilde{\psi}$ is autonomous with respect to
$\widetilde{\mathcal{F}}$ such that $\widetilde{\psi}(1)=1$ and
$\widetilde{\psi}(\Gamma)=\Gamma$. So, using the previous corollary, we get
$\widetilde{\psi}\in\sf{Aut}(G)$. As a conclusion we have:

######  Corollary 2.3.

Let $\mathcal{F}$ be a framing of $G/\Gamma$ induced by projecting a right
invariant framing $\widetilde{\mathcal{F}}$ of $G$. Assume
$\varphi:G/\Gamma\to G/\Gamma$ is an autonomous diffeomorphism with respect to
$\mathcal{F}$. Then there exists $A\in\sf{Aut}(G)$ satisfying
$A(\Gamma)=\Gamma$, and $g\in G$ such that $\varphi$ is covered by the affine
map $L_{g}\circ A$.

######  Remark 2.4.

Recall that the cocycle matrix $M_{L_{g}}=M_{\sf{Ad}(g)}$ and
$M_{A}=D_{1}A:\sf{Lie}(G)\approx\mathbb{R}^{n}\to\sf{Lie}(G)$ with respect to
$\mathcal{F}$. Hence, we have that $M_{\varphi}=M_{\sf{Ad}(g)}M_{A}$.

###  2.3. Integration of infinitesimal actions of Lie algebras

Let $G$ be a connected Lie group with a Lie algebra $\sf{Lie}(G)$ seen as the
algebra of right invariant vector fields. Assume that $\psi:G\times M\to M$ is
a smooth action of $G$ on $M$. For any right invariant vector field
$X\in\sf{Lie}(G)$, there exists a unique one-parameter subgroup
$(g^{t})=\exp(tX)$ of $G$ whose tangent vector at $t=0$ is the vector $X(1)$.
So, this defines an action of $\mathbb{R}$ (a flow) on $M$ as follows:

$\mathbb{R}\times M\to M$ $(t,x)\to\psi(\exp(tX(1)),x)$

This flow is obtained by integrating the vector field:

$X_{M}(x)=\left.\frac{\partial}{\partial t}\right|_{t=0}\psi(\exp(tX(1)),x)$

That is, $X_{M}(x)\in T_{x}M$ is the tangent vector of the curve
$\psi(\exp(tX(1)),x)$ at time $t=0$. This defines a map

$d\psi:\sf{Lie}(G)\to\Gamma^{\infty}(M),\ \ \ X\to X_{M}$

where $\Gamma^{\infty}(M)$ is the space of all the smooth vector fields of
$M$. The map $d\psi$ is a homomorphism of Lie algebras and the action $\psi$
is uniquely determined by $d\psi$, that is, if $\psi^{{}^{\prime}}$ is another
action of $G$ on $M$ such that $d\psi=d\psi^{{}^{\prime}}$, then
$\psi=\psi^{{}^{\prime}}$. We call $d\psi$ the infinitesimal action.

It is natural to ask whether a homomorphism of Lie algebras
$\sf{Lie}(G)\to\Gamma^{\infty}(M)$ corresponds to an infinitesimal action of
the Lie group $G$?

######  Theorem 2.5 (Palais 1957).

Let $G$ be a simply connected Lie group with Lie algebra $\sf{Lie}(G)$ of
right invariant vector fields. Let $M$ be a manifold and assume that
$\phi:\sf{Lie}(G)\to\Gamma^{\infty}(M)$ is a Lie algebra homomorphism. Assume
furthermore that $\phi(X)\in\Gamma^{\infty}(M)$ is complete for every
$X\in\sf{Lie}(G)$. Then $\phi$ is the infinitesimal action of a unique action
of $G$ on $M$.

Assume $M$ is a compact connected manifold of dimension $n$ and $\mathcal{F}$
is a smooth framing of $M$ generated by the smooth vector fields
$X_{1},X_{2},\ldots,X_{n}$. Assume, in addition, that:

$[X_{i},X_{j}]=\sum_{k=1}^{n}a_{k}^{ij}X_{k}$

for every $i,j\in\\{1,2,\ldots,n\\}$ where
$a_{1}^{ij},a_{2}^{ij},\ldots,a_{n}^{ij}$ are constant real numbers. Let
$\mathcal{G}\subset\Gamma^{\infty}(M)$ be the vector sub-space generated by
$X_{1},X_{2},\ldots,X_{n}$. Then $\mathcal{G}$ is a $n-$dimensional real Lie
sub-algebra. Let $G$ be the associated $n-$dimensional real simply connected
Lie group. Since the inclusion
$i:\mathcal{G}=\sf{Lie}(G)\to\Gamma^{\infty}(M)$ is a Lie algebra
homomorphism, then, according to the Palais’s theorem, this corresponds to the
infinitesimal action of a unique action of $G$ on $M$.

The condition that $X_{1},X_{2},\ldots,X_{n}$ define a basis of $T_{x}M$ at
any $x\in M$ implies that the action of $G$ on $M$ is locally free. Thus each
orbit of this action is open. This implies that $G$ acts transitively on $M$
which gives that $M=G/\Gamma$ where $\Gamma=\sf{Stab}(x)$ is the stabilizer of
a chosen point $x\in M$. In addition, the vector fields
$X_{1},X_{2},\ldots,X_{n}$ on $M=G/\Gamma$ correspond to the projection of
right invariant vector fields of $G$. Combining with the previous results we
obtain:

######  Corollary 2.6.

Assume $M$ is a compact connected manifold of dimension $n$ and $\mathcal{F}$
is a smooth framing as indicated above. Let $\varphi:M\to M$ be an autonomous
diffeomorphism with respect to $\mathcal{F}$. Then $M=G/\Gamma$ and $\varphi$
is covered by an affine map $L_{g}\circ A:G\to G$ with $A\in\sf{Aut}(G)$
satisfying $A(\Gamma)=\Gamma$ (where $G$, $\Gamma$, and the identification
$M=G/\Gamma$ are as discussed above).

##  3\. Proof of Theorem 1.2 in the non-parabolic case

Assume that $M$ is a two dimensional compact surface and a framing
$\mathcal{F}$ given by two vector fields $X$ and $Y$ (this implies that $M$ is
diffeomorphic to the torus). Let $\varphi$ be an autonomous diffeomorphism
with matrix:

$M_{\varphi}=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$

We have seen that ${\sf{det}}(M_{\varphi})=\pm 1$. Let $[X,Y]$ be the Lie
bracket of $X$ and $Y$, and $\varphi_{*}([X,Y])$ be its push forward by
$\varphi$. Then we have:

$\varphi_{*}([X,Y])=[aX+cY,bX+dY]=(ad-bc)[X,Y]=\pm[X,Y]$

In particular, if $\varphi$ preserves the orientation then $[X,Y]$ is
$\varphi-$invariant, that is, $\varphi_{*}([X,Y])=[X,Y]$.

###  3.1. The hyperbolic case:

######  Proposition 3.1.

Consider a framing $\mathcal{F}$ given by two vector fields $X$ and $Y$ and
assume that $M_{\varphi}$ is hyperbolic. Then $[X,Y]=0$.

Proof: We have that $Z=[X,Y]$ is $\varphi-$invariant (as observed above). Let
$x\in\mathbb{T}^{2}$ and suppose that $Z(x)\neq 0$. So, since $M_{\varphi}$ is
hyperbolic, either $D\varphi^{n}(Z(x))$ or $D\varphi^{-n}(Z(x))$ goes to
infinity. But this contradicts the fact that $Z$ is $\varphi-$invariant and
bounded.

$\square$

######  Corollary 3.1.

The diffeomorphism $\varphi$ is smoothly conjugate to an Anosov linear
automorphism of $\mathbb{T}^{2}$.

Proof: because in this case we have a transitive locally free action of
$\mathbb{R}^{2}$ on the torus $\mathbb{T}^{2}$ generated by the flows of the
vector fields $X$ and $Y$. Hence $\mathbb{T}^{2}=\mathbb{R}^{2}/\Gamma$ and
$\mathcal{F}$ is obtained by projecting a parallel framing
$\widetilde{\mathcal{F}}$ of $\mathbb{R}^{2}$. Furthermore, there is a
diffeomorphism $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ covering
$\varphi$ and has a constant derivative (with respect to the framing
$\widetilde{\mathcal{F}}$) and equals $M_{\varphi}$. Hence
$\widetilde{\varphi}$ is affine. But since $\widetilde{\varphi}$ is
hyperbolic, then it must have at least one fixed point. This implies that it
is conjugate by a translation to its linear part.

$\square$

###  3.2. The elliptic case

Suppose that the cocycle matrix $M_{\varphi}$ of $\varphi$ with respect to
$\mathcal{F}$ is elliptic. Then $M_{\varphi}$ belongs to the isometry group of
a Euclidean inner product on $\mathbb{R}^{2}$, denoted by $q_{0}$. Since each
tangent space $T_{x}\mathbb{T}^{2}$ is identified to $\mathbb{R}^{2}$ via the
basis $\mathcal{F}(x)$, then, we can define the smooth Riemannian metric $g$
on $\mathbb{T}^{2}$ by inducing $q_{0}$ on each tangent plane via the
identification. In this case, $\varphi$ is an isometry of the smooth
Riemannian metric $g$.

######  Proposition 3.2.

The metric $g$ is conformally equivalent to a flat metric.

Proof: this is exactly the content of the uniformization theorem (see for
instance [20] for more details).

$\square$

######  Corollary 3.2.

The map $\varphi$ is an automorphism of a complex torus, hence, it is affine.

Proof: Let $f:\mathbb{T}^{2}\to\mathbb{R}_{>0}$ be smooth map such that the
metric $fg$ is flat. We have that $\varphi$ is conformal with respect to the
metric $fg$, because it is an isometry of $g$. The flat metric $fg$ defines a
unique complex structure of $\mathbb{T}^{2}$ for which $\varphi$ is
biholomorphic. Automorphisms of complex tori are known to be affine.

$\square$

######  Remark 3.3.

In this case we have $\mathbb{T}^{2}=\mathbb{C}/\Gamma$ and $\varphi$ is
covered by an automorphism $\widetilde{\varphi}:\mathbb{C}\to\mathbb{C}$ of
the form $z\to az+b$ such that $a\Gamma=\Gamma$. This implies that
$a\in\mathbb{S}^{1}$ (and rational). Hence $\varphi$ is, in fact, an isometry
of the flat metric.

##  4\. Proof of Theorem 1.2 in the parabolic case

Up to changing the framing, we can suppose that $\mathcal{F}$ is a smooth
framing of $\mathbb{T}^{2}$ for which the diffeomorphism $\varphi$ is
autonomous with cocycle matrix:

$M_{\varphi}=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$

The framing $\mathcal{F}$ is given by two independent vector fields $X$ and
$Y$. Thus both $X$ and $[X,Y]$ are $\varphi-$invariant.

######  Proposition 4.1.

We have $[X,Y]=\beta X$ such that $\beta$ is a smooth $\varphi-$invariant
function defined on $\mathbb{T}^{2}$.

Proof: if a vector field $Z$ is $\varphi-$invariant and for some
$a\in\mathbb{T}^{2}$ we have $Z(a)\neq\lambda X(a)$ then the norm of the
iterates of $Z(a)$ under $\varphi$ (in the associated Riemannian metric) goes
to infinity. This is impossible because $Z$ is $\varphi-$invariant. The fact
that $\beta$ is $\varphi-$invariant comes from the fact that $[X,Y]$ is
$\varphi-$invariant.

$\square$

######  Lemma 4.1.

Suppose that $\beta$ is constant non null. Then there is a locally free
transitive action of the affine Lie group $\sf{Aff}(\mathbb{R})$ on
$\mathbb{T}^{2}$.

Proof: we can suppose that $\beta=1$ (up to changing the framing), that is,
$[X,Y]=X$. This defines a finite Lie subalgebra of the algebra of all the
smooth vector fields generated by $X$ and $Y$ isomorphic to the Lie algebra of
$\sf{Aff}(\mathbb{R})$. Hence, applying the Palais’ theorem, we conclude that
this defines an action of $\sf{Aff}(\mathbb{R})$ on $\mathbb{T}^{2}$ which is
transitive and locally free (because $X$ and $Y$ define a framing).

$\square$

######  Corollary 4.2.

If the function $\beta$ is constant then it vanishes and $\varphi$ is smoothly
conjugate to an affine parabolic automorphism of the torus.

Proof: the fact that ”if $\beta$ is constant then it must vanish” comes from
the fact that there is no locally free transitive action of the group
$\sf{Aff}(\mathbb{R})$ on $\mathbb{T}^{2}$ because $\sf{Aff}(\mathbb{R})$
doesn’t contain any discrete subgroup $\Gamma$ for which
$\sf{Aff}(\mathbb{R})/\Gamma$ is compact (since it is not unimodular). Thus we
are in the flat case, which we already investigated.

$\square$

######  Lemma 4.3.

We have that $d\beta(X)=0$, that is, the line field generated by $X$ is
contained in $\ker(d\beta)$.

Proof: Let $z\in\mathbb{T}^{2}$ be a recurrent point for $\varphi$. Suppose
that $X(z)\notin\ker(d_{z}\beta)$ that is $X(z)$ is transversal to the line
$\ker(d_{z}\beta)$. In this case, we have that the line $\ker(d_{z}\beta)$
converges under the iterates of $\varphi$ to be tangent to $X$ because
$\varphi$ is autonomous with respect to $X$ and $Y$ with cocycle matrix
$M_{\varphi}=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$. But on the other side
$\varphi$ preserves $\ker(d\beta)$ because $\beta$ is $\varphi-$invariant. So,
since $z$ is recurrent, we conclude that $\ker(d_{z}\beta)$ is tangent to $X$.
Which contradicts our assumption. Hence $X(z)\in\ker(d_{z}\beta)$. But almost
every point is recurrent (because $\varphi$ preserves a finite Riemannian
volume), this implies $d\beta(X)=0$.

$\square$

Suppose now that $\beta$ is not constant, then for a generic point $a$ in the
image of $\beta$, the level $\beta^{-1}(a)$ (which is $\varphi-$ and
$X-$invariant) is a compact one dimensional manifold ($i.e$ union of circles)
tangent to the vector field $X$.

The equality $[X,Y]=\beta X$ implies that the flow generated by $Y$ sends
$X-$orbits to $X-$orbits (this is true only geometrically, one forgets about
parametrizations). More accurately, the flow $\phi^{t}_{Y}$ preserves the
$X-$foliation.

We want to show that for a generic $\beta-$level, its saturation by the
$Y-$flow covers the whole $\mathbb{T}^{2}$.

######  Proposition 4.2.

Let $F$ be a codimension one foliation on a compact manifold $N$ with at least
one compact leaf. assume there is a flow $\phi^{t}$ transversal to $F$ and
sends leaf to leaf. Then $F$ corresponds to a fibration $N\to\mathbb{S}^{1}$.

Proof: Let $L_{0}$ be a compact leaf. Hence $L_{0}$ is a compact embedded
submanifold of $M$. Define the map $i:L_{0}\times\mathbb{R}\to M$ by
$i(x,t)=\phi^{t}(x)$. Since $\phi^{t}$ is transversal to $F$ and sends leaf to
leaf then the map $i$ is an immersion and for every $t\in\mathbb{R}$ there is
$\epsilon>0$ such that the restriction of $i$ on
$L_{0}\times]t-\epsilon,t+\epsilon[$ is an embedding (because $L_{0}$ is
compact). The map $i$ cannot be injective, in fact if $L_{x}$ is any leaf,
define

$O_{x}=\\{\phi^{t}(y)\mid t\in\mathbb{R},y\in L_{x}\\}$

then $O_{x}$ is open (because $\phi^{t}$ is transversal to $F$) and either
$O_{x_{1}}\cap O_{x_{2}}=\emptyset$ or $O_{x_{1}}=O_{x_{2}}$ (because
$\phi^{t}$ sends leaf to leaf). So we have a partition of $M$ into open sets.
But $M$ is connected, and this implies that we have only one open set. Hence
the map $i$ is surjective. If furthermore it is injective then it is a
diffeomorphism which is impossible. Let $t_{0}\in\mathbb{R}$ be defined as
follows:

$t_{0}=\inf\\{t>0\mid\phi^{t}(L_{0})=L_{0}\\}$

So $t_{0}>0$ because $i$ is a local diffeomorphism and $t_{0}\neq\infty$
otherwise $i$ is injective. Thus $M\approx L_{0}\times[0,t_{0}]$ with
$(x,0)\sim(\phi^{t_{0}}(x),t_{0})$.

$\square$

######  Remark 4.4.

Note that the assumption that the foliation has at least one compact leaf is
crucial. It is needed to assure that the image of this compact leaf $L_{0}$ by
the transversal flow, in a small neighbourhood of the time $0$, defines a
foliation by disjoint leaves in a neighbourhood of $L_{0}$. A counterexample
without the compactness assumption is the foliation of the torus by dense
parallel lines with a transversal translation flow.

######  Corollary 4.5.

In this case $\varphi$ preserves a fibration $F$ by circles tangent to the
vector field $X$.

Proof: We have that $\varphi$ preserves the foliation $F$ defined by $X$.
Since $\beta$ is not constant, a generic level $\beta^{-1}(a)$ is a finite
union of circles tangent to $X$. Hence, applying the previous proposition we
conclude that $F$ is a fibration by circles all tangent to $X$ and preserved
by $\varphi$.

$\square$

######  Lemma 4.6.

Let $f:\mathbb{S}^{1}\to\mathbb{S}^{1}$ be an orientation preserving
diffeomorphism that preserves a framing of $\mathbb{S}^{1}$, then $f$ is
smoothly conjugate to the rotation $R_{\alpha}$ (where $\alpha$ is the
rotation number of $f$).

Proof: this happens because $f$ is an isometry of the associated metric. In
fact let $x\in\mathbb{S}^{1}$ and let $\gamma:[0,1]\to\mathbb{S}^{1}$ be the
simple arc joining $x$ and $f(x)$ and moving in the direction of the framing.
Let $l(\gamma)$ be the length of $\gamma$ in the associated metric and $d$ be
the total length of $\mathbb{S}^{1}$. Put $\alpha=2\pi(l(\gamma)/d)$. Then
clearly the isometry $f$ is conjugate to $R_{\alpha}$.

$\square$

Let $X$ be a smooth non-vanishing vector field on
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ such that $X(z_{1},z_{2})$ is tangent to
the circle $\\{z_{1}\\}\times\mathbb{S}^{1}$ for every
$(z_{1},z_{2})\in\mathbb{S}^{1}\times\mathbb{S}^{1}$. Denote by
$(\mathbb{S}^{1}\times\mathbb{S}^{1},X)$ to mean the space
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ with the vector field $X$ on it, and let

$\varphi:(\mathbb{S}^{1}\times\mathbb{S}^{1},X)\to(\mathbb{S}^{1}\times\mathbb{S}^{1},X)$

be a diffeomorphism preserving $X$ and preserves each factor
$\\{z\\}\times\mathbb{S}^{1}=S_{z}$. Hence, for each $z\in\mathbb{S}^{1}$, the
restriction of $\varphi$ on $S_{z}$ is smoothly conjugate to the associated
rotation, denoted $R_{\alpha(z)}$. In this case we have the following:

######  Lemma 4.7.

There exists a global diffeomorphism
$\phi:(\mathbb{S}^{1}\times\mathbb{S}^{1},X)\to\mathbb{S}^{1}\times\mathbb{S}^{1}$
preserving each factor $\\{z\\}\times\mathbb{S}^{1}=S_{z}$ such that the
restriction of $\phi\circ\varphi\circ\phi^{-1}$ on each $S_{z}$ equals
$R_{\alpha(z)}$.

Proof: Let $(z_{1},z_{2})\in\mathbb{S}^{1}\times\mathbb{S}^{1}$ and let
$l(z_{1},z_{2})$ be the length of the unique curve starting from $(z_{1},1)$
and ending at $(z_{1},z_{2})$ and moving in the direction of the restriction
of the vector field $X$ on $S_{z}$ (where the length is computed with respect
to the metric given by $X$ on $S_{z}$). Let $d(z_{1})$ be the total length of
$S_{z_{1}}$ with respect to this metric. Define
$f_{z_{1}}:\\{z_{1}\\}\times\mathbb{S}^{1}=\mathbb{S}^{1}\to\mathbb{S}^{1}$
by:

$f_{z_{1}}(z_{2})=e^{i(l(z_{2})/d(z_{1}))2\pi}$

Put now $\phi(z_{1},z_{2})=(z_{1},f_{z_{1}}(z_{2}))$. From the construction we
observe that $f_{z_{1}}$ sends the restriction of $X$ on
$\\{z_{1}\\}\times\mathbb{S}^{1}$ to the usual unitary framing of
$\mathbb{S}^{1}$. Hence $\phi$ sends the vector field $X$ to the constant
unitary vector field on $\mathbb{S}^{1}\times\mathbb{S}^{1}$ tangent to the
fibers $\\{z\\}\times\mathbb{S}^{1}$ and in the positive orientation on each
fiber (the fibers are identified with $\mathbb{S}^{1}$). This implies that
$\phi$ satisfies the desired condition.

$\square$

Now let
$\varphi:\mathbb{S}^{1}\times\mathbb{S}^{1}\to\mathbb{S}^{1}\times\mathbb{S}^{1}$
be a diffeomorphism, preserving each factor $\\{z\\}\times\mathbb{S}^{1}$, and
$X$ be a non-vanishing vector field on $\mathbb{S}^{1}\times\mathbb{S}^{1}$
all as defined above. Let $\alpha:\mathbb{S}^{1}\to\mathbb{S}^{1}$ be the map
satisfying that the restriction of $\varphi$ on each
$\\{z\\}\times\mathbb{S}^{1}$ is smoothly conjugate to the associated rotation
$R_{\alpha(z)}$. From the previous lemma, we observe that $\alpha$ is smooth.
We have:

######  Lemma 4.8.

If the map $\alpha:\mathbb{S}^{1}\to\mathbb{S}^{1}$ is a local diffeomorphism
(hence a covering) then $\varphi$ is smoothly conjugate to the linear
automorphism on $\mathbb{R}^{2}/\mathbb{Z}^{2}$ generated by the matrix
$\begin{pmatrix}1&k\\\ 0&1\end{pmatrix}$ where $k$ is the degree of this
covering.

Proof: Let $z_{1},z_{2},\ldots,z_{k}$ be the points of the fiber
$\alpha^{-1}(1)$ ordered in the positive direction. Let
$\gamma:[0,1]\to\mathbb{S}^{1}$ be the curve defined by
$\gamma(t)=z_{1}e^{(2\pi t)i}$. Hence $\gamma$ is a simple closed curve such
that $\gamma(0)=\gamma(1)=z_{1}$. Put $\pi:\mathbb{R}\to\mathbb{S}^{1}$ to be
the universal covering given by $\pi(x)=e^{(2\pi x)i}$ and let
$\widetilde{\alpha\circ\gamma}:[0,1]\to\mathbb{R}$ be the lifting of the curve
$\alpha\circ\gamma$ such that $\widetilde{\alpha\circ\gamma}(0)=0$. Thus
$\widetilde{\alpha\circ\gamma}(1)=k$. In addition we have
$\widetilde{\alpha\circ\gamma}(t_{i})=i-1$ for each $t_{i}$ satisfying
$\gamma(t_{i})=z_{i}$, $1\leq i\leq k$. Define the map
$f:\mathbb{S}^{1}\to\mathbb{R}/\mathbb{Z}$ as follows:

$f(z)=\frac{\widetilde{\alpha\circ\gamma}(\gamma^{-1}(z))}{k}\mod(1)$

note that this map is well-defined because $\gamma^{-1}(z_{1})=\\{0,1\\}$ has
two choices but they are the same $\mod(1)$. Consider the map (as defined in
the proof of the previous lemma)
$l:\mathbb{S}^{1}\times\mathbb{S}^{1}\to\mathbb{R}_{\geq 0}$ such that
$l(z_{1},z_{2})$ is the length of the unique curve starting from $(z_{1},1)$
and ending at $(z_{1},z_{2})$ and moving in the direction of the restriction
of the vector field $X$ on $S_{z}$. Similarly, let
$d:\mathbb{S}^{1}\to\mathbb{R}_{>0}$ such that $d(z)$ is the total length of
$\\{z\\}\times\mathbb{S}^{1}$. The maps $l$ and $d$ are smooth. Define finally
the map
$\phi:\mathbb{S}^{1}\times\mathbb{S}^{1}\to\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$
as follows:

$\phi(z_{1},z_{2})=(f(z_{1}),\frac{l(z_{1},z_{2})}{d(z_{1})})\mod(1)$

Hence $\phi\circ\varphi\circ\phi^{-1}$ is equal to $\begin{pmatrix}1&k\\\
0&1\end{pmatrix}$ because one sees that the restriction of the diffeomorphism
$\phi\circ\varphi\circ\phi^{-1}$ on each segment $\\{t\\}\times[0,1]$ for
$t\in[0,1]$ is exactly the vertical translation by the value $kt\mod(1)$.

$\square$

######  Remark 4.9.

If the map $\alpha$ is supposed to be only a covering map (without being a
local diffeomorphism) then $\varphi$ is still conjugate to a linear parabolic
automorphism of $\mathbb{R}^{2}/\mathbb{Z}^{2}$. But, in this case, the
conjugacy is just $C^{0}$ $i.e$ the conjugating map is just a homeomorphism.

Let us now come back to our situation. Suppose
$\varphi:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be an autonomous diffeomorphism with
respect to a smooth framing $\mathcal{F}$, given by the vector fields $X$ and
$Y$, and having a parabolic cocycle matrix $M_{\varphi}=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}$. We have seen that $\varphi$ preserves a fibration $F$ by
circles tangent to the vector field $X$. The fibration $F$ is equivalent (by a
diffeomorphism) to the trivial fibration of
$\mathbb{S}^{1}\times\mathbb{S}^{1}$ given by the vertical circles
$\\{z\\}\times\mathbb{S}^{1}$, $z\in\mathbb{S}^{1}$. Hence the quotient
$\mathbb{T}^{2}/F$ is diffeomorphic to $\mathbb{S}^{1}$ and $\varphi$ induces
a diffeomorphism $\widehat{\varphi}:\mathbb{T}^{2}/F\to\mathbb{T}^{2}/F$. The
flow $\phi_{Y}^{t}$ generated by $Y$ preserves $F$, $i.e$ sends circle to
circle. This implies that $Y$ projects to a smooth vector field $\widehat{Y}$
on $\mathbb{T}^{2}/F$. Furthermore, $\widehat{\varphi}$ preserves
$\widehat{Y}$ because we have:

$\widehat{\varphi}_{*}(\widehat{Y})=\pi_{*}(\varphi_{*}(Y))=\pi_{*}(X+Y)$

Where $\pi:\mathbb{T}^{2}\to\mathbb{T}^{2}/F$ is the quotient map. So, since
$X$ is tangent to $F$, we have $\pi_{*}(X)=0$ which implies
$\pi_{*}(X+Y)=\pi_{*}(Y)=\widehat{Y}$. Hence
$\widehat{\varphi}_{*}(\widehat{Y})=\widehat{Y}$ and, consequently,
$\widehat{\varphi}$ is smoothly conjugate to a rotation.

Since the function $\beta$ is not constant (recall that $[X,Y]=\beta X$ and
$\beta$ is $\varphi-$invariant) then $\varphi$ preserves a generic fiber
$\beta^{-1}(a)$ which is a finite union of circles tangent to $F$. This
implies that $\widehat{\varphi}$ is smoothly conjugate to a periodic rotation
of period $n\geq 1$. Thus the diffeomorphism
$\varphi^{n}:\mathbb{T}^{2}\to\mathbb{T}^{2}$ preserves each circle of the
fibration $F$ and has cocycle matrix $M_{\varphi^{n}}=\begin{pmatrix}1&n\\\
0&1\end{pmatrix}$ with respect to the framing $\mathcal{F}$. In this case we
have:

######  Lemma 4.10.

Let $\varphi^{n}$ be the diffeomorphism as above and let
$R:\mathbb{T}^{2}/F\to\mathbb{S}^{1}$ be the map associating the rotation
number of the restriction of $\varphi^{n}$ on each circle of the fibration
$F$. Then $R$ is a local diffeomorphism.

Proof: let $\alpha:\mathbb{T}^{2}\to\mathbb{R}$ be a positive non vanishing
smooth function which is $\varphi-$invariant, that is, $\alpha$ is constant
along each leaf of the fibration $F$. Define $X^{\prime}=\alpha X$ and
$Y^{\prime}=\alpha Y$. Then it is clear that $\varphi^{n}$ is still autonomous
with the framing given by $X^{\prime}$ and $Y^{\prime}$ and having the same
parabolic cocycle matrix because $\varphi^{n}$ preserves each fiber of the
fibration $F$. In addition, we have that:

$[X^{\prime},Y^{\prime}]=[\alpha X,\alpha Y]=\alpha^{2}\beta X-\alpha
Y(\alpha)X=\alpha(\alpha\beta-Y(\alpha))X=(\alpha\beta-Y(\alpha))X^{\prime}$

where $[X,Y]=\beta X$ and we have that $(\alpha\beta-Y(\alpha))$ is constant
along each fiber just as we discussed before in Lemma 4.3. So
$[X^{\prime},Y^{\prime}]=0$ if and only if $\alpha\beta-Y(\alpha)=0$. Let now
$p\in\mathbb{T}^{2}$ and $\gamma:]-\epsilon,\epsilon[\to\mathbb{T}^{2}$ be an
embedded smooth small curve such that $\gamma(0)=p$ and tangent to $Y$. Then
$\beta$ restricted to $\gamma$ is a well defined function and the differential
equation $\alpha\beta-Y(\alpha)=0$ on $\gamma$ has a local solution around
$p$, that is, there exists a unique
$\alpha_{x_{0}}:\gamma(]-\epsilon^{\prime},\epsilon^{\prime}[)\to\mathbb{R}$
with $\epsilon^{\prime}<\epsilon$ satisfying $\alpha_{x_{0}}(p)=x_{0}$ for
$x_{0}\in\mathbb{R}_{>0}$. Then $\alpha_{x_{0}}$ is positive around $p$.
Consequently, we can construct a global function
$\alpha:\mathbb{T}^{2}\to\mathbb{R}_{>0}$ which is constant along each fiber
of $F$ and coincides with $\alpha_{x_{0}}$ around $p$. Since
$(\alpha\beta-Y(\alpha))$ is constant along fibers, then it vanishes on a
cylinder $C$ containing the fiber of $p$. So, the restriction of the framing
given by $X^{\prime}$ and $Y^{\prime}$ on the cylinder $C$ is flat, which
implies that $C$ has a translation structure where the flow given by
$X^{\prime}$ is by translations. consider the segment $I_{p}$ in $C$ which is
around $p$ and tangent to $Y^{\prime}$, then its image under the
$X^{\prime}-$flow comes back to itself identically since each point is
periodic and the $X^{\prime}-$flow on $C$ sends a line to a line parallel to
it (with respect to the translation structure). Thus $C$ is universally
covered by a strip band $B=]a,b[\times\mathbb{R}$ with a translation acting on
it. This implies that the diffeomorphism $\varphi^{n}$ restricted to $C$ can
be lifted to diffeomorphism of $B$ preserving each level
$\\{x\\}\times\mathbb{R}$ and having the same cocycle matrix with respect to
the canonical framing. Consequently, the function on $C$ associating rotation
number of $\varphi^{n}$ restricted to each fiber of $F$ in $C$ is a local
diffeomorphism because this is reduced to the linear case (locally).

$\square$

As observed by Pierre Mounoud, we can, with another argument, verify that the
differential equation in the proof of the previous lemma has a global solution
on $\mathbb{S}^{1}$. This simplifies the situation and reduces the problem to
the flat case.

######  Corollary 4.11.

The diffeomorphism $\varphi^{n}$ is smoothly conjugate to the parabolic linear
automorphism of $\mathbb{R}^{2}/\mathbb{Z}^{2}$ given by the matrix
$A_{k}=\begin{pmatrix}1&k\\\ 0&1\end{pmatrix}$ where $k$ is the degree of the
covering $R:\mathbb{T}^{2}/F\to\mathbb{S}^{1}$.

$\square$

This completes the proof of Theorem 1.2

##  5\. Proof of Theorem 1.1

###  5.1. $C^{\infty}-$Regularity:

For the sake of clarity and ease of reading we present here a proof assuming
smooth regularity. Then, in the next section 5.3, we deal with the $C^{1}$
case. So, assume that everything is smooth, particularly, the vector fields
fields $X_{s},X_{c}$ and $X_{u}$

######  Lemma 5.1.

Assume that $\lambda_{c}\neq\pm 1$, then $[X_{i},X_{j}]=0$ for
$i,j\in\\{s,c,u\\}$.

Proof: let $i,j\in\\{s,c,u\\}$. We have that
$\varphi_{*}([X_{i},X_{j}])=[\lambda_{i}X_{i},\lambda_{j}X_{j}])=\lambda_{i}\lambda_{j}[X_{i},X_{j}]$.
Suppose that $[X_{i},X_{j}]\neq 0$ and let $x\in M$ such that
$[X_{i},X_{j}](x)\neq 0$. Then the vector field $[X_{i},X_{j}]$ does not
vanish on an open neighbourhood $V$ of $x$. Put $U$ to be the open subset
obtained as the saturation of $V$ (the union of all the $\varphi-$iterates of
$V$). Then $[X_{i},X_{j}]$ does not vanish on $U$ and the claim is that
$[X_{i},X_{j}]$ (restricted to $U$) is proportional to one of the vector
fields $X_{s},X_{c}$, or $X_{u}$. Indeed, let $l$ be the line field on $U$
generated by $[X_{i},X_{j}]$, then $l$ is $\varphi-$invariant and for a
generic recurrent point $y\in V$ the iterates of the line $l(y)$ come back
sufficiently close to $l(y)$. This forces $l(y)$ to be proportional to one of
the directions $X_{s},X_{c}$, or $X_{u}$. Since we can choose $V$ small
enough, we have that $[X_{i},X_{j}]$ is proportional to one of the vector
fields $X_{s},X_{c}$, or $X_{u}$ and, so, the same holds on $U$. Thus
$[X_{i},X_{j}]=\beta X_{k}$ for some function $\beta$ and $k\in\\{s,c,u\\}$.
Furthermore, on $U$ we have:

$\varphi_{*}([X_{i},X_{j}](x))=\lambda_{i}\lambda_{j}[X_{i},X_{j}](\varphi(x))=\lambda_{i}\lambda_{j}\beta(\varphi(x))X_{k}(\varphi(x))$

On the other hand:

$\varphi_{*}(\beta(x)X_{k}(x))=\lambda_{k}\beta(x)X_{k}(\varphi(x))$

So, $\lambda_{i}\lambda_{j}\beta(\varphi(x))=\lambda_{k}\beta(x)$, or, in
other words:

$\beta(\varphi(x))=\frac{\lambda_{k}}{\lambda_{i}\lambda_{j}}\beta(x)$

So $\beta$ must be identically zero on $U$ since
$\left|\frac{\lambda_{k}}{\lambda_{i}\lambda_{j}}\right|\neq 1$ and almost
every point recurrent.

$\square$

######  Corollary 5.2.

If $\lambda_{c}\neq\pm 1$ then $\varphi$ is smoothly conjugate to a linear
Anosov automorphism of the three dimensional torus
$\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$.

Proof: the framing given by $X_{s},X_{c}$, and $X_{u}$ is flat and the cocycle
matrix is hyperbolic. This implies the desired conclusion.

$\square$

######  Remark 5.3.

Note that, in this case, the same results still hold if we assume only that
the vector fields $X_{s},X_{c}$, and $X_{u}$ are just $C^{1}$. The only
difference is that the conjugacy in this situation is $C^{1}$.

It remains to treat the case $\lambda_{c}=\pm 1$. But we can restrict
ourselves to the case $\lambda_{c}=1$ and $\lambda_{s},\lambda_{u}>0$ by
considering $\varphi^{2}=\varphi\circ\varphi$. So, from now on, we suppose
$\lambda_{c}=1$, $\lambda_{s},\lambda_{u}>0$.

######  Lemma 5.4.

We have $[X_{c},X_{s}]=\beta X_{s}$ and $[X_{c},X_{u}]=\alpha X_{u}$ where
$\beta$ and $\alpha$ are smooth $\varphi-$invariant functions on $M$.

Proof: we have that
$\varphi_{*}([X_{c},X_{s}])=[X_{c},\lambda_{s}X_{s}]=\lambda_{s}[X_{c},X_{s}]$.
Hence, the vector field $[X_{c},X_{s}]$ is contracting under the action of
$\varphi$ which implies that $[X_{c},X_{s}]=\beta X_{s}$ where $\beta$ is a
smooth function on $M$. We have that
$D_{x}\varphi(\beta(x)X_{s}(x))=\lambda_{s}\beta(x)X_{s}(\varphi(x))$. On the
other hand, since $\varphi_{*}([X_{c},X_{s}])=\lambda_{s}[X_{c},X_{s}]$, we
have that:

$D_{x}\varphi(\beta(x)X_{s}(x))=\lambda_{s}\beta(\varphi(x))X_{s}(\varphi(x))$

So $\beta(x)=\beta(\varphi(x))$. The same holds for $X_{c}$ and $X_{u}$ by
considering $\varphi^{-1}$.

$\square$

######  Proposition 5.1.

Suppose that $\beta$ is not constant. Then for a generic regular point $a$ in
the image of $\beta$ we have that the level surface $\beta^{-1}(a)$ is tangent
to the plane distribution given by the vector fields $X_{s}$ and $X_{u}$.

######  Lemma 5.5.

If a continuous function $f:M\to Y$ is $\varphi-$invariant then $f$ is
invariant by the flows $\phi_{s}^{t}$ and $\phi_{u}^{t}$ generated by $X_{s}$
and $X_{u}$ respectively.

Proof: let $x\in M$ and $\gamma:[0,1]\to M$ be a $C^{1}$ curve tangent to the
vector field $X_{s}$ with $\gamma(0)=x$ and $\gamma(1)=y$. Let $l(\gamma)$ be
the length of this curve with respect to the Riemannian metric given by the
framing. Since $\gamma^{{}^{\prime}}(t)=\alpha(t)X_{s}(\gamma(t))$ we obtain
that
$(\varphi\circ\gamma)^{{}^{\prime}}(t)=D_{\gamma(t)}(\gamma^{{}^{\prime}}(t))=\lambda_{s}\gamma^{{}^{\prime}}(t)$.
This implies that $l(\varphi\circ\gamma)=\lambda_{s}l(\gamma)$. Hence the
curve $\gamma$ is getting smaller under the iterates of $\varphi$ which
implies that $x$ and $y$ are getting closer. But $\beta$ is
$\varphi-$invariant, thus, $\beta(\varphi^{n}(x))=\beta(x)$ and
$\beta(\varphi^{n}(y))=\beta(y)$. Since $\varphi^{n}(x)$ and $\varphi^{n}(y)$
keep getting closer and closer, we obtain $\beta(x)=\beta(y)$ (using the
compactness of $M$). Consequently, $\beta$ is $\phi_{s}^{t}-$invariant and the
same argument holds for the unstable flow $\phi_{u}^{t}$ by considering
$\varphi^{-1}$ instead. This proves the lemma and the proposition since the
level surface $\beta^{-1}(x)$ is, hence, $\phi_{s}^{t}$ and $\phi_{u}^{t}$
invariant.

$\square$

So $\varphi$ preserves the level surface $\beta^{-1}(a)$. Suppose,
furthermore, that $\varphi$ preserves each connected component of
$\beta^{-1}(a)$ (otherwise consider a finite power of $\varphi$). Thus,
$\varphi$ preserves a two dimensional torus $\mathbb{T}^{2}\subset M$ tangent
to the plane distribution given by $X_{s}$ and $X_{u}$. Consequently,
$\varphi$ acts as a linear Anosov diffeomorphism on $\mathbb{T}^{2}$. In this
case we have:

######  Proposition 5.2.

The image of this torus $\mathbb{T}^{2}$ by the central flow $\phi_{c}^{t}$
defines a fibration of $M$ over $\mathbb{S}^{1}$ where each fiber is
diffeomorphic to $\mathbb{T}^{2}$ and preserved by $\varphi$. Furthermore,
$\phi_{c}^{t}$ sends fiber to fiber by conjugating the dynamics of $\varphi$
on each one.

Proof: consider the saturation of this torus $\mathbb{T}^{2}$ by the central
flow. This is clearly an open set $U$ since $\mathbb{T}^{2}$ is transversal to
$\phi_{c}^{t}$. The set $U$ is also invariant by the flows $\phi_{s}^{t}$ and
$\phi_{u}^{t}$ because $\phi_{c}^{t}$ preserves the orbits of $\phi_{s}^{t}$
and $\phi_{u}^{t}$ respectively and $\mathbb{T}^{2}$ is tangent to the stable
and the unstable directions. But the subsets of $M$ saturated by
$\phi_{c}^{t}$, $\phi_{s}^{t}$, and $\phi_{u}^{t}$ define a partition of $M$
by open subsets. So we have $U=M$ and the torus $\mathbb{T}^{2}$ must come
back since $M$ is compact. Hence the image of the torus $\mathbb{T}^{2}$ by
the central flow $\phi_{c}^{t}$ defines a fibration of $M$ over
$\mathbb{S}^{1}$ where each fiber is diffeomorphic to $\mathbb{T}^{2}$. Thus,
we have a projection $p:M\to\mathbb{S}^{1}$ where $\mathbb{S}^{1}$ denotes the
space of fibers on which $\varphi$ induces an autonomous diffeomorphism with
respect to the projection of the central vector field. So, this induced
diffeomorphism is conjugate to a rotation, but, it has a fixed point
(corresponding to the invariant torus). Consequently, $\varphi$ preserves each
fiber and, since $\varphi$ commutes with the central flow, then $\phi_{c}^{t}$
sends fiber to fiber by conjugating the dynamics of $\varphi$ on each one.

$\square$

Let $A:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be an Anosov linear automorphism and
consider the diffeomorphism
$\widetilde{\varphi}:\mathbb{T}^{2}\times[0,1]\to\mathbb{T}^{2}\times[0,1]$
defined by $\widetilde{\varphi}(x,t)=(A(x),t)$. Let
$f:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be a diffeomorphism that commutes with $A$
and consider $M$ to be the three-manifolds defined by identifying $(x,0)$ to
$(f(x),1)$ in the product space $\mathbb{T}^{2}\times[0,1]$. Then
$\widetilde{\varphi}$ induces a diffeomorphism $\varphi$ on $M$. We call the
diffeomorphism $\varphi$ a suspension of an Anosov linear diffeomorphism $A$
of $\mathbb{T}^{2}$ by the diffeomorphism $f$.

######  Corollary 5.6.

The diffeomorphism $\varphi$ (up to finite power) is conjugate to a suspension
of an Anosov linear diffeomorphism $A$ of $\mathbb{T}^{2}$ by a diffeomorphism
that commutes with $A$.

###  5.2. Algebraic classification

######  Proposition 5.3.

We have that $[X_{s},X_{u}]=\delta X_{c}$ where $\delta$ is a constant real
number.

Proof: we have $\varphi([X_{s},X_{u}])=[X_{s},X_{u}]$. So the vector field
$[X_{s},X_{u}]$ must be tangent to the central direction (otherwise it cannot
be preserved). Hence $[X_{s},X_{u}]=\delta X_{c}$ and $\delta$ is
$\varphi-$invariant. Suppose $\delta$ is not constant and let $a$ in the image
of $\delta$ be a regular value. Thus all the points in a neighbourhood of $a$
are regular values and their level sets define a foliation of an open subset
$O=\delta^{-1}(]a-\epsilon,a+\epsilon[)$ of $M$. This implies that the plane
field generated by $X_{s}$ and $X_{u}$ is integrable on $O$ which implies that
$[X_{s},X_{u}]$ is tangent to this plane field. But in the same time it is
tangent to the central direction. Consequently $\delta$ equals zero on $O$
which contradicts the fact that $a$ is a regular value. Thus $\delta$ is
constant.

$\square$

We have seen in the previous subsection that $[X_{c},X_{s}]=\beta X_{s}$ and
$[X_{c},X_{u}]=\alpha X_{u}$ and we understood the diffeomorphism $\varphi$ in
the case where $\beta$ is not constant. The same holds if $\alpha$ is not
constant. So it remains to classify in the case where
$[X_{c},X_{s}]=a_{cs}X_{s}$, $[X_{c},X_{u}]=a_{cu}X_{u}$ and
$[X_{s},X_{u}]=a_{su}X_{c}$ such that $a_{cs}$, $a_{cu}$ and $a_{su}$ are
constant real numbers.

In this case, assuming $\alpha$ and $\beta$ constant, the vector fields
$X_{s}$, $X_{c}$, and $X_{u}$ generate a three dimensional real Lie subalgebra
of the Lie algebra of all smooth vector fields. Put
$\mathcal{G}=<X_{s},X_{c},X_{u}>$ to be the subalgebra spanned by these vector
fields and let $G$ be the (unique) associated connected simply connected Lie
group (that is, $\sf{Lie}(G)=T_{1}G\approx\mathcal{G}$). Palais’ theorem
implies that $G$ acts on $M$ transitively and locally freely (because $X_{s}$,
$X_{c}$, and $X_{u}$ define a basis at each point of the compact manifold $M$)
and that $M$ is identified with $G/\Gamma$ where $\Gamma$ is a cocompact
lattice ($i.e$ $\Gamma$ is a discrete subgroup of $G$ with $G/\Gamma$
compact). In addition, the vector fields $X_{s}$, $X_{c}$, and $X_{u}$ are the
projection of right invariant vector fields $\widetilde{X_{s}}$,
$\widetilde{X_{c}}$, and $\widetilde{X_{u}}$ of $G$ on the quotient
$G/\Gamma=M$. Thus our diffeomorphism $\varphi:M=G/\Gamma\to G/\Gamma$ is
covered by an affine automorphism $\widetilde{\varphi}:G\to G$, that is,
$\widetilde{\varphi}=L_{g}\circ A$ where $L_{g}$ is a left translation on $G$
and $A\in\sf{Aut}(G)$ satisfying $A(\Gamma)=\Gamma$. The cocycle matrix
$M_{\varphi}$ of $\varphi$ with respect to the framing $\mathcal{F}$ generated
by $X_{s}$, $X_{c}$, and $X_{u}$ is equal to the cocycle matrix
$M_{\widetilde{\varphi}}$ of $\widetilde{\varphi}=L_{g}\circ A$ with respect
to the framing $\widetilde{\mathcal{F}}$ generated by $\widetilde{X_{s}}$,
$\widetilde{X_{c}}$, and $\widetilde{X_{u}}$. And we have:
$M_{\widetilde{\varphi}}=M_{\sf{Ad}(g)}M_{A}$ (where $M_{\sf{Ad}(g)}$ and
$M_{A}$ are the cocycle matrices of $\sf{Ad}(g)$ and $A$ respectively).

So, our situation is the following: we have a three-dimensional connected
simply connected Lie group $G$ and $\Gamma\subset G$ is a discrete subgroup
with $G/\Gamma$ is compact. Furthermore, we have $A\in\sf{Aut}(G)$ and $g\in
G$ such that $A(\Gamma)=\Gamma$ and the matrix $M_{\sf{Ad}(g)}M_{A}$ is
partially hyperbolic with respect to a basis $(X_{s},X_{c},X_{u})$ of the
algebra $\sf{Lie}(G)$ of right invariant vector fields. In this case we have a
classification.

It is known that if $G$ is a simply connected Lie group of dimension three
having a cocompact lattice $\Gamma\subset G$, then $G$ must be one of the
following six groups (for more details see, for example, the paper [27]):

  * •

The Abelian group $\mathbb{R}^{3}=\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}$.

  * •

The nilpotent group $\sf{Heis}_{3}$ (the Heisenberg group) of $3\times 3$
upper triangular real matrices of the form $\begin{pmatrix}1&x&z\\\ 0&1&y\\\
0&0&1\end{pmatrix}$.

  * •

The solvable group $\widetilde{\sf{Euc}}(\mathbb{R}^{2})$: the universal cover
of the Euclidean group of orientation preserving isometries of
$\mathbb{R}^{2}$ ($i.e$ $\sf{SO}(2)\ltimes\mathbb{R}^{2}$).

  * •

The solvable group $\sf{Lor}(1,1)={\sf{SO}}(1,1)\ltimes\mathbb{R}^{2}$ (also
called $\sf{SOL}$): the identity component of the isometry group (the
Lorentzian group) of the Minkowski space $(\mathbb{R}^{2},q)$ where $q$ is the
quadratic form $q(x,y)=xy$. It is isomorphic to the semi-direct product
$\mathbb{R}\ltimes\mathbb{R}^{2}$ where $\mathbb{R}$ acts on $\mathbb{R}^{2}$
via the representation $t\to\begin{pmatrix}e^{t}&0\\\ 0&e^{-t}\end{pmatrix}$.

  * •

The simple compact group $\sf{SU}(2)\approx\mathbb{S}^{3}$.

  * •

The simple non-compact group $\widetilde{\sf{SL}}(2,\mathbb{R})$: The
universal cover of the special linear group $\sf{SL}(2,\mathbb{R})$. This is
the richest case.

######  Lemma 5.7.

Let $G$ be either $\sf{SU}(2)$ or $\widetilde{\sf{Euc}}(\mathbb{R}^{2})$. Let
$\widetilde{\varphi}=L_{g}\circ A:G\to G$ be an affine automorphism. Then the
cocycle matrix of $\widetilde{\varphi}$ cannot be partially hyperbolic with
respect to a right invariant vector field.

Proof: if $G=\sf{SU}(2)$ then the Lie algebra of $G$ is isomorphic to
$\mathbb{R}^{3}$ endowed with the bilinear map given by the usual cross
product. The automorphisms of this algebra are exactly the linear Euclidean
isometries. Hence $\sf{Aut}(G)$ is compact. Therefore the affine group
$\sf{Aff}(G)=\\{L_{g}\circ A\mid A\in\sf{Aut}(G),g\in G\\}$ is compact. Let
$\mathcal{F}$ be a right invariant framing and consider the cocycle
representation:

$\sf{Aff}(G)\to\sf{Lie}(G)={\sf{GL}}(3,\mathbb{R})$ $L_{g}\circ A\to
M_{\sf{Ad}(g)}M_{A}$

Thus, the image of this representation is compact. So, it does not contain any
partially hyperbolic matrix.

Let now $\widetilde{\sf{Euc}}(\mathbb{R}^{2})=\mathbb{R}\ltimes\mathbb{R}^{2}$
where $\mathbb{R}$ acts on $\mathbb{R}^{2}$ via the representation
$\rho:t\in\mathbb{R}\to\exp(tA)$. where $A$ is the matrix
$\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}$.
Claim: all one-parameter subgroups
$\mathbb{R}\subset\widetilde{\sf{Euc}}(\mathbb{R}^{2})$ which are transversal
to $\mathbb{R}^{2}$ (the subgroup of translations) are conjugate to each
other. To see this we observe that such a one-parameter subgroup acts on
$\mathbb{R}^{2}$ by rotations around some fixed point, hence, the conjugacy
between two one-parameter subgroups transversal to $\mathbb{R}^{2}$ can be
done by the translation between the two fixed points.
The Lie algebra $\mathcal{G}$ of $\widetilde{\sf{Euc}}(\mathbb{R}^{2})$ is
generated by $T,X,Y$ where $T$ corresponds to $\mathbb{R}$ and $X,Y$
correspond to $\mathbb{R}^{2}$ with bracket relations:

$[T,X]=AX,\ \ [T,Y]=AY$

Let $\varphi:\mathcal{G}\to\mathcal{G}$ be an automorphism such that
$\varphi(T)=T$. Then:

$\varphi(AX)=\varphi([T,X])=[\varphi(T),\varphi(X)]=[T,\varphi(X)]=A(\varphi(X))$

Hence $\varphi\circ A=A\circ\varphi$. Let now
$\varphi:\mathcal{G}\to\mathcal{G}$ be partially hyperbolic. We can assume
that $\varphi(T)=\alpha T$ (because all one-parameter subgroups transversal to
$\mathbb{R}^{2}$ are conjugate). Suppose $\alpha\neq 1$, then:

$\varphi(AX)=\varphi([T,X])=[\varphi(T),\varphi(X)]=[\alpha
T,\varphi(X)]=\alpha A(\varphi(X))$

which is impossible because ${\sf{det}}(A)\neq 0$. Thus, $T$ is the central
direction and $\mathbb{R}^{2}=E^{s}\oplus E^{u}$. Since $\varphi\circ
A=A\circ\varphi$ we have (up to conjugacy) that $A=\begin{pmatrix}a&0\\\
0&b\end{pmatrix}$ with $a+b=0$. This implies that $\mathcal{G}$ is the Lie
algebra of $\sf{Lor}(1,1)$, contradiction. So
$\widetilde{\sf{Euc}}(\mathbb{R}^{2})$ does not contain any partially
hyperbolic automorphism.

$\square$

So, it remains to classify in terms of the four remaining cases:
$\mathbb{R}^{3},\ \sf{Heis}_{3},\ \sf{Lor}(1,1)$, and
$\widetilde{\sf{SL}}(2,\mathbb{R})$.

  * •

The Abelian group $\mathbb{R}^{3}$: In this case any discrete cocompact
subgroup $\Gamma\subset\mathbb{R}^{3}$ is equivalent to
$\mathbb{Z}^{3}=\mathbb{Z}\oplus\mathbb{Z}\oplus\mathbb{Z}$. The automorphisms
of $\mathbb{R}^{3}$ preserving the lattice $\mathbb{Z}^{3}$ are exactly the
elements of $\sf{GL}(3,\mathbb{Z})$. An affine automorphism $A+v$ with
$A(\mathbb{Z}^{3})=\mathbb{Z}^{3}$ has cocycle matrix equal to $A$ (with
respect to the canonical basis). Hence $A+v$ induces a map on
$\mathbb{T}^{3}=\mathbb{R}^{3}/\mathbb{Z}^{3}$ which is partially hyperbolic
if and only if $A\in\sf{GL}(3,\mathbb{Z})$ is partially hyperbolic. Suppose
$A$ is such a map, then we have the splitting of $\mathbb{R}^{3}$ into the
three directions $E_{s}$, $E_{c}$, and $E_{u}$ (the stable, central, and
unstable directions). Suppose $E_{c}$ is associated to the eigenvalue
$\lambda_{c}=\pm 1$ (otherwise $A$ is Anosov). In this case $E_{c}$ always
meets the lattice $\mathbb{Z}^{3}$ in a point other than the origin. Hence the
foliation of $\mathbb{R}^{3}$ by the parallel lines to $E_{c}$ projects to a
fibration by circles on $\mathbb{T}^{3}$. But the foliation of
$\mathbb{R}^{3}$ by the planes parallel to $E_{s}\oplus E_{u}$ doesn’t, in
general, project to a foliation by $2-$dimensional tori of $\mathbb{T}^{3}$.
In fact, the so defined foliation on $\mathbb{T}^{3}$ has either all of its
leaves compact, or all of them are dense in $\mathbb{T}^{3}$.

  * •

The Heisenberg group $\sf{Heis}_{3}$: Define, for $k\in\mathbb{N}$, the
discrete subgroup $\Gamma_{k}\subset\sf{Heis}_{3}$:

$\Gamma_{k}=\left\\{\begin{pmatrix}1&m&l/k\\\ 0&1&n\\\ 0&0&1\end{pmatrix}\mid
m,n,l\in\mathbb{Z}\right\\}$

Then $\Gamma_{k}$ is a (uniform) lattice and it is known that every lattice of
$\sf{Heis}_{3}$ is equivalent to exactly one $\Gamma_{k}$ (see for example
[22]). The quotient space $\sf{Heis}_{3}/\Gamma_{k}$ is diffeomorphic to a
circle extension of the torus $\mathbb{T}^{2}$ where the circle correspond to
the central direction in $\sf{Heis}_{3}$.

Let $C\approx\mathbb{R}$ be the center of $\sf{Heis}_{3}$. Then we have an
exact sequence:

$1\to C\to\sf{Heis}_{3}\to\mathbb{R}^{2}\to 1$

Let now $\sf{Aut}(\sf{Heis}_{3})$ be the automorphism group and
$\sf{Inn}(\sf{Heis}_{3})$ be the subgroup of inner automorphisms. Hence,
$\sf{Inn}(\sf{Heis}_{3})\approx\sf{Heis}_{3}/C\approx\mathbb{R}^{2}$ and we
obtain a natural map $\sf{Aut}(\sf{Heis}_{3})\to\sf{GL}(2,\mathbb{R})$ since
$C$ is preserved by automorphisms. This defines a sequence:

$1\to\sf{Inn}(\sf{Heis}_{3})\approx\mathbb{R}^{2}\to\sf{Aut}(\sf{Heis}_{3})\to\sf{GL}(2,\mathbb{R})\to
1$

This sequence is, in fact, exact and it splits, $i.e$,
$\sf{Aut}(\sf{Heis}_{3})\approx\sf{GL}(2,\mathbb{R})\ltimes\mathbb{R}^{2}$.

It is also known (see [22]) that we have a similar decomposition of the
automorphism group of a (uniform) lattice:

$\sf{Aut}(\Gamma_{k})\approx\sf{GL}(2,\mathbb{Z})\ltimes\mathbb{Z}^{2}$

In particular, the group $\sf{Aut}(\Gamma_{k})$ contains partially hyperbolic
automorphisms (with central direction tangent to the center $C$).

  * •

The Lorentzian group $\sf{Lor}(1,1)={\sf{SO}}(1,1)\ltimes\mathbb{R}^{2}$: We
have that $\sf{Lor}(1,1)=\mathbb{R}\ltimes\mathbb{R}^{2}$ where $\mathbb{R}$
acts on $\mathbb{R}^{2}$ via the representation $t\to\begin{pmatrix}e^{t}&0\\\
0&e^{-t}\end{pmatrix}$. Let $\Gamma\subset\sf{Lor}(1,1)$ be a uniform lattice.
Then $\Gamma_{0}=\Gamma\cap\mathbb{R}^{2}$ is a uniform lattice of
$\mathbb{R}^{2}$. And also the projection
$p:\Gamma\to\sf{Lor}(1,1)/\mathbb{R}^{2}\approx\mathbb{R}$ is a lattice ($i.e$
discrete). Let $\Gamma_{1}=p^{-1}(p(\Gamma))$. We have that
$\Gamma_{0}\subset\Gamma$ is normal and $\Gamma_{1}\ltimes\Gamma_{0}=\Gamma$
($\Gamma_{1}$ acts by conjugacy on $\Gamma_{0}$).

Let $(h^{t})$ be the one-parameter subgroup of
$\sf{Lor}(1,1)=\mathbb{R}\ltimes\mathbb{R}^{2}$ consisting of diagonal
matrices and let $h^{\alpha}\in(h^{t})$ such that $h^{\alpha}$ is conjugate to
an element of $\sf{SL}(2,\mathbb{Z})$. Then $h^{\alpha}$ preserves some
lattice $\Gamma_{0}$ of $\mathbb{R}^{2}$ and
$\Gamma=\\{h^{n\alpha}\\}\ltimes\Gamma_{0}\approx\mathbb{Z}\ltimes\mathbb{Z}^{2}$
is a uniform lattice of $\sf{Lor}(1,1)$. Left translations on $\sf{Lor}(1,1)$
by elements of $(h^{t})$ are partially hyperbolic and they induce partially
hyperbolic autonomous diffeomorphisms on $\sf{Lor}(1,1)/\Gamma$.

  * •

The group $\sf{\widetilde{SL}}(2,\mathbb{R})$: Let
$\pi:\widetilde{\sf{SL}}(2,\mathbb{R})\to\sf{PSL}(2,\mathbb{R})$ be the
covering projection. Then
$\pi_{1}({\sf{PSL}}(2,\mathbb{R}))=\ker(\pi)\approx\mathbb{Z}$ is a central
discrete subgroup. Let
$\widetilde{\Gamma}\subset\widetilde{\sf{SL}}(2,\mathbb{R})$ be a uniform
lattice and $\Gamma=\pi(\widetilde{\Gamma})$. Our claim is that $\Gamma$ is a
uniform lattice in ${\sf{PSL}}(2,\mathbb{R})$ (this seems to be a folkloric
fact but not easy to localise in the literature. So, let us give a hint).

We have a natural map
$\rho:\widetilde{\Gamma}\setminus\widetilde{\sf{SL}}(2,\mathbb{R})\to\Gamma\setminus{\sf{PSL}}(2,\mathbb{R})$
and all the point is to show that $\Gamma$ is discrete (because in this case
$\rho$ is a local diffeomorphism and
$\widetilde{\Gamma}\setminus\widetilde{\sf{SL}}(2,\mathbb{R})$ is compact
which implies that $\Gamma\setminus{\sf{PSL}}(2,\mathbb{R})$ is compact and
$\rho$ is a covering map).

The first observation is that $\Gamma$ is discrete if and only if
$\widetilde{\Gamma}\cap\ker(\pi)$ has finite index in $\ker(\pi)$, that is,
there is $N\in\mathbb{N}$ such that $\sigma^{N}\in\widetilde{\Gamma}$ where
$\sigma$ is a generator of $\ker(\pi)$. Put
$X=\widetilde{\Gamma}\setminus\widetilde{\sf{SL}}(2,\mathbb{R})$ then $\sigma$
acts on $X$ by right multiplication. Endow $\widetilde{\sf{SL}}(2,\mathbb{R})$
with a left invariant Riemannian metric $m$. Then $m$ is well defined on $X$
and the right action of $\sigma$ (which is also a left multiplication because
$\sigma$ is central) preserves $m$, that is, $\sigma\in\sf{Isom}(X,m)$. Since
$X$ is compact then $\sf{Isom}(X,m)$ is compact. If $\sf{Isom}(X,m)$ is finite
then for some power $N$ we have $\sigma^{N}=Id$ and this implies that
$\sigma^{N}\in\widetilde{\Gamma}$. Suppose now that the closure of
$\\{\sigma^{n},\ n\in\mathbb{Z}\\}$ contains a one-parameter subgroup, then
there is a corresponding one-parameter subgroup $(h^{t})$ in
$\sf{Isom}(\widetilde{\sf{SL}}(2,\mathbb{R}),m)$. One can choose $m$ such that
$\sf{Isom}(\widetilde{\sf{SL}}(2,\mathbb{R}),m)=\widetilde{\sf{SL}}(2,\mathbb{R})$
acting on itself by left translations. Hence, the one-parameter group
$(h^{t})$ is a subgroup of left multiplications which implies that
$\widetilde{\Gamma}$ is centralized by all elements of $(h^{t})$. In
particular, $\widetilde{\Gamma}\subset\sf{centralizer}((h^{t}))$. But, the
observation is that $\sf{centralizer}((h^{t}))=(h^{t}).Z$ where $Z=\ker(\pi)$
and $\pi:\widetilde{\sf{SL}}(2,\mathbb{R})\to{\sf{PSL}}(2,\mathbb{R})$ is the
covering projection). We have that:

$(h^{t}).Z\setminus\widetilde{\sf{SL}}(2,\mathbb{R})=(h^{t})\setminus{\sf{PSL}}(2,\mathbb{R})$

which is non-compact and this contradicts the fact that
$\widetilde{\Gamma}\subset\sf{centralizer}((h^{t}))$. We conclude that
$\Gamma=\pi(\widetilde{\sf{SL}}(2,\mathbb{R}))$ is a uniform lattice of
${\sf{PSL}}(2,\mathbb{R})$.

Let now $\Gamma\subset{\sf{PSL}}(2,\mathbb{R})$ be a uniform lattice and
$\varphi\in{\sf{Aut}}({\sf{PSL}}(2,\mathbb{R}))$ preserving $\Gamma$. One fact
is that
$\sf{Inn}({\sf{PSL}}(2,\mathbb{R}))\subset{\sf{Aut}}({\sf{PSL}}(2,\mathbb{R}))$
has finite index, so, up to a finite power, we may assume that
$\varphi\in\sf{Inn}({\sf{PSL}}(2,\mathbb{R}))$. But an inner automorphism
$L_{g}R_{g^{-1}}$ and the left translation $L_{g}$ have the same cocycle
matrix with respect to a right invariant framing (which is the matrix of
$\sf{Ad}(g)$) and it is partially hyperbolic if and only if $g$ is a
hyperbolic element of ${\sf{PSL}}(2,\mathbb{R})$. In particular, $\Gamma$
contains many hyperbolic elements and the associated inner automorphism
preserves $\Gamma$ and induces a partially hyperbolic diffeomorphism on
${\sf{PSL}}(2,\mathbb{R})/\Gamma$.

An other example is the left action of the one-parameter subgroup of diagonal
matrices of ${\sf{PSL}}(2,\mathbb{R})$ on ${\sf{PSL}}(2,\mathbb{R})/\Gamma$ by
partially hyperbolic left translations (this is no other than the geodesic
flow of $\Gamma\setminus\mathbb{H}$).

###  5.3. The $C^{1}-$regularity

Assume now that the vector fields $X_{s}$, $X_{c}$, and $X_{u}$ are only
$C^{1}$. We have, as before, that $[X_{c},X_{s}]=\beta X_{s}$ and
$[X_{c},X_{u}]=\alpha X_{u}$ but in this case $\beta$ and $\alpha$ are only
continuous $\varphi-$invariant functions on $M$. The case where
$\left|\lambda_{c}\right|\neq 1$ is noted in Remark 5.3. So, in the remaining
part, we suppose $\lambda_{c}=1$.

####  5.3.1. Non-constant structure coefficients:

Let $\mathcal{A}$ denote the set of all the closed subsets of $M$ that are
$\varphi-$invariant and invariant under the action of the flows $\phi_{s}^{t}$
and $\phi_{u}^{t}$. Clearly if $a$ is a point in the image of
$\varphi-$invariant function $\beta$, then the set $\beta^{-1}(a)$ belongs to
$\mathcal{A}$ (for the same argument as in the $C^{\infty}$ regularity). In
addition, the central flow $\phi_{c}^{t}$ preserves $\mathcal{A}$ because if
$F\in\mathcal{A}$ then
$\varphi(\phi_{c}^{t}(F))=\phi_{c}^{t}(\varphi(F))=\phi_{c}^{t}(F)$ and
$\phi_{c}^{t}$ preserves the stable and the unstable foliations (because of
the equalities $[X_{c},X_{s}]=\beta X_{s}$ and $[X_{c},X_{u}]=\alpha X_{u}$).
Recall that $\phi_{c}^{t}$ and $\varphi$ commute since
$\varphi_{*}(X_{c})=X_{c}$.

Suppose now that $\beta$ is not constant and let $x\in M$. Let $D_{x}$ be the
minimal subset containing $x$ and belongs to $\mathcal{A}$, that is, the
intersection of all the elements of $\mathcal{A}$ containing $x$. It is clear
that $D_{x}$ is a proper subset of $M$ since $\beta^{-1}(\beta(x))$ is in
$\mathcal{A}$, containing $x$, and it is a proper subset of $M$ (because
$\beta$ is not constant). The aim is to prove that $D_{x}$ is a compact
$C^{1}$ embedded surface.

For this, let $\Lambda_{x}=\\{t\in\mathbb{R}\mid\phi_{c}^{t}(D_{x})=D_{x}\\}$.
Clearly $\Lambda_{x}$ is a closed subgroup of $\mathbb{R}$. Suppose that
$\Lambda_{x}=\mathbb{R}$, then $D_{x}$ is invariant under the flows
$\phi_{s}^{t},\phi_{c}^{t}$, and $\phi_{u}^{t}$. This implies that $D_{x}=M$
but this is impossible since $D_{x}$ is a proper subset of $M$. Hence
$\Lambda_{x}=d_{x}\mathbb{Z}$ where $d_{x}$ is a positive real number. Note
that if for some $t\in\mathbb{R}$ we have $\phi_{c}^{t}(D_{x})\cap
D_{x}\neq\emptyset$ then $\phi_{c}^{t}(D_{x})=D_{x}$ because
$\phi_{c}^{t}(D_{x})\cap D_{x}$ belongs to $\mathcal{A}$ since both
$\phi_{c}^{t}(D_{x})$ and $D_{x}$ do, and $D_{x}$ is minimal.

######  Lemma 5.8.

For every $x\in M$, the subset $D_{x}$ is a compact embedded $C^{1}$ surface
tangent to the plane field generated by $X_{s}$ and $X_{u}$.

Proof: Let $y\in D_{x}$ be any point and let
$I_{\epsilon}=]-\epsilon,\epsilon[$. Let $\gamma_{x}^{s}(I_{\epsilon})$ be the
image of the curve $\gamma$ tangent to $X_{s}$ and satisfies $\gamma(0)=x$,
that is,
$\gamma_{x}^{s}(I_{\epsilon})=\\{\phi_{s}^{t}(x)\mid-\epsilon<t<\epsilon\\}$.
Let now:

$S_{x}^{\epsilon}=\\{\phi_{u}^{t}(\gamma_{x}^{s}(I_{\epsilon}))\mid-\epsilon<t<\epsilon\\}$

Then for $\epsilon$ small enough, $S_{x}^{\epsilon}$ is a small piece of a
$C^{1}$ surface tangent to $X_{s}$ and $X_{u}$ contained in $D_{x}$. The
central flow $\phi_{c}^{t}$ is transversal to $S_{x}^{\epsilon}$ and for a
small time $\alpha$ the sets $\phi_{c}^{t}(S_{x}^{\epsilon})$ for
$-\alpha<t<\alpha$ define a $C^{1}$ foliation of an open subset $B_{x}$ around
$x$. But for $\alpha<d_{x}$ the subsets $\phi_{c}^{t}(S_{x}^{\epsilon})$ are
all disjoint and do not intersect $D_{x}$ except $S_{x}^{\epsilon}$. This
shows $S_{x}^{\epsilon}$ is the only piece of $D_{x}$ inside $B_{x}$ which
proves the lemma.

$\square$

######  Corollary 5.9.

The plane field generated by $X_{s}$ and $X_{u}$ is integrable and defines a
$C^{1}$ foliation by compact leaves.

Proof: for any $x\in M$ the subset $D_{x}$ is a closed embedded
$\varphi-$invariant surface tangent to $X_{s}$ and $X_{u}$.

$\square$

Consider, as before, $D_{x}$ to be a closed minimal $\varphi-$invariant and
preserved by the flows $\phi_{s}^{t}$ and $\phi_{u}^{t}$. We proved that
$D_{x}$ is a closed compact $C^{1}$ embedded surface. It is clear that each
connected component of $D_{x}$ is $C^{1}$ diffeomorphic to the torus
$\mathbb{T}^{2}$, because its tangent bundle is necessarily trivial. The
diffeomorphism $\varphi$ acts by permuting this components in a cyclic
transitive way, otherwise $D_{x}$ wouldn’t be minimal. Let $C$ be such a
component, then a finite power of $\varphi$ preserves $C$ (and each other
component). The central flow $\phi_{c}^{t}$ preserves the foliation generated
by the vector fields $X_{s}$ and $X_{u}$. Since $\phi_{c}^{t}$ is transversal
to each leaf, then the orbit of every leaf is an open subset of $M$. But $M$
is connected, hence, The image of $C$ by central flow covers $M$. Following
the same idea in the proof of proposition (?? cite it) we conclude that $M$ is
$C^{1}$ diffeomorphic to a suspension of $\mathbb{T}^{2}$, in other words, the
foliation generated by $X_{s}$ and $X_{u}$ corresponds to a $C^{1}$ fibration
$M\to\mathbb{S}^{1}$ with $\mathbb{T}^{2}$ as a typical fiber. More precisely,
we have:

######  Corollary 5.10.

A finite power of the diffeomorphism $\varphi$ is $C^{1}$ conjugate to a
suspension of an Anosov linear diffeomorphism $A$ of $\mathbb{T}^{2}$ by a
diffeomorphism that commutes with $A$.

Proof: let $C$ be as indicated above (so $C$ is minimal). Then $C$ is $C^{1}$
diffeomorphic to $\mathbb{T}^{2}$ and preserved by a finite power
$\varphi^{k}$ of $\varphi$. The action of $\varphi^{k}$ on $C$ is clearly
hyperbolic and, hence, $C^{1}$ conjugate to a linear Anosov automorphism of
$\mathbb{T}^{2}$. Furthermore, the image of $C$ by the central flow defines a
$C^{1}$ foliation of $M$ that corresponds to a fibration over $\mathbb{S}^{1}$
as follows: let $t_{0}>0$ be the smallest time for which
$\phi_{c}^{t_{0}}(C)=C$ (note that $t_{0}$ exists otherwise we would have an
embedding of $C\times\mathbb{R}$. Which is surjective because the orbit of
each leaf is open and since $M$ is connected we must have only one orbit).
Then the action of the central flow on the space of leaves is periodic, with
period $t_{0}$, and this implies that this corresponds to a fibration and the
space of leaves is $C^{1}$ diffeomorphic to $\mathbb{S}^{1}$ on which the
diffeomorphism $\varphi^{k}$ acts by preserving the framing induced by
projecting $X_{c}$, hence it acts isometrically with respect to the
corresponding metric. Since $\varphi^{k}$ preserves $C$ then it acts on the
space of leaves trivially (preserving each leaf) because this action is
isometric (since it preserves the projection of $X_{c}$) with a fixed point.
Thus $\varphi^{k}$ preserves each leaf of the fibration. In addition, the
central flow $\phi_{c}^{t}$ commutes with $\varphi^{k}$, hence $\phi_{c}^{t}$
sends leaf to leaf by conjugating the induced dynamics on each one. In
particular, $\phi_{c}^{t_{0}}$ commutes with the restriction of $\varphi^{k}$
on $C$ (where $t_{0}$ is, as mentioned before, the smallest time for which
$\phi_{t_{0}}^{c}(C)=C$). This proves the result.

$\square$

####  5.3.2. Case of constant structure coefficients:

We have the equality:

$\varphi_{*}([X_{s},X_{u}])=[\lambda_{s}X_{s},\lambda_{u}X_{u}]=[X_{s},X_{u}]$

Hence $[X_{s},X_{u}]$ is $\varphi-$invariant and, for the same reasons as
before, this implies that $[X_{s},X_{u}]$ is tangent to the central direction
$i.e$ $[X_{s},X_{u}]=\delta X_{c}$ where $\delta$ is a $\varphi-$invariant
continuous function on $M$. If $\delta$ is not constant, then we are in the
previous situation which we understood. So, it remains to treat the case where
the functions $\alpha$, $\beta$, and $\delta$ are constant. Put
$\alpha=a_{cu}$, $\beta=a_{cs}$, and $\delta=a_{su}$. Thus, we have:
$[X_{s},X_{u}]=a_{su}X_{c}$, $[X_{c},X_{s}]=a_{cs}X_{s}$, and
$[X_{c},X_{u}]=a_{cu}X_{u}$ where $a_{su},a_{cs},a_{cu}\in\mathbb{R}$. Then we
have:

######  Lemma 5.11.

The vector fields $X_{s}$, $X_{c}$, and $X_{u}$ generate a three-dimensional
Lie algebra, that is, the Jacobi identity is satisfied.

######  Remark 5.12.

Observe that we cannot consider derivation of brackets since the vector fields
are only $C^{1}$.

Proof: note that it is enough to verify the identity just for the triplet
$X_{s}$, $X_{c}$, and $X_{u}$. We have the following equality:

$[[X_{s},X_{c}],X_{u}]+[[X_{c},X_{u}],X_{s}]+[[X_{u},X_{s}],X_{c}]=-a_{su}(a_{cs}+a_{cu})X_{c}$

So, in order to have the Jacobi identity, we must have either $a_{su}=0$ or
$a_{cs}+a_{cu}=0$. But the central flow is autonomous and preserves the stable
and the unstable directions and for some time $t_{0}$ it has determinant equal
to $e^{(a_{cs}+a_{cu})}$ which must equal $1$. Hence $a_{cs}+a_{cu}=0$.

$\square$

In this case, we have that the $C^{1}$ vector fields $X_{s}$, $X_{c}$, and
$X_{u}$ generate an abstract three dimensional real Lie algebra denoted
$\mathcal{G}$. Let $G$ be the associated connected simply connected Lie group.
Here one observes that Palais’s theorem holds in the $C^{1}$ regularity case.
Indeed, its proof is based upon Frobenius’ theorem which is true in the
$C^{1}$ case (in fact even in the Lipschitz case (see [26])). So, there is a
unique $C^{1}$ action of $G$ on the compact manifold $M$ such that the
embedding of $\sf{Lie}(G)=\mathcal{G}$ into the space of $C^{1}$ vector fields
on $M$ sending $\mathcal{G}$ onto $<X_{s},X_{c},X_{u}>$ is seen as the
corresponding infinitesimal action. More precisely, $G$ acts (a $C^{1}$
action) on $M$ transitively and locally freely such that $X_{s}$, $X_{c}$, and
$X_{u}$ are the projections of right invariant vector fields
$\widetilde{X_{s}}$, $\widetilde{X_{c}}$, and $\widetilde{X_{u}}$ of $G$ on
the quotient $M=G/\Gamma$ where $\Gamma$ is the stabilizer of a chosen point.
The autonomous diffeomorphism $\varphi:M=G/\Gamma\to M$ is induced by an
affine map $L_{g}\circ A:G\to G$ with $A\in\sf{Aut}(G)$ satisfying
$A(\Gamma)=\Gamma$. As a conclusion we have:

######  Corollary 5.13.

The diffeomorphism $\varphi:M=G/\Gamma\to M$ is covered by an affine
automorphism $L_{g}\circ A:G\to G$ such that $A\in\sf{Aut}(G)$ satisfying
$A(\Gamma)=\Gamma$. Thus, we have the same classification as in the
$C^{\infty}$ regularity, the only difference is that the conjugacy is just
$C^{1}$.

##  6\. Counterexamples to Theorem 1.1 in the $C^{0}-$regularity case

We have seen that in the $C^{1}$ regularity case we have (essentially) an
algebraic classification of partially hyperbolic autonomous dynamical systems.
The situation when assuming only the $C^{0}$ regularity of the framing is
different. We show here how to construct autonomous partially hyperbolic
diffeomorphisms (called skew-product of Anosov diffeomorphism) which are,
generically, non-algebraic (this implies, according to Theorem 1.1, that the
framing is not $C^{1}$).

Let $\varphi:M\to M$ be a diffeomorphism and let $\rho:M\to G$ be a $C^{k}$
map where $G$ is a compact connected Lie group. Define the map:

$\varphi_{\rho}:M\times G\to M\times G,\ \ \varphi_{\rho}(x,g)=(x,\rho(x)g)$

So, $\varphi_{\rho}$ sends the fiber $\\{x\\}\times G$ to the fiber
$\\{\varphi(x)\\}\times G$ by applying the left translation $L_{\rho(x)}$ on
the second factor $i.e$ on $G$. In particular, we have
$\pi\circ\varphi_{\rho}=\varphi\circ\pi$ where $\pi:M\times G\to M$ is the
projection on $M$. The diffeomorphism $\varphi_{\rho}$ is called a
$G-$extension (or a skew-product) of $\varphi$. If $G=\mathbb{S}^{1}$ then it
is simply called a ”circle extension” of $\varphi$.

Suppose that $\mathcal{F}_{G}$ is a left invariant framing on $G$. Then we can
define a family of independent global vector fields on $M\times G$ by inducing
$\mathcal{F}_{G}$ on each fiber $\\{x\\}\times G$. Furthermore, these vector
fields are preserved by $\varphi_{\rho}$ because it acts by left translations
on the fibers.

Suppose now that $\varphi:M\to M$ is Anosov and $\varphi_{\rho}:M\times G\to
M\times G$ as before. Suppose in addition that there exists a horizontal
$\varphi_{\rho}-$invariant distribution $\mathcal{H}$ of $T(M\times G)$. That
is, $\mathcal{H}$ is a sub-bundle of $T(M\times G)$ such that
$\mathcal{H}(x)\oplus T_{x}G_{x}=T_{x}(M\times G)$ and
$D_{x}\varphi_{\rho}(\mathcal{H}(x))=\mathcal{H}(\varphi_{\rho}(x))$ for every
$x\in M\times G$ where $G_{x}$ is the fiber $\\{\pi(x)\\}\times G$. In this
case the diffeomorphism $\varphi_{\rho}$ is partially hyperbolic such that the
$\varphi-$invariant splitting $E_{s}\oplus E_{u}=TM$ is lifted to
$\varphi_{\rho}-$invariant splitting
$\widetilde{E}_{s}\oplus\widetilde{E}_{u}=\mathcal{H}$ and the Whitney sum:

$\widetilde{E}_{s}\oplus\widetilde{E}_{c}\oplus\widetilde{E}_{u}=T(M\times G)$

satisfies the partial hyperbolicity conditions (where
$\widetilde{E}_{c}=\ker(d\pi)$ is the vertical distribution tangent to the
fibers $G_{x}$, $x\in G$).

######  Fact 6.1.

A circle extension $\varphi_{\rho}:M\times\mathbb{S}^{1}\to
M\times\mathbb{S}^{1}$ of an Anosov diffeomorphism $\varphi:M\to M$ is always
partially hyperbolic where the central distribution is the vertical line sub-
bundle tangent to the fibers.

For further discussions and more details see [8]. Here is an idea to prove
this fact (this is known as the invariant cone method): let $E_{u}$ be the
unstable line sub-bundle on $M$ and define $F_{u}=\pi^{-1}(E_{u})$ (that is,
$F_{u}$ is the inverse image of $E_{u}$ by the projection). So, $F_{u}$ is a
$\varphi_{\rho}-$invariant plane sub-bundle of $T(M\times\mathbb{S}^{1})$
containing the central line sub-bundle $\widetilde{E}_{c}$ tangent to the
fibers. The aim is to construct a $\varphi_{\rho}-$invariant line sub-bundle
$\widetilde{E}_{u}$ such that
$\widetilde{E}_{c}\oplus\widetilde{E}_{u}=F_{u}$. For this, we start with any
line sub-bundle $E$ in $F_{u}$ supplementary to $\widetilde{E}_{c}$ (that is,
$\widetilde{E}_{c}\oplus E=F_{u}$) and prove that the images of $E$ by action
of the iterates of $\varphi_{\rho}$ converge to an invariant line sub-bundle.
The way to do this is to show that the action of $\varphi_{\rho}$ on the space
of continuous line sub-bundles $E$ in $F_{u}$ and supplementary to
$\widetilde{E}_{c}$ is in some sense a ”contraction” and, therefore, it has a
unique fixed point and every orbit converges to it. Constructing
$\widetilde{E}_{s}$ is similar by using $\varphi_{\rho}^{-1}$ instead and we
have that $\widetilde{E}_{u}$ and $\widetilde{E}_{s}$ verify the desired fact.

######  Corollary 6.2.

Let $A:\mathbb{T}^{2}\to\mathbb{T}^{2}$ be a linear Anosov automorphism and
$\rho:\mathbb{T}^{2}\to\mathbb{S}^{1}$ be a smooth map. Let
$A_{\rho}:\mathbb{T}^{2}\times\mathbb{S}^{1}\to\mathbb{T}^{2}\times\mathbb{S}^{1}$
be the circle extension of $A$. Then $A_{\rho}$ is partially hyperbolic
autonomous diffeomorphism (with respect to a framing).

Proof: we have seen (in the previous fact) that $A_{\rho}$ is partially
hyperbolic with central direction tangent to the fibers, and, the stable and
the unstable directions define a $\varphi_{\rho}-$invariant horizontal plane
field $\mathcal{H}$. Each plane $\mathcal{H}(x)$ projects isomorphically onto
$T_{\pi(x)}\mathbb{T}^{2}$. Let $\mathcal{F}$ be a framing of $\mathbb{T}^{2}$
for which $A$ is autonomous, then its pullback $\pi_{*}(\mathcal{F})$ defines
two independent vector fields on $\mathbb{T}^{2}\times\mathbb{S}^{1}$
contained in $\mathcal{H}$. Let $X$ be a ”constant” non-null vector fields on
$\mathbb{T}^{2}\times\mathbb{S}^{1}$ tangent to the fibers. Then $X$ is
preserved by $A_{\rho}$ and $X$ together with $\pi_{*}(\mathcal{F})$ define a
framing of $\mathbb{T}^{2}\times\mathbb{S}^{1}$ for which $A_{\rho}$ is
autonomous with cocycle matrix equal to $\begin{pmatrix}1&0&0\\\ 0&a&b\\\
0&c&d\end{pmatrix}$ where $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$ is the
cocycle matrix of $A$ with respect to $\mathcal{F}$.

$\square$

Since the map $\rho:\mathbb{T}^{2}\to\mathbb{S}^{1}$ can be anything (for
example constant inside some open set and non-constant outside) we deduce that
the partially hyperbolic dynamics
$A_{\rho}:\mathbb{T}^{2}\times\mathbb{S}^{1}\to\mathbb{T}^{2}\times\mathbb{S}^{1}$
is not algebraic. Thus, the constructed framing for which $A_{\rho}$ is
autonomous cannot be $C^{1}$ because this would contradict Theorem 1.1. In
fact, the lack of regularity occurs in the horizontal plane field generated by
the stable and unstable directions.

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