Classification of Special Anosov Endomorphisms of Nil-manifolds

Abstract

In this paper we give a classification of special endomorphisms of nil-manifolds: Let f:N/Γ→N/Γf:N/\Gamma \rightarrow N/\Gamma be a covering map of a nil-manifold and denote by A:N/Γ→N/ΓA:N/\Gamma \rightarrow N/\Gamma the nil-endomorphism which is homotopic to f. If f is a special TA-map, then A is a hyperbolic nil-endomorphism and f is topologically conjugate to A.

Full text

arXiv:1702.08821v4 [math.DS] 6 Feb 2018
Classification of special Anosov
endomorphisms of nil-manifolds
Seyed Mohsen Moosavi and Khosro Tajbakhsh
Abstract. In this paper we give a classification of special endo-
morphisms of nil-manifolds: Let f:N/Γ→N/Γ be a covering
map of a nil-manifold and denote by A:N/Γ→N/Γ the nil-
endomorphism which is homotopic to f. If fis a special T A-map,
then Ais a hyperbolic nil-endomorphism and fis topologically
conjugate to A.
1. Introduction
Finding a universal model for Anosov diffeomorphisms has been
an important problem in dynamical systems. In this general context,
Franks and Manning proved that every Anosov diffeomorphism of an
infra-nil-manifold is topologically conjugate to a hyperbolic infra-nil-
automorphism [7, 8, 12, 13] (According to Dekimpe’s work [5], some
of their results are incorrect). Based on this result, Aoki and Hiraide
has been studied the dynamics of covering maps of a torus [2]. The
importance of infra-nil-manifolds comes from the following Conjecture
1.1 and Theorem 1.2 :
The first non-toral example of an Anosov diffeomorphism was con-
structed by S. Smale in [16]. He conjectured that, up to topologically
conjugacy, the construction in Smale’s example gives every possible
Anosov diffeomorphism on a closed manifold.
Conjecture 1.1.Every Anosov diffeomorphism of a closed mani-
fold is topologically conjugate to a hyperbolic affine infra-nil-automorphism.
Theorem 1.2 (Gromov [9]).Every expanding map on a closed
manifold is topologically conjugate to an expanding affine infra-nil-
endomorphism.
The conjecture has been open for many years (see [6] page 48). An
interesting problem is to consider the conjecture for endomorphisms of
2010 Mathematics Subject Classification. Primary 37D05, 37D20.
Key words and phrases. Hyperbolicity, Anosov endomorphisms, special Anosov
endomorphisms, TA maps, nil-manifolds.
1

2 S.M.MOOSAVI AND K.TAJBAKHSH
a closed manifold. Our main theorem is a partial answer to the con-
jecture.
In this paper we give a classification of special endomorphisms of nil-
manifolds. Infact, Aoki and Hiraide [2] in 1994 proposed two problems:
Problem 1.3.Is every special Anosov differentiable map of a torus
topologically conjugate to a hyperbolic toral endomorphism?
Problem 1.4.Is every special topological Anosov covering map of
an arbitrary closed topological manifold topologically conjugate to a
hyperbolic infra-nil- endomorphism of an infra-nil-manifold ?
Aoki and Hiraide answered problem 1.3 partially as follows:
Theorem 1.5 ([2] Theorem 6.8.1).Let f:Tn→Tnbe a T A-
covering map of an n-torus and denote by A:Tn→Tnthe toral
endomorphism homotopic to f. Then Ais hyperbolic. Furthermore the
inverse limit system of (Tn, f )is topologically conjugate to the inverse
limit system of (Tn, A).
Theorem 1.6 ([2] Theorem 6.8.2).Let fand Abe as Theorem 1.5.
Suppose fis special, then the following statements hold:
(1) if fis a T A-homeomorphism, then Ais a hyperbolic toral au-
tomorphism and fis topologically conjugate to A,
(2) if fis a topological expanding map, then Ais an expanding
toral endomorphism and fis topologically conjugate to A,
(3) if fis a strongly special T A-map, then Ais a hyperbolic toral
endomorphism and fis topologically conjugate to A.
In [17], Sumi has altered the condition ” strongly special ” (part
(3) of Theorem 1.6) to just ” special ” as follows:
Theorem 1.7 ([17]).Let fand Abe as Theorem 1.5. If fis a
special T A-map, then Ais a hyperbolic toral endomorphism and fis
topologically conjugate to A.
In [18], Sumi generalized (incorrectly) Theorem 1.5 and parts (1)
and (2) of Theorem 1.6 for infra-nil-manifolds as follows:
Theorem 1.8 ([18] Theorem 1).Let f:N/Γ→N/Γbe a cov-
ering map of an infra-nil-manifold and denote as A:N/Γ→N/Γ
the infra-nil-endomorphism homotopic to f. If fis a TA-map, then A
is hyperbolic and the inverse limit system of (N/Γ, f )is topologically
conjugate to the inverse limit system of (N/Γ, A).
Theorem 1.9 ([18] Theorem 2).Let fand Abe as in Theorem
1.8. Then the following statements hold:
(1) if fis a TA-homeomorphism, then Ais a hyperbolic infra-nil-
automorphism and fis topologically conjugate to A,

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 3
(2) if fis a topological expanding map, then Ais an expanding
infra-nil-endomorphism and fis topologically conjugate to A.
Dekimpe [5], expressed that there might exist (interesting) diffeo-
morphisms and self-covering maps of an infra-nil-manifold which are
not even homotopic to an infra-nil-endomorphism. Dekimpe [5] in
§4, gave an expanding map not topologically conjugate to an infra-
nil-endomorphism. And in §5, he gave an Anosov diffeomorphism
not topologically conjugate to an infra-nil-automorphism. According
to [5], Theorem 1.8 and Theorem 1.9 are true for nil-manifolds. Of
course, if in Sumi’s works, the map fhas a desired homotopic infra-
nil-endomorphism, then the theorems hold.
Since, nil-manifolds are included in infra-nil-manifolds, we consider
[18] for nil-manifolds.
In the paper, by using Theorem 1.7, we partially answer problem
1.4 of Aoki and Hiraide as follows:
Theorem 1.10 (Main Theorem).Let f:N/Γ→N/Γbe a cov-
ering map of a nil-manifold and denote as A:N/Γ→N/Γthe nil-
endomorphism homotopic to f(according to [5], such a unique homo-
topy exists for nil-manifolds). If fis a special T A-map, then Ais a
hyperbolic nil-endomorphism and fis topologically conjugate to A.
Corollary 1.11.If f:N/Γ→N/Γis a special Anosov endo-
morphism of a nil-manifold then it is conjugate to a hyperbolic nil-
endomorphism.
2. Preliminaries
Let Xand Ybe compact metric spaces and let f:X→Xand
g:Y→Ybe continuous surjections. Then fis said to be topologically
conjugate to gif there exists a homeomorphism ϕ:Y→Xsuch that
f◦ϕ=ϕ◦g.
Let Xbe a compact metric space with metric d. For f:X→X
a continuous surjection, we let
Xf={˜x= (xi) : xi∈Xand f(xi) = xi+1, i ∈Z},
σf((xi)) = (f(xi)).
The map σf:Xf→Xfis called the shift map determined by f.
We call (Xf, σf) the inverse limit of (X, f). A homeomorphism f:
X→Xis called expansive if there is a constant e > 0 (called an
expansive constant) such that if xand yare any two distinct points of
Xthen d(fi(x), fi(y)) > e for some integer i. A continuous surjection
f:X→Xis called c-expansive if there is a constant e > 0 such that

4 S.M.MOOSAVI AND K.TAJBAKHSH
for ˜x, ˜y∈Xfif d(xi, yi)≤efor all i∈Zthen ˜x= ˜y. In particular, if
there is a constant e > 0 such that for x, y ∈Xif d(fi(x), f i(y)) ≤e
for all i∈Nthen x=y, we say that fis positively expansive. A
sequence of points {xi:a < i < b}of Xis called a δ-pseudo orbit of f
if d(f(xi), xi+1)< δ for i∈(a, b −1). Given ǫ > 0 a δ-pseudo orbit of
{xi}is called to be ǫ-traced by a point x∈Xif d(fi(x), xi)< ǫ for every
i∈(a, b−1) . Here the symbols aand bare taken as −∞ ≤ a < b ≤ ∞
if fis bijective and as −1≤a < b ≤ ∞ if fis not bijective. fhas the
pseudo orbit tracing property (abbrev. POTP) if for every ǫ > 0 there
is δ > 0 such that every δ-pseudo orbit of fcan be ǫ-traced by some
point of X.
We say that a homeomorphism f:X→Xis a topological Anosov
map (abbrev. T A-map) if fis expansive and has POTP. Analogously,
We say that a continuous surjection f:X→Xis a topological Anosov
map if fis c-expansive and has POTP, and say that fis a topological
expanding map if fis positively expansive and open. We can check
that every topological expanding map is a T A-map (see [2] Remark
2.3.10).
Let Xand Ybe metric spaces. A continuous surjection f:X→Y
is called a covering map if for y∈Ythere exists an open neighborhood
Vyof yin Ysuch that
f−1(Vy) = [
i
Ui(i6=i′⇒Ui∩U′
i=∅)
where each of Uiis open in Xand f|Ui:Ui→Vyis a homeomorphism.
A covering map f:X→Yis especially called a self-covering map if
X=Y. We say that a continuous surjection f:X→Yis a local
homeomorphism if for x∈Xthere is an open neighborhood Ux, of xin
Xsuch that f(Ux) is open in Yand f|Ux:Ux→f(Ux) is a homeomor-
phism. It is clear that every covering map is a local homeomorphism.
Conversely, if Xis compact, then a local homeomorphism f:X→Y
is a covering map (see [2] Theorem 2.1.1).
Let π:Y→Xbe a covering map. A homeomorphism α:Y→Y
is called a covering transformation for πif π◦α=πholds. We denote
as G(π) the set of all covering transformations for π. It is easy to see
that G(π) is a group, which is called the covering transformation group
for π.
Let Mbe a closed smooth manifold and let C1(M, M ) be the set of
all C1maps of Mendowed with the C1topology. A map f∈C1(M, M)
is called an Anosov endomorphism if fis a C1regular map and if there
exist C > 0 and 0 < λ < 1 such that for every ˜x= (xi)∈Mf={˜x=

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 5
(xi) : xi∈Mand f(xi) = xi+1, i ∈Z}there is a splitting
TxiM=Es
xi⊕Eu
xi, i ∈Z
(we show this by T˜xM=Si(Es
xi⊕Eu
xi)) so that for all i∈Z:
(1) Dxif(Eσ
xi) = Eσ
xi+1 where σ=s, u,
(2) for all n≥0
kDxifn(v)k≤ Cλnkvkif v∈Es
xi,
kDxifn(v)k≥ C−1λ−nkvkif v∈Eu
xi.
If, in particular, T˜xM=SiEu
xifor all ˜x= (xi)∈Mf, then fis said to
be expanding differentiable map, and if an Anosov endomorphism fis
injective then fis called an Anosov diffeomorphism. We can check that
every Anosov endomorphism is a TA-map, and that every expanding
differentiable map is a topological expanding map (see [2] Theorem
1.2.1).
A map f∈C1(M, M ) is said to be C1-structurally stable if there is
an open neighborhood N(f) of fin C1(M, M) such that g∈ N (f) im-
plies that fand gare topologically conjugate. Anosov [1] proved that
every Anosov diffeomorphism is C1-structurally stable, and Shub [15]
showed the same result for expanding differentiable maps. However,
Anosov endomorphisms which are not diffeomorphisms nor expanding
do not be C1-structurally stable ([11],[14]).
A map f∈C1(M, M ) is said to be C1-inverse limit stable if there
is an open neighborhood N(f) of fin C1(M, M) such that g∈ N (f)
implies that the inverse limit (Mf, σf) of (M, f) and the inverse limit
(Mg, σg) of (M, g) are topologically conjugate. Man´e and Pugh [11]
proved that every Anosov endomorphism is C1-inverse limit stable.
We define special TA-maps as follows. Let f:X→Xbe a con-
tinuous surjection of a compact metric space. Define the stable and
unstable sets
Ws(x) = {y∈X: lim
n→∞ d(fn(x), f n(y)) = 0},
Wu(˜x) = {y0∈X:∃˜y= (yi)∈Xfs.t. lim
i→∞ d(x−i, y−i) = 0}.
for x∈Xand ˜x∈Xf. A TA-map f:X→Xis special if fsatisfies
the property that Wu(˜x) = Wu(˜y) for every ˜x, ˜y∈Xfwith x0=y0.
Every hyperbolic nil-endomorphism is a special TA-covering map (See
[18] Remark 3.13). By this and Theorem 1.10 We have the following
corollary:
Corollary 2.1.A TA-covering map of a nil-manifold is special if
and only if it is conjugate to a hyperbolic nil-endomorphism.

6 S.M.MOOSAVI AND K.TAJBAKHSH
ALie group is a smooth manifold obeying the group properties
and that satisfies the additional condition that the group operations are
differentiable. Let Nbe a Lie group. A vector field Xon Nis said to be
invariant under left translations if for each g, h ∈N, (dlg)h(Xh) = Xgh,
where (dlg)h:ThN→TghNand lg:N→N;x7→ gx. A Lie algebra
gis a vector space over some field Ftogether with a binary operation
[·,·] : g×g→gcalled the Lie bracket, that satisfies:
(1) Bilinearity: [ax+by, z] = a[x, z]+b[y, z],[z, ax+by] = a[z, x]+
b[z, y]∀x, y, z ∈g,
(2) Alternativity: [x, x] = 0 ∀x∈g,
(3) The Jacobi Identity: [x, [y, z]]+[z, [x, y]]+[y, [z, x]] = 0 ∀x, y, z ∈
g
Let Lie(N) be the set of all left-translation-invariant vector fields on
N. It is a real vector space. Moreover, it is closed under Lie bracket.
Thus Lie(N) is a Lie subalgebra of the Lie algebra of all vector fields
on Nand is called the Lie algebra of N. A nilpotent Lie group is a
Lie group which is connected and whose Lie algebra is a nilpotent Lie
algebra. That is, its Lie algebra’s central series eventually vanishes.
A group Gis a torsion group if every element in Gis of finite or-
der. Gis called torsion free if no element other than identity is of
finite order. A discrete subgroup of a topological group Gis a sub-
group Hsuch that there is an open cover of Hin which every open
subset contains exactly one element of H. In other words, the subspace
topology of Hin Gis the discrete topology. A uniform subgroup H
of Gis a closed subgroup such that the quotient space G/H is compact.
We bring here the definitions of nil-manifolds and infra-nil-manifolds
from Karel Dekimpe in [4] and [5].
Let Nbe a Lie group and Aut(N) be the set of all automorphisms
of N. Assume that A∈Aut(N) is an automorphism of N, such that
there exists a discrete and cocompact subgroup Γ of N, with A(Γ) ⊆Γ.
Then the space of left cosets N/Γ is a closed manifold, and Ainduces
an endomorphism A:N/Γ→N/Γ, gΓ7→ A(g)Γ. If we want this
endomorphism to be Anosov, Amust be hyperbolic (i.e. has no eigen-
value with modulus 1). It is known that this can happen only when
Nis nilpotent. So we restrict ourselves to that case, where the result-
ing manifold N/Γ is said to be a nil-manifold. Such an endomorphism
Ainduced by an automorphism Ais called a nil-endomorphism and
is said to be a hyperbolic nil-automorphism, when Ais hyperbolic. If
in the above definition, A(Γ) = Γ, the induced map is called a nil-
automorphism.
All tori, Tn=Rn/Znare examples of nil-manifolds.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 7
Let Xbe a topological space and let Gbe a group. We say that
Gacts (continuously) on Xif to (g, x)∈G×Xthere corresponds a
point g·xin Xand the following conditions are satisfied:
(1) e·x=xfor x∈Xwhere eis the identity,
(2) g·(g′·x) = gg′·xfor x∈Xand g, g′∈G,
(3) for each g∈Ga map x7→ g·xis a homeomorphism of X.
When Gacts on X, for x, y ∈Xletting
x∼y⇔y=g·xfor some g∈G
an equivalence relation ∼in Xis defined. Then the identifying space
X/ ∼, denoted as X/G, is called the orbit space by Gof X. It follows
that for x∈X, [x] = {g·x:g∈G}is the equivalence class.
An action of Gon Xis said to be properly discontinuous if for each
x∈Xthere exists a neighborhood U(x) of xsuch that U(x)∩gU (x) = ∅
for all g∈Gwith g6=eG. Here gU (x) = {g·y:y∈U(x)}.
Now we give an extended definition of nil-manifolds. Let Nbe
a connected and simply connected nilpotent Lie group and Aut(N)
be the group of continuous automorphisms of N. Then Aff (N) =
N⋊Aut(N) acts on Nin the following way:
∀(n, γ)∈Af f(N),∀x∈N: (n, γ).x =nγ(x).
So an element of Aff (N) consists of a translational part n∈N
and a linear part γ∈Aut(N) (as a set Af f(N) is just N×Aut(N))
and Aff (N) acts on Nby first applying the linear part and then
multiplying on the left by the translational part). In this way, Af f (N)
can also be seen as a subgroup of Diff(N).
Now, let Cbe a compact subgroup of Aut(N) and consider any
torsion free discrete subgroup Γ of N⋊C, such that the orbit space
N/Γ is compact. Note that Γ acts on Nas being also a subgroup of
Aff (N). The action of Γ on Nwill be free and properly discontinuous,
so N/Γ is a manifold, which is called an infra-nil-manifold.
Klein bottle is an example of infra-nil-manifolds.
In what follows, we will identify Nwith the subgroup N× {id}of
N⋊Aut(N) = Aff (N), Fwith the subgroup {id} × Fand Aut(N)
with the subgroup {id} × Aut(N).
It follows from Theorem 1 of L. Auslander in [3], that Γ ∩Nis a
uniform lattice of Nand that Γ/(Γ ∩N) is a finite group. This shows
that the fundamental group of an infra-nil-manifold N/Γ is virtually
nilpotent (i.e. has a nilpotent normal subgroup of finite index). In fact
Γ∩Nis a maximal nilpotent subgroup of Γ and it is the only normal
subgroup of Γ with this property. (This also follows from [3]).
If we denote by p:N⋊C→Cthe natural projection on the second
factor, then p(Γ) = Γ ∩Nis a uniform lattice of Nand that Γ/(Γ∩N).

8 S.M.MOOSAVI AND K.TAJBAKHSH
Let Fdenote this finite group p(Γ), then we will refer to Fas being
the holonomy group of Γ (or of the infra-nil-manifold N/Γ). It follows
that Γ ⊆N⋊F. In case F={id}, so Γ ⊆N, the manifold N/Γ is a
nil-manifold. Hence, any infra-nil-manifold N/Γ is finitely covered by
a nil-manifold N/(Γ ∩N). This also explains the prefix ”infra”.
Fix an infra-nil-manifold N/Γ, so Nis a connected and simply
connected nilpotent Lie group and Γ is a torsion free, uniform discrete
subgroup of N⋊F, where Fis a finite subgroup of Aut(N). We will
assume that Fis the holonomy group of Γ (so for any µ∈F, there
exists an n∈Nsuch that (n, µ)∈Γ).
We can say that an element of Γ is of the form nµ for some n∈N
and some µ∈F. Also, any element of Af f (N) can uniquely be written
as a product nψ, where n∈Nand ψ∈Aut(N). The product in
Aff (N) is then given as
∀n1, n2∈N, ∀ψ1, ψ2∈Aut(N) : n1ψ1n2ψ2=n1ψ1(n2)ψ1ψ2.
Now we can define infra-nil-endomorphisms as follows:
Let Nbe a connected, simply connected nilpotent Lie group and
F⊆Aut(N) a finite group. Assume that Γ is a torsion free, discrete
and uniform subgroup of N⋊F. Let A:N⋊F→N⋊Fbe an
automorphism, such that A(F) = Fand A(Γ) ⊆Γ, then, the map
A:N/Γ→N/Γ,Γ·n7→ Γ·A(n).
is the infra-nil-endomorphism induced by A. In case A(Γ) = Γ, we call
Aan infra-nil-automorphism.
In the definition ab ove, Γ ·ndenotes the orbit of n under the action
of Γ. The computation above shows that Ais well defined. Note
that infra-nil-automorphisms are diffeomorphisms, while in general an
infra-nil-endomorphism is a self-covering map.
The following theorem shows that the only maps of an infra-nil-
manifold, that lift to an automorphism of the corresponding nilpotent
Lie group are exactly the infra-nil-endomorphisms defined above.
Theorem 2.2 ([5] Theorem 3.4).Let Nbe a connected and simply
connected nilpotent Lie group, F⊆Aut(N)a finite group and Γa
torsion free discrete and uniform subgroup of N⋊Fand assume that
the holonomy group of Γis F. If A:N→Nis an automorphism for
which the map
A:N/Γ→N/Γ,Γ·n7→ Γ·A(n).
is well defined (meaning that Γ·A(n) = Γ ·A(γ·n)for all γ∈Γ), then
A:N⋊F→N⋊F:x7→ φxφ−1(conjugation in Aff(N))
is an automorphism of N⋊F, with A(F) = Fand A(Γ) ⊆Γ. Hence,
Ais an infra-nil-endomorphism.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 9
Let Xbe a topological space. We write Ω(X;x0) the family of all
closed paths from x0to x0. Let Ω(X;x0)/∼be the identifying space
with respect to the equivalence relation ∼by homotopty. We write
this set
π1(X;x0) = Ω(X;x0)/∼.
The group π1(X;x0) is called the fundamental group at a base point x0
of X. If, in particular, π1(X;x0) is a group consisting of the identity,
then Xis said to be simply connected with respect to a base point x0.
Let x0and x1be points in X. If there exists a path wjoining x0
and x1, then we can define a map
w♯: Ω(X;x1)→Ω(X;x0),by w♯(u) = (w·u)·w,
where u∈Ω(X;x1), (w·u) is the concatenation of wand uand w
is win reverse direction. For u, v ∈Ω(X;x1) suppose u∼v. Then
w♯(u)∼w♯(v) and thus w♯induces a map
w∗:π1(X;x1)→π1(X;x0),by w∗([u]) = [w♯(u)],
this map is an isomorphism (see [2] Lemma 6.1.4).
Remark 2.3.If Xis a path connected space then we can remove
the base point and write π1(X;x0) = π1(X).
Let f, g :X→Ybe homotopic and Fa homotopy from fto g(f∼
g(F)). Then for x0∈Xwe can define a path w∈Ω(Y;f(x0), g(x0))
by
w(t) = F(x0, t)t∈[0,1],
and the relation between homomorphisms f∗:π1(X;x0)→π1(Y;f(x0))
and g∗:π1(X;x0)→π1(Y;g(x0)) is: g∗=w∗◦f∗(see [2] Lemma 6.1.9).
Let Xand Ybe topological spaces and f:X→Ya continuous
map. Take x0∈Xand let y0=f(x0). It is clear that f u =f◦u∈
Ω(X, y0) for u∈Ω(X;x0). Thus we can find a map
f♯: Ω(X;x0)→Ω(Y;Y0),by f♯(u) = fu,
where u∈Ω(X;x0). If u∼v(F) for u, v ∈Ω(X;x0), then we have
fu ∼f v (f◦F), from which the following map will be induced:
f∗:π1(X;x0)→π1(Y;y0),by f∗([u]) = [f♯(u)] = [fu],
It is easy to check that f∗is a homomorphism. We say that f∗is a
homomorphism induced from a continuous map f:X→Y.
Lemma 2.4 ([2] Remark 6.7.9).Let f, g :N/Γ→N/Γbe continu-
ous maps of a nil-manifold and let f(x0) = g(x0)for some x0∈N/Γ.
Then fand gare homotopic if and only if f∗=g∗:π1(N/Γ, x0)→
π1(N/Γ, f (x0)).
Theorem 2.5 ([2] Theorem 6.3.4).If π:Y→Xis the universal
covering, then for each b∈Y

10 S.M.MOOSAVI AND K.TAJBAKHSH
(1) the map α7→ α(b)is a bijection from G(π)onto π−1(π(b)),
(2) the map ψb:G(π)→π1(X, π(b)) by α7→ [π◦uα(b)]is an
isomorphism where uα(b)is a path from bto α(b).
Furthermore, the action of G(π)on Yis properly discontinuous and
Y/G(π)is homeomorphic to X.
Theorem 2.6 ([2] Theorem 6.3.7).Let Gbe a group and Xa
topological space. Suppose that Gacts on Xand the action is properly
discontinuous. Then
(1) the natural projection π:X→X/G is a covering map,
(2) if Xis simply connected, then the fundamental group π1(X/G)
is isomorphic to G.
Corollary 2.7.Let N/Γbe an infra-nil-manifold and π:N→
N/Γbe the natural projection. Then
Γ∼
=π1(N/Γ) ∼
=G(π).
Proof. Since Γ acts on Nproperly discontinuous, the natural pro-
jection π:N→N/Γ is a covering map. Since Nis simply connected,
by Theorem 2.6 we have Γ ∼
=π1(N/Γ).
On the other hand, since Nis simply connected and Γ acts on N
properly discontinuous the natural projection π:N→N/Γ is the
universal covering map. So by Theorem 2.5 we have Γ ∼
=G(π). 
From now on we only consider N/Γ as a nil-manifold.
Lemma 2.8.Let f:N/Γ→N/Γbe a continuous map of a nil-
manifold, and A:N/Γ→N/Γbe the unique nil-endomorphism homo-
topic to f, then f∗=A∗: Γ →Γ.
Proof. By corollary 2.7, f∗and A∗are two maps on Γ. For [e] =
{x∈N:γ(x) = γ.x =efor some γ∈Γ}, we have
f([e]) = f◦π(e) = π◦f(e) = π(f(e)) = π(e) = [e] = A([e]).
So according to lemma 2.4, f∗=A∗.
Lemma 2.9 ([18] Lemma 1.3).Let f:N/Γ→N/Γbe a self-
covering map of a nil-manifold and A:N/Γ→N/Γdenote the nil-
endomorphism homotopic to f. If fis a TA-covering map, then Ais
hyperbolic.
Lemma 2.10 ([18] Lemma 1.5).Let f:N/Γ→N/Γbe a self-
covering map and let f:N→Nbe a lift of fby the natural projection
π:N→N/Γ. If fis a TA-covering map then fhas exactly one fixed
point.
For continuous maps fand gof Nwe define D(f, g) = sup{D(f(x), g(x)) :
x∈N}where Ddenotes a left invariant, Γ-invariant Riemannian dis-
tance for N. Notice that D(f, g) is not necessary finite.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 11
Suppose that f:N/Γ→N/Γ is a TA-covering map. Let A:
N/Γ→N/Γ be the nil-endomorphism homotopic to f, and let A:
N→Nbe the automorphism which is a lift of Aby the natural
projection π. Since DeAis hyperbolic by Lemma 2.9, the Lie algebra
Lie(N) of Nsplits into the direct sum Lie(N) = Es
e⊕Eu
eof subspaces
Es
eand Eu
esuch that DeA(Es
e) = Es
e,DeA(Eu
e) = Eu
eand there are
c > 1,0< λ < 1 so that for all n≥0
||DeAn(v)|| ≤ cλn||v|| (v∈Es
e),
||DeA−n(v)|| ≤ cλn||v|| (v∈Eu
e),
where || · || is the Riemannian metric. Let Lσ(e) = exp(Eσ
e) (σ=s, u)
and let Lσ(x) = x·Lσ(e)(σ=s, u) for x∈N. Since left translations
are isometries under the metric D, it follows that for all x∈N
Ls(x) = {y∈N:D(Ai(x), Ai(y)) →0 (i→ ∞)},
Lu(x) = {y∈N:D(Ai(x), Ai(y)) →0 (i→ −∞)}.
Lemma 2.11 ([10] Lemma 3.2).For x, y ∈N,Ls(X)∩Lu(y)con-
sists of exactly one point.
For x, y ∈Ndenote as β(x, y) the point in Ls(X)∩Lu(y) .
Lemma 2.12 ([10] Lemma 3.2).(1) For L > 0and ǫ > 0there
exists J > 0such that for x, y ∈Nif D(Ai(x), Ai(y)) ≤Lfor all i
with |i| ≤ J, then D(x, y)≤ǫ.
(2) For given L > 0, if D(Ai(x), Ai(y)) ≤Lfor all i∈Z, then
x=y(x, y ∈N).
Lemma 2.13 ([18] Lemma 2.3).Under the assumptions and nota-
tions as above, there is a unique map h:N→Nsuch that
(1) A◦h=h◦f,
(2) D(h, idN)is finite,
where idN:N→Nis the identity map of N. Furthermore his
surjective and uniformly continuous under D.
In addition, if fis not an expanding map then his a homeomor-
phism i.e. his D-biuniformly continuous. (See [2]Proposition 8.4.2)
Lemma 2.14 ([18] Lemma 2.4).For the semiconjugacy hof lemma
2.13, we have the following properties:
(1) There exists K > 0such that D(h(xγ), h(x)γ)< K for x∈N
and γ∈Γ.
(2) For any λ > 0, there exists L∈Nsuch that D(h(xγ), h(x)γ)<
λfor x∈Nand γ∈AL
∗(Γ) .
(3) For x∈Nand γ∈T∞
i=0 Ai
∗(Γ) , we have h(xγ) = h(x)γ.
(4) For x∈Nand γ∈Γ, we have h(xγ)∈Ls(h(x)γ).

12 S.M.MOOSAVI AND K.TAJBAKHSH
Remark 2.15.By part (2) of theorem 2.13, there is a δK>0 such
that D(h(x), x)< δKfor x∈N, we have (see [2] page 270 (8.5))
Ws(x)⊂UδK(Ls(h(x))) and Wu(x;e)⊂UδK(Lu(h(x))).
By lemma 2.10 if f:N/Γ→N/Γ is a T A-map and f:N→Na lift
of it, then there exists a unique fixed point say bsuch that f(b) = b.
For simplisity we can suppose that b=e. Indeed, we can choose a
homeomorphism ϕof Nsuch that ϕ(π(b)) = e. Then ϕ◦f◦ϕ−1is a
T A-covering map such that ϕ◦f◦ϕ−1(e) = e.
Let x∈N, we define the stable set and unstable sets of xfor fand
Aas follow (for more details see [2]):
Ws(x) = {y∈N: lim
i→∞ D(fi(x), f i(y)) = 0},
Wu(x, e) = {y∈N: lim
i→−∞ D(fi(x), f i(y)) = 0},
Where e= (. . . , e, e, e, . . .).
Remark 2.16.By lemma 2.13, since his D-uniformly continuous
then h(Ws(x)) = Ls(h(x)) and h(Wu(x;e)) = Lu(h(x)).
Lemma 2.17.The following statements hold:
(1) Ws(x)γ=Ws(xγ)for γ∈Γand x∈N,
(2) Wu(x;e)γ=Wu(xγ;e)for γ∈Γand x∈N,
Proof. It is an easy corollary of lemma 6.6.11 of [2]. According
to corollary 2.7, in the mentioned lemma put Γ instead of G(π) and N
instead of X.
Lemma 2.18.The following statements hold:
(1) Ls(x)γ=Ls(xγ)for γ∈Γand x∈N,
(2) Lu(x)γ=Lu(xγ)for γ∈Γand x∈N,
Proof. Proof is the same as in lemma 2.17. 
Lemma 2.19 ([18] Lemma 5.4).Let N/Γbe a nil-manifold. If f:
N/Γ→N/Γis a T A-covering map, then the nonwandering set Ω(f)
coincides with the entire space N/Γ.
Lemma 2.20 ([2] Lemma 8.6.2).For ǫ > 0there is δ > 0such that
if D(x, y)< δ, x, y ∈Nthen Ws(x)⊂Uǫ(Ws(y)) and Wu(x;e)⊂
Uǫ(Wu(y;e)). Where for a set S, Uǫ(S) = {y∈N:D(y, S)< ǫ}.
3. Proof of Main Theorem
In this section we suppose that f:N/Γ→N/Γ is a special T A-
covering map of a nil-manifold which is not injective or expanding, and
A:N/Γ→N/Γ is the unique nil-endomorphism homotopic to f.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 13
Sketch of proof. By lemma 2.13, there is a unique semiconju-
gacy h:N→Nbetween fand A, such that by proposition 3.2.(3),
h(vγ) = h(v)γ, for each γ∈Wu(e;e)∩Γ and v∈Wu(e;e). Through
proposition 3.3 to proposition 3.13 we show that for all γ∈Γ and
x∈N,h(xγ) = h(x)γ. Based on this result, hinduces a homeomor-
phism h:N/Γ→N/Γ which is the conjugacy between fand A.
To prove the main theorem we need some consequential lemmas
and propositions.
Lemma 3.1.The following statements hold:
(1) Let Dbe the metric of Nas above. for each x∈N,D(x−1, e) =
D(x, e).
(2) If x∈Wu(e;e), then Wu(x;e) = Wu(e;e).
(3) If x∈Lu(e), then Lu(x) = Lu(e).
Proof. (1)
D(x−1, e) = D(x−1e, x−1x)
(Dis left invariant) = D(e, x)
=D(x, e)
(2) Since x∈Wu(e;e), we have D(fi(x), f i(e)) →0 as i→ −∞. Let
y∈Wu(e;e), then D(fi(y), fi(e)) →0 as i→ −∞. We have,
D(fi(y), fi(x)) < D(fi(y), f i(e)) + D(fi(e), fi(x)) →0 as i→ −∞.
So, y∈Wu(x;e), i.e. Wu(e;e)⊂Wu(x;e). Conversely, if y∈
Wu(x;e) then D(fi(y), fi(x)) →0 as i→ −∞, and
D(fi(y), fi(e)) < D(fi(y), f i(x)) + D(fi(x), fi(e)) →0 as i→ −∞.
So, y∈Wu(e;e), i.e. Wu(x;e)⊂Wu(e;e).
(3) Its proof is the same as part (2). 
For simplicity, let Γf=Wu(e;e)∩Γ and ΓA=Lu(e)∩Γ.
Proposition 3.2.The following statements hold:
(1) ΓAand Γfare subgroups of Γ.
(2) Γf⊂ΓA.
(3) h(vγ) = h(v)γ, for each γ∈Γfand v∈Wu(e;e).
(4) If Wu(γ1;e) = Wu(γ2;e), for some γ1, γ2∈Γ, then we have
h(xγ−1
1)γ1=h(xγ−1
2)γ2,for x∈Wu(γ1;e).
Proof. (1) Let γ1, γ2∈ΓA. Since Γ is a group we have γ1γ−1
2∈Γ.
Now consider that γ1, γ2∈Lu(e), since Ai(e) = e, for all i, then by

14 S.M.MOOSAVI AND K.TAJBAKHSH
definition,
lim
i→−∞ D(Ai(γ1), e) = lim
i→−∞ D(Ai(γ1), Ai(e)) = 0
lim
i→−∞ D(Ai(γ2), e) = lim
i→−∞ D(Ai(γ2), Ai(e)) = 0.(3.1)
As Dis left invariant we have
0≤lim
i→−∞ D(Ai(γ1γ−1
2), Ai(e)) = lim
i→−∞ D(Ai(γ1)Ai(γ−1
2), e)
= lim
i→−∞ D(Ai(γ1)A−i(γ2), Ai(γ1)A−i(γ1))
(D is left invariant) = lim
i→−∞ D(A−i(γ2), A−i(γ1))
≤lim
i→−∞ D(A−i(γ2), e) + D(e, A−i(γ1))
(Lemma 3.1.(1) and (3.1)) = lim
i→−∞ D(Ai(γ2), e) + D(Ai(γ1), e) = 0
Thus γ1γ−1
2∈Lu(e) and Lu(e) is a subgroup of N. So Lu(e)∩Γ is a
subgroup of Γ.
For the second part, Let γ1, γ2∈Γf. Since Γ is a group we have
γ1γ−1
2∈Γ. Now consider that γ1, γ2∈Wu(e;e). Then,
Wu(e;e)γ1
(Lemma 2.17) = Wu(eγ1;e)
=Wu(γ1;e)
(Lemma 3.1) (2) = Wu(e;e).
Similarly, Wu(e;e)γ2=Wu(e;e). So we have Wu(e;e)γ1=Wu(e;e)γ2
and then γ1γ−1
2∈Wu(e;e), and we have the result.
(2) Take γ∈Γf, such that γ /∈ΓA. So, γ /∈Lu(e) and for each
n∈Z, n 6= 0, γn/∈Lu(e). On the other hand, by part (1), remark
2.15 and the fact that h(e) = e, for all n∈Z, we have γn∈Wu(e;e)⊂
UδK(Lu(e)), which is impossible.
(3)Let γ∈Γfand v∈Wu(e;e). We have
vγ ∈Wu(e;e)γ
(Lemma 2.17) = Wu(eγ;e)
=Wu(γ;e)
(Lemma 3.1 (2)) = Wu(e;e),
so,
h(vγ)∈h(Wu(e;e))
(Remark 2.16) = Lu(e).

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 15
By part (2), γ∈ΓA, Thus
h(v)γ∈h(Wu(e;e))γ
(Remark 2.16) = Lu(e)γ
(Lemma 2.18) = Lu(eγ)
=Lu(γ)
(Lemma 3.1 (3)) = Lu(e).
Again by Lemma 3.1 (3) and last part of the above relation, Lu(h(v)γ) =
Lu(e), and
h(vγ)∈Lu(e) = Lu(h(v)γ).
On the other hand, by part (4) of lemma 2.14, h(vγ)∈Ls(h(v)γ).
Since Lu(h(v)γ)∩Ls(h(v)γ) = {h(v)γ}(see [18] Lemma 2.1), then
h(vγ) = h(v)γ.
(4) Let x∈Wu(γ1;e) = Wu(γ2;e). For γ1, γ2∈Γ, we have γ2∈
Wu(γ2;e) = Wu(γ1;e) = Wu(e;e)γ1. Thus, γ2γ−1
1∈Wu(e;e), and
then γ2γ−1
1∈Γf. Similarly, xγ−1
1, xγ−1
2∈Wu(e;e). Now, by part (3),
h(xγ−1
1)γ1=h(xγ−1
2γ2γ−1
1)γ1
=h(xγ−1
2)γ2γ−1
1γ1
=h(xγ−1
2)γ2.

According to part (4) of proposition 3.2, we can define a map h′:
Sγ∈ΓWu(γ;e)→Sγ∈ΓLu(γ), by
h′(x) = h(xγ−1)γ x ∈Wu(γ;e) (γ∈Γ).
Next lemma shows some properties of h′:
Proposition 3.3.The following statements hold:
(1) A◦h′=h′◦fon Sγ∈ΓWu(γ;e),
(2) D(h′, id|Sγ∈ΓWu(γ;e))<∞,
(3) h′(γ) = γfor γ∈Γ,
(4) if x∈Wu(γ;e) (γ∈Γ), then h′(x)∈Lu(γ)and h′(x)∈
Ls(h(x)),
(5) if y∈Ws(x)for x, y ∈Sγ∈ΓWu(γ;e), then h′(y)∈Ls(h′(x)).
Proof. (1) Suppose that x∈Wu(γ;e) = Wu(e;e)γ, for some
γ∈Γ. Then
(3.2) xγ−1∈Wu(e;e).

16 S.M.MOOSAVI AND K.TAJBAKHSH
By [2] page 205, we have fWu(γ;e)=Wu(f(γ); e). Here f(γ) means
f∗(γ) which by lemma 2.8 is equal to A∗(γ) and A∗(γ)∈Γ. Therefore,
f(x)∈fWu(γ;e)=Wu(A∗(γ); e) = Wu(e;e)A∗(γ)
so,
(3.3) (f(x))(A∗(γ))−1∈Wu(e;e).
Thus we have
A◦h′(x) = Ah(xγ−1)γ
((3.2) and proposition 3.2.(3)) = Ah(xγ−1γ)
=A◦h(x)
(lemma 2.13) = h◦f(x)
=h(f(x))(A∗(γ))−1(A∗(γ))
((3.3) and proposition 3.2.(3)) = h(f(x))(A∗(γ))−1(A∗(γ))
=h′(f(x))
=h′◦f(x).
(2) Let x∈Wu(γ;e), for some γ∈Γ, and let δK>0 be satisfying
D(h, idN)< δK. Then
D(h′(x), x) = Dh(xγ−1)γ , x
=Dh(xγ−1)γ, xγ−1γ
(D is Γ-invariant) = Dh(xγ−1), xγ−1
< δK.
(3) For any γ∈Γ, by definition we have
h′(γ) = h(γγ−1)γ=h(e)γ=eγ =γ.
(4) Let x∈Wu(γ;e), for some γ∈Γ. We have
h′(x) = h(xγ−1)γ∈h(Wu(γ;e)γ−1)γ
(lemma 2.17) = h(Wu(e;e)γγ−1)γ
=h(Wu(e;e))γ
(remark 2.16) = Lu(e)γ
(lemma 2.18) = Lu(γ),

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 17
and
h′(x) = h(xγ−1)γ
(lemma 2.14.(4)) ∈Ls(h(x)γ−1)γ
(lemma 2.18.(1)) = Ls(h(x))γ−1γ
=Ls(h(x)).
(5) By the second part of proof of (4), we have
Ls(h′(y)) = Lsh(y)=h(Ws(y)) = h(Ws(x)) = Lsh(x)=Ls(h′(x)),
so, h′(y)∈Ls(h′(x)). 
Lemma 3.4 ([18] Lemma 7.6).For each u, v ∈N,Wu(u;e)∩Ws(v)
is the set of one point.
According to the above lemma, define ι(u, v) = Wu(u;e)∩Ws(v).
Lemma 3.5.For ǫ > 0, there is δ > 0such that
D(u, v)< δ ⇒max{D(ι(u, v), u), D(ι(u, v), v)}< ǫ (u, v ∈N)
Proof. Let ǫ > 0 be given. Since his D-biuniformly contiuous
there exists ǫ′>0 such that
D(x, y)< ǫ′⇒D(h−1(x), h−1(y)) < ǫ (x, y ∈N).
By [2] theorem 6.6.5 or [18] lemma 7.2, since Nis simply connected,
Ahas local product structure (for definition and details, see [2]), and
then for ǫ > 0 there exists δ′>0 such that
D(u, v)< δ′⇒max{D(β(u, v), u), D(β(u, v), v)}< ǫ′(u, v ∈N)
Again since his D-biuniformly continuous, there exists δ > 0 such that
D(u, v)< δ ⇒D(h(u), h(v)) < δ′(u, v ∈N).
We know that hι(u, v)=β(h(u), h(v)) therefore
D(u, v)< δ ⇒D(h(u), h(v)) < δ′
⇒max{D(β(h(u), h(v)), h(u)), D(β(h(u), h(v)), h(v))}< ǫ′
⇒max{D(h(ι(u, v)), h(u)), D(h(ι(u, v)), h(v)) < ǫ′
⇒max{D(ι(u, v), u), D(ι(u, v), v)}< ǫ.

Proposition 3.6.h′is D-uniformly continuous.
Proof. Suppose that the statement is false. So there is ǫ0>0, for
all δ > 0, there are x, y ∈Sγ∈ΓWu(γ;e) such that
(3.4) D(x, y)< δ and D(h′(x), h′(y)) > ǫ0.

18 S.M.MOOSAVI AND K.TAJBAKHSH
By definition of Lσ(x) (x∈N, σ =s, u), for w∈Ls(v) there is ǫ1>0
such that
(3.5) D(v, w)< ǫ0/2⇒D(Lu(v), Lu(w)) > ǫ1.
Take n > 0 and δ1>0 such that ǫn≥2δKand δn
1≤2δK.
By Lemma 2.20, there exists δ2>0 such that
(3.6) D(v, w)< δ2⇒Wu(w, e)⊂Uδ1Wu(v, e).
Since his continuous, take δ3>0 such that
(3.7) D(u, v)< δ3⇒D(h(u), h(v)) < ǫ0/2.
By lemma 3.5, there is 0 < δ < δ2such that
(3.8)
D(x, y)< δ ⇒D(yγ−1
y, ι(y, x)γ−1
y) = D(y, ι(y, x)) < δ3(x, y ∈N).
Now consider x, y ∈Sγ∈ΓWu(γ;e) satisfy (3.4). There exist γx, γy∈
Γ such that x∈Wu(γx,e) and y∈Wu(γy,e). We have
Dh′(x), h′(ι(y, x))≥Dh′(x), h′(y)−Dh′(y), h′(ι(y, x))
(by (3.4)) ≥ǫ0−Dh(yγ−1
y)γy, h(ι(y, x)γ−1
y)γy
(D is Γ-invariant) = ǫ0−Dh(yγ−1
y),h(ι(y, x)γ−1
y)
(by (3.7) and (3.8)) > ǫ0−ǫ0/2 = ǫ0/2.(3.9)
By proposition 3.3.(4)
x∈Wu(γx,e)⇒h′(x)∈Lu(γx),
ι(y, x)∈Wu(γy,e)⇒h′(ι(y, x)) ∈Lu(γy).
Thus by proposition 3.3.(5), (3.9) and (3.5) we have
DLu(γx), Lu(γy)> ǫ1.
Suppose γ=γyγ−1
x. We have
γγx=γy6∈ Uǫ1Lu(γx)⇒γ6∈ Uǫ1Lu(γxγ−1
x)=Uǫ1Lu(e)
⇒γn(e)6∈ Uǫn
1Lu(e)⊃U2δKLu(e).(3.10)
On the other hand,
Wu(γγx;e) = Wu(γy;e)
(y∈Wu(γy;e)) = Wu(y;e)
(by (3.6)) ⊂Uδ1Wu(x;e)
(x∈Wu(γx;e)) = Uδ1Wu(γx;e)
=Uδ1Wu(e;e)γx.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 19
Now we have
Wu(γγx;e)γ−1
x⊂Uδ1Wu(e;e)⇒Wu(γγxγ−1
x;e)⊂Uδ1Wu(e;e)
⇒Wu(γ;e)⊂Uδ1Wu(e;e)
⇒Wu(γ2;e)⊂Uδ1Wu(γ;e)
(by induction) ⇒Wu(γn;e)⊂Uδ1Wu(γn−1;e)
⇒Wu(γn;e)⊂Uδn
1Wu(e;e)⊂U2δKLu(e)
⇒γn∈Wu(γn;e)⊂U2δKLu(e).(3.11)
Finally, (3.10) and (3.11) make a contradiction. 
Let ˜u= (ui)∈Nfand for each i∈Z,fui,ui+1 be the lift of fby π
such that f(ui) = ui+1 and define
fi
˜u=


fui−1,ui◦...◦fu0,u1for i > 0,
(fui,ui+1 )−1◦...◦(fu−1,u0)−1for i < 0,
idNfor i= 0.
We define a map τ˜u=τf
˜u:N→(N/Γ)fby
τ˜u(x) = (π◦fi
˜u(x))∞
i=−∞ (x∈N).
Since f(e) = e, then τe(e) = τ˜e(e) = (π(e))∞
i=−∞.
Lemma 3.7 ([2] Lemma 6.6.8 (1)).If x∈Xand ˜u∈Nfthen
π(Wu(x; ˜u)) = Wu(τ˜u(x)).
Let Xbe a compact metric set and f:X→Xa continuous
surjection. A point x∈Xis said to be a nonwandering point if for any
neighborhood Uof xthere is an integer n > 0 such that fn(U)∩U6=∅.
The set Ω(f) of all nonwandering points is called the nonwandering set.
Clearly Ω(f) is closed in Xand invariant under f.
fis said to be topologically transitive (here Xmay be not necessarily
compact), if there is x0∈Xsuch that the orbit O+(x0) = {fi(x0) :
i∈Z≥0}is dense in X. It is easy to check that if Xis compact,
a continuous surjection f:X→Xis topologically transitive if and
only if for any U, V nonempty open sets there is n > 0 such that
fn(U)∩V6=∅.
A continuous surjection f:X→Xof a metric space is topologically
mixing if for nonempty open sets U, V there exists N > 0 such that
fn(U)∩V6=∅for all n > N. Topological mixing implies topological
transitivity.
For continuing, we need next theorem for which proof one can see
[2] Theorem 3.4.4.

20 S.M.MOOSAVI AND K.TAJBAKHSH
Theorem 3.8 (Topological decomposition theorem).Let f:X→
Xbe a continuous surjection of a compact metric space. If f:X→X
is a T A-map, then the following properties hold:
(1) (Spectral decomposition theorem due to Smale) The nonwan-
dering set, Ω(f), contains a finite sequence Bi(1 ≤i≤l)of
f-invariant closed subsets such that
(i) Ω(f) = Sl
i=1 Bi(disjoint union),
(ii) f|Bi:Bi→Biis topologically transitive.
Such the subsets Biare called basic sets.
(2) (Decomposition theorem due to Bowen) For Ba basic set there
exist a > 0and a finite sequence Ci(0 ≤i≤a−1) of closed
subsets such that
(i) Ci∩Cj=∅(i6=j), f(Ci) = Ci+1 and fa(Ci) = Ci,
(ii) B=Sa−1
i=0 Ci,
(iii) fa
|Ci:Ci→Ciis topologically mixing,
Such the subsets Ciare called elementary sets.
Lemma 3.9 ([18] Lemma 5.4).Ω(f) = N/Γ.
Lemma 3.10.N/Γis indeed an elementary set.
Proof. By lemma 2.10, let f:N→Nbe the lift of fsuch that
f(e) = e. By the commuting diagram:
Nf
//
π

N
π

N/Γf
//N/Γ
we have,
f([e]) = f(π(e)) = π(f(e)) = π(e) = [e].
Therefore, [e] is a fixed point of f.
By lemma 3.9, Ω(f) = N/Γ. As Nis connected and πis a con-
tinuous surjection then N/Γ is connected. In the proof of part (1) of
spectral decomposition theorem, they prove that basic sets are close
and open. Hence by connectedness of Ω(f) = N/Γ, it consists of only
one basic set, say B. On the other hand, by part (2) of spectral de-
composition theorem, N/Γ = Bis the union of elementary sets. There
is an elementary set, say C, such that [e]∈C. Since elementary sets
are disjoint, by condition f(Ci) = Ci+1,N/Γ = Bconsists of only one
elementary set. 
Lemma 3.11 ([2] Remark 5.3.2 (2)).Let f:X→Xbe a T A-map
of a compact metric space and let Cbe an elementary set of f. If
˜u= (ui)∈Nfand xi∈Cfor all i∈Zthen Wu(˜x)∩Cis dense in C.
Lemma 3.12.Sγ∈ΓWu(γ;e)is dense in N.

CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 21
Proof. By lemma 2.17 and lemma 3.7 we have
(3.12) [
γ∈Γ
Wu(γ;e) = [
γ∈Γ
(Wu(e;e))γ=π−1(Wu(τe(e))).
We have τe(e) = π(e)∞
i=−∞ ∈(N/Γ)f. On the other hand, Since
by lemma 3.10, Ω(f) = N/Γ is an elementary set, say C, and for
π(e)∞
i=−∞ we have π(e)∈N/Γ = Cfor all i∈Z, by lemma 3.11 we
have
Wu(τe(e)) = Wu(τe(e)) ∩(N/Γ) = Wu(τe(e)) ∩C
is dense in C=N/Γ. By relation (3.12), we have the desired result. 
By lemma 3.6, h′is extended to a continuous map ˜
h:N→N.
From proposition 3.3 (1), (2) and (3), and lemma 2.13, we have h=˜
h
and h(γ) = γfor all γ∈Γ.
Proposition 3.13.For all γ∈Γand x∈N,h(xγ) = h(x)γ.
Proof. According to lemma 2.14.(4), we have
(3.13) h(xγ)∈Ls(h(x)γ).
Suppose that x∈Sγ∈ΓWu(γ;e). Then there is γx∈Γ such that
x∈Wu(γx;e). For each γ∈Γ we have
xγ ∈Wu(γx;e)γ=Wuγxγ;e.
Thus
h(xγ)∈hWuγxγ;e
(by Remark 2.16) = Luh(γxγ)
=Lu(γxγ).(3.14)
On the other hand,
h(x)γ∈h(Wu(γx;e))γ
(by Remark 2.16) = Lu(h(γx))γ
=Lu(γx)γ
(by Lemma 2.18) = Lu(γxγ).(3.15)
By (3.15), we have Lu(γxγ) = Luh(x)γ. Therefore, by (3.14) we have
(3.16) h(xγ)∈Luh(x)γ.
By (3.13) and (3.16) we have
(3.17) h(xγ)∈Luh(x)γ∩Lsh(x)γ={h(x)γ}.
Thus for each x∈Sγ∈ΓWu(γ;e) we have h(xγ) = h(x)γ. Since h
is continuous and Sγ∈ΓWu(γ;e) is dense in N, we have the desired
result. 

22 S.M.MOOSAVI AND K.TAJBAKHSH
The end of main theorem’s proof: According to proposition
3.13, hinduces a homeomorphism h:N/Γ→N/Γ such that h◦π=
π◦h. i.e. the following diagram commutes:
Nh//
π

N
π

N/Γh
//N/Γ
his the conjugacy between fand A. For if x∈N/Γ then there is
y∈Nsuch that x=π(y) and
h◦f(x) = h◦f(π(y)) = h(f◦π(y)) = h(π◦f(y))
=h◦π(f(y)) = π◦h(f(y)) = π(h◦f(y))
=π(A◦h(y)) = π◦A(h(y)) = A◦π(h(y))
=A(π◦h(y)) = A(h◦π(y)) = A◦h(π(y))
=A◦h(x).
So the Main Theorem is proved.
Proof of Corrollary 1.11. As mentioned in section 2, every endo-
morphism of a compact metric space is a covering map. Every Anosov
endomorphism is a T A-map (see [2] Theorem 1.2.1). Every diffeo-
morphism is special (since it is injective). For every diffepmrphism or
special expanding map of a nil-manifold, by (repaired for nil-manifolds)
Theorem 1.9, it is conjugate to a hyperbolic nil-automorphism or an
expanding nil-endomorphism, respectively, which are hyperbolic nil-
endomorphisms. In Theorem 1.10, we prove the case that fis not
injective or expanding. So in this case fis conjugate to a hyperbolic
nil-endomorphism too.
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CLASSIFICATION OF SPECIAL ANOSOV ENDOMORPHISMS 23
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Department of Mathematics, Faculty of Mathematical Sciences,
Tarbiat Modares University, Tehran 14115-134, Iran
E-mail address:seyedmohsen.moosavi@modares.ac.ir, smohsenmoosavi2009@gmail.com
Department of Mathematics, Faculty of Mathematical Sciences,
Tarbiat Modares University, Tehran 14115-134, Iran
E-mail address:khtajbakhsh@modares.ac.ir, arash@cnu.ac.kr