# Affine Anosov Diffeomorphims of Affine Manifolds

Authors: Aristide Tsemo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

We show that a compact affine manifold endowed with an affine Anosov transformation is finitely covered by a complete affine nilmanifold. This is a partial answer of a conjecture of Franks for affine manifolds.


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## Body

## 1. Introduction

Ann-affine manifold (M,∇M) is an n-differentiable manifold M endowed with a locally flat connection ∇M,
that is a connection ∇M whose curvature and torsion forms vanish
identically. The connection ∇M defines on M an atlas (affine) whose coordinates change
is locally affine maps. The pull-back ∇M^ of the connection ∇M to the universal cover M^ of M is a locally flat connection. The affine
structure of M^ is defined by a local diffeomorphism DM:M^→ℝn called the developing map. The developing map
gives rise to a representation hM:π1(M)→Aff(ℝn) called the holonomy. The linear part L(hM) of hM is the linear holonomy. The affine manifold (M,∇M) is complete if and only if DM is a diffeomorphism. This means also that the
connection ∇M is geodesically complete.A diffeomorphismf of M is called an Anosov diffeomorphism f if and only if there exists a norm ∥⋅∥ on M associated to a differentiable metric 〈⋅,⋅〉,
a real number 0<λ<1 such that the tangent bundle TM of M is the direct summand of two bundles TMs and TMu called, respectively, the stable bundle and
the unstable bundle such that
(1.1)∥dfm(v)∥≤cλm∥v∥,v∈TMs,∥dfm(w)∥≥cλ−m∥w∥,w∈TMu,
where d is the usual differential, c is a positive real number, and m is a positive integer.The stable distributionTMs (resp., the unstable distribution TMu) is tangent to a topological foliation ℱs (resp., ℱu).The property for a diffeomorphism to be Anosov is
independent of the choice of the differentiable metric ifM is compact. In this case we can suppose that c is 1.A conjecture of
Franks asserts that an Anosov diffeomorphismf defined on a compact manifold M is C0-conjugated to an hyperbolic
infranilautomorphism. This conjecture is proved in [1] with the assumptions
that f is topologically transitive, the stable and
unstable foliations of f are C∞,
and f preserves a symplectic form or a connection.The goal of this paper is to characterize compact
affine manifolds endowed with affine Anosov transformations. More precisely, we
show the following.Theorem 1.1.
Let(M,∇M) be a compact affine manifold, and f an affine Anosov transformation of M,
then (M,∇M) is finitely covered by a complete affine
nilmanifold.


## 2. The Proof of The Main Theorem

The main goal of this part is to show Theorem1.1.
In the sequel, (M,∇M) will be an n-compact affine manifold endowed with an
affine Anosov diffeomorphism f.
The stable foliation ℱs (resp., the unstable foliation ℱu) pulls-back on the universal cover M^ to a foliation ℱ^s,
(resp., ℱ^u).Let(U,ϕ) be an affine chart of M,
we say that the restriction of a differentiable metric of M to U is an Euclidean metric adapted to the affine
structure of U if this restriction is the pull-back by ϕ to U,
of the restriction to ϕ(U) of an Euclidean metric of ℝn.Proposition 2.1.
The stable and the unstable distributions off define on (M,∇M) foliations whose leaves are immersed affine
submanifolds.Proof.
Letx be an element of M, ∥∥ a norm associated to a differentiable metric 〈,〉 of M.
Let v be an element of TxMs the subspace of TxM tangent to ℱs;
we have ∥dfm(v)∥≤λm∥v∥ where 0<λ<1 and m∈ℕ.
Let (U,ϕ) be an affine chart which contains an
accumulation point of the sequence (fp(x))p∈ℕ.
We can suppose that the restriction of 〈,〉 to U is Euclidean and adapted to the affine
structure. Let p>p′ such that fp(x) and fp′(x) are elements of U,
and v∈Tfp′(x)Ms;
we have ∥dfp−p′(v)∥≤λp−p′∥v∥.
This implies that every element y=fp′(x)+w,
such that w∈Tfp′(x)Ms and fp−p′(y)∈U is an element of the stable leaf of fp′(x) since the distance between fnq(x) and fnq(y) converges towards zero for a subsequence nq>p such that fnq(x) is an element of U.
We deduce that this leaf is an immersed submanifold. The result for the
unstable foliation is deduced by considering f−1.Proposition 2.2.
The affine structure induced by∇M on a leaf of ℱs is geodesically complete.Proof.
Letx be an element of M;
since M is compact, the sequence (fm(x))m∈ℕ has an accumulation point y.
Let Uy be an open set containing y such that there is a strictly positive number r,
such that for every z∈Uy, v∈TzM whose norm is less than r for a given differentiable metric, a (affine)
geodesic from x whose derivative at 0 is v is defined at 1.
Let w be an element of TxMs such that ∥dfp(w)∥≤r.
Without loss of generality, we can suppose that fp(x)∈Uy,
the (affine) geodesic from fp(x) whose derivative is dfp(w) at 0 is defined at 1.
This implies that the geodesic from x whose derivative at 0 is w is defined at 1,
since f−1 is an affine map. This shows the result.It is a well-known fact that an Anosov's
diffeomorphism of a compact manifold as a periodic fixed point. We can replacef by an iterated fp,p∈ℕ and suppose that f has a fixed point x.
This implies the existence of a map F:M^→M^ over f which fixed the element x^ over x.Proposition 2.3.
Lety^ and z^ be two elements of ℱ^t^u,
where t^ is an element of M^,
then the images of ℱ^y^s and ℱ^z^s by the developing map are parallel affine
subspaces.Proof.
Lety^ be an element of ℱ^t^u,
it is sufficient to show that D(ℱ^t^s) and D(ℱ^y^s) are parallel affine subspaces.
We know that the tangent bundle of a simply connected
affine manifold is trivial. We haveTM^=M^×Tt^M^.
Let 〈,〉′ be differentiable metric on M which pulls back on M^ is 〈,〉.
The map F is Anosov relatively to 〈,〉.
Without restricting the generality, we can assume that the distributions
tangent to ℱ^s and ℱ^u are orthogonal.
Letw be a vector of TM^y^ tangent to ℱ^y^s.
We can write w=s+u where s is a vector of TM^t^ tangent to ℱ^s and u is a vector of TM^t^ tangent to ℱ^u.
The vector u is also an element of TM^y^ tangent to ℱ^y^u since the unstable foliation is affine (we can
identify canonically the vector spaces TM^y^ and TM^t^ since TM^ is trivial). This implies that ∥dFy^l(u)∥≥λ−l∥u∥ for 0<λ<1,l∈ℕ.
But on the other hand we have ∥dFy^l(s+u)∥≤λl∥(s+u)∥,
which implies that the limit of ∥dFy^l(s+u)∥ is zero when l converges towards the infinity. We have
supposed that the distributions tangent to ℱ^s and ℱ^u are orthogonal, this implies that ∥dFy^(s+u)∥=∥dFy^(s)∥+∥dFy^(u)∥.
We deduce that u=0.The images byDM of the leaf ℱ^t^s of ℱ^s and ℱ^t^u of ℱ^u are affine subspaces of ℝn whose direction are supplementary subspaces of ℝn.
Since for every element z of ℱ^t^s the leaf of ℱ^u passing by z is complete, we deduce that the developing map
is surjective.Proposition 2.4.
The affine manifold(M,∇M) is complete.Proof.
Lett^ be an element of M^,
and E^t^ the set of elements y of M^ such that there is and element z in ℱ^t^u such that y is an element of ℱ^zs.
The image of E^t^ by DM is ℝn,
and the restriction of DM to E^t^ is injective. The set {E^t^,t^∈M^} is a partition of M^ by disjoint open sets. It has only one element
since M^ is connected. We deduce that (M,∇M) is complete.Remark 2.5.
The existence of an affine Anosov transformationf^ on the universal cover of a compact affine
manifold (M,∇M) which pushes forward to a diffeomorphism f of M does not imply that f is an Anosov diffeomorphism as the following
example shows.
LetNn be the quotient of ℝn/{0} by an homothetic transformation hλ whose ratio λ is such that 0<λ<1.
It is a compact affine manifold. Every homothetic transformation hc which ratio c is positive and different from 1 and λ is an Anosov diffeomorphism of ℝn endowed with an Euclidean metric. But the push
forward of hc to Nn is an isometry of Nn endowed with the push forward of the
differentiable metric of ℝn/{0} defined by
(2.1)1∥x∥2〈u,v〉,where x is an element of ℝn/{0},
and u, v are elements of its tangent space.Proof of Theorem1.1.
First, we show that the module of the eigenvalues of
the elements of the linear holonomy of(M,∇M) is 1.
Let(A,a) be an element of π1(M) such that A has an eigenvector u associated to an eigenvalue b (which may be a complex number) whose norm is
different from 1.
The argument used before Proposition2.3, allows us to
suppose that f has a fixed point x,
and there exists a map F over f which has also a fixed point x^.
Since (M,∇M) is complete, we can assume without restricting
the generality that x^=0.
Consider a differentiable metric〈,〉′ on M whose restriction to an affine neighborhood N of x is Euclidean adapted to the affine structure.
Expressing on N the fact that F is an Anosov diffeomorphism using the metric 〈,〉′,
one obtains that ℝn=U⊕V,
where U and V are two subvector
spaces such that there exists a number 0<λ<1 such that
(2.2)∥Fl(u)∥≤λl∥u∥,u∈U,∥Fl(v)∥≥λ−l∥u∥,v∈V,
where u and v are, respectively, elements of U and V and ∥∥ is a norm associated to an Euclidean metric 〈,〉 of ℝn.
The images of the subvector spaces U and V by the covering map are, respectively, TxMs and TxMu.
We will assume that they are orthogonal with respect to 〈,〉.
The vectors spaces U and V are stable by the linear holonomy (see Proposition 2.3).
Putu=u1+u2,
where u1 and u2 are, respectively, elements of U′ and V′,
the complexified vectors spaces, respectively, associated to U and V.
Without restricting the generality, one can assume
after eventually having changedγ=(A,a) by γ−1 and (or) f by f−1 that u1 is not zero and that the norm of b is strictly superior to 1.
Letq be a positive integer. We denote by ∥Aq∥=sup{∥x∥=1}∥Aq(x)∥ the norm of the linear operator Aq associated to the Euclidean metric 〈,〉.
For every element u∈U and every integer n,
since A preserves U,
we have
(2.3)∥Fn(Aq(u))∥≤λn∥Aq(u)∥≤λn∥Aq∥⋅∥u∥.Since λ<1,
there exists an integer n0 such that for every n>n0, ∥Fn(Aq(u))∥<∥u∥ for every element u in U.
We know thatA is invertible; let l be the dimension of V;
since the l−1-dimensional sphere is compact, there exists a
strictly positive real number aq such that inf{v∈V,∥v∥=1}∥Aq(v)∥≥aq.
For every element v in V,
and every integer n,
we have
(2.4)∥Fn(Aq(v))∥≥λ−n∥Aq(v)∥≥λ−naq∥v∥.
This implies the existence of integern1,
such that for every n>n1, ∥Fnq(Aq(v))∥>∥v∥.
This implies the existence of an integer nq such that Fnq∘γq has a fixed point m^q;
take for example n=sup(n0,n1).
In the sequel, we suppose that nq is the smallest integer such that Fnq∘γq has a fixed point.
Lety be a point of accumulation of the sequence p(m^q) which exists since M is compact. Up to the replacement of (m^q) by a subsequence and the replacement of m^q by another element over mq,
we can suppose that the sequence (m^q) converges to y^ over y.
This can be done by considering a neighborhood U of y^ in the universal cover of M on which the restriction of the covering map p is bijective onto its image; there exists q0 such that q>q0 implies that mq is an element of p(U).
We can then lift mq to U for q>q0 to obtain the desired sequence. The linear
part of the elements over fnq which fix the elements over mq has the same eigenvalues since they are
conjugated. Consider a differentiable metric of M which pullback on ℝn coincides on a neighborhood of y^ with an Euclidean metric.
(1) The Sequence(nq) Is Bounded
The restriction of the linear part of the map overfnq conjugated to Fnq∘γq which fixes m^q at U is a contracting map (for the last Euclidean
metric) with ratio λnq (0<λ<1), since it coincides on the neighborhood of y^ with the lift of a metric on M,
and f is an Anosov map. This is not possible since the
restriction of the linear part of this element to U′ (the tensor product of U with the complex line) has the same
eigenvalues than the restriction of Fnq∘Aq to U′,
and the limit when q goes to infinity of the norm of Fnq∘Aq(u1) for the Hermitian metric associated to the
last Euclidean metric is infinity, since the sequence nq is bounded and the norm of b>1 (recall that b is an eigenvalue of A).
(2) The Sequence(nq) Is Not Bounded
Up to the change of(nq) to a subsequence, we can suppose that (nq) goes to infinity. The linear map Fnq−1∘γq does not have a fixed point. Its linear part
has the eigenvalue 1 associated to the eigenvector vq.
Write vq=v1q+v2q where v1q and v2q are, respectively, elements of U and V.
If v1q is not zero, then ∥Fnq∘Aq(v1q)∥=∥F(v1q)∥ (the norm considered is the precedent
Euclidean norm); the restriction of Fnq∘Aq cannot be contracting with ratio λnq for q big enough since the sequence (nq) goes to infinity.
Ifv2q is not zero, then ∥Fnq∘Aq(v2q)∥=∥F(v2q)∥,
the restriction of Fnq∘Aq to V cannot be dilating (for the last Euclidean
metric) with ratio λ−nq for q big enough since nq goes to infinity. There is a contradiction
since the eigenvalues of the restriction of Fnq∘Aq to U′ (resp., to V′) coincide with the eigenvalues of the
restriction to U′ of the linear part of the conjugated to Fnq∘γq which fixes m^q (resp., the eigenvalues of the restriction to V′ of the linear part of the element conjugated
to Fnq∘γq which fixes m^q). This implies that the eigenvalues of A have norm 1.
We deduce that the linear holonomy of M is distal (see [2, paragraph 5.2]), it follows
from [3, Theorem 3] that M is finitely covered by a nilmanifold, see also
[4, Theorem 2].Remark 2.6.
Let(M′,∇′) be the finite cover of (M,∇) whose fundamental
group is G,
and f′ the pulls-back of f to M′.
The map Hn(M′,ℝ)→Hn(M′,ℝ), α→(f′2)*α is the identity since f′2 is a diffeomorphism which preserves the
orientation. We deduce that f′2 preserves the parallel volume form of (M′,∇′);
then a well-known result of Anosov implies that f′2 is ergodic and its periodic points are dense.



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