Endomorphisms of Compact Differentiable Manifolds
Author(s): Michael Shub
Source: American Journal of Mathematics, Vol. 91, No. 1 (Jan., 1969), pp. 175-199
Published by: The Johns Hopkins University Press
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ENDOMORPHISMS OF COMPACT DIFFERENTIABLE
MANIFOLDS.*
By MICHAEL S:EUB.
Introduction. A endomorphism of a differentiable manifold llI is a
differentiable function f: ll I -> M of class Cr, r ? 1. An endomorphism f is
an automorphism if f has a differentiable inverse: i. e., if f is a diffeomorphism.
The orbit of x C M relative to the endomorphism f is the subset (ftm (x) I m E Z+)
where Z+ denotes the integers ? 0. If fn (x) = x for some integer m> 0,
then x is called a periodic point and the minimal such m the period of x.
If the period of x is 1, then x is a fixed point. In (20) Smale has exhibited
very powerful techniques for the study of the global orbit structure of auto-
morphisms of compact manifolds. It is the purpose of this paper to provide
an introduction to a similar study of endomorphisms of compact manifolds.
The concepts and terminology of (20) are used throughout. A connected
finite dimensional C? manifold will be simply called a manifold.
The strongest useful equivalence relation for the study of the orbit
structure of endomorphisms appears to be topological conjugacy; f: l -> Mll
is topclogically conjugate to g: N -* N if there exists a homeomorphism
h: M -> N such that hf = gh. Let M be a compact manifold and Er (M) be
the space of Cr endomorphism of M with the topology of uniform convergence
of the first r derivatives; then f C Er (M) is called structurally stable if there
exists a neighborhood U of f in Er (M) such that any g C U is topologically
conjugate to f. Some of the most natural endomorphisms to investigate
which are not automorphisms are the endomorphisms of the complex numbers
of absolute value one: fn: S -> S1 defined by f (z) zn nE Z, I n I > 1.
We may naturally ask, are these fX structurally stable? The answer is yes,
alnd is given by the following more general proposition.
PROPOSITION. Let f: S1 3 S1 be a C1 endomorphism such that the
absolute value of the derivative of f is greater than 1 everywhere, then f is
to7opologically conjugate to fn , where n = deg f.
Received October 9, 1967.
* Partially submitted as a Ph. D. thesis.
175
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176
MICHAEL
SHUB.
The endomorphisms fn: S1 -> *1 are examples of expanding endomorphisms and the above proposition is a special case of a more general theorem.
Definition. Let Mi be a complete Riemannian Manifold, OM =+, then
a CO endomorphism f: M -> M is expanding if 3c > 0, X > 1 such that
11 Tfmv 11 ? cAm 1 v 11 for all v C TM and for all m C Z+ such that m > 0. (Note
that if M is compact this definition is independent of the metric.)
THEOREM (a). Any two homotopic expanding endomorphisms of a
compact manifold are topologically conjugate.
As the expanding endomorphisms of a compact manifold M are clearly
open in E1 (M) we have the following corollary.
COROLLARY. Any expanding endomorphism of a compact manifold is
structurally stable.
Examples of expanding endomorphisms as well as the proof of Theorem
(a) are given in I. Anosov endomorphisms are also considered there, and
new examples of Anosov diffeomorphisms are given. They are slight modifi-
cations of the general construction given in [20] and were suggested by the
situation for expanding endomorphisms. Smale [20] has used the notion
of expanding endomorphism to construct the D-E series of indecomposable
pieces of non-wandering sets and R. F. Williams [24] has considered
expanding endomorphisms of branched one-manifolds, called expansions, in
the study of one dimensional non-wandering sets of diffeomorphisms.
In 11 an extension of the 1?upka-Smale approximation theorem for
diffeomorphisms (see [12], [22], and [17]) is given.
Definition. Let x C M be a periodic point of period p for the endo-
morphisms f: MK - M. Then x is called hyperbolic if TfP: T,M -> T,,M has
no eigenvalue of absolute value 0 or 1.
If x is a hyperbolic periodic point of the endomorphism f: KM- K then
as in [22] for example we may define the local stable and local unstable
manifolds of f at p Wio0s(p) and Wl0,,u(p) respectively.
Definition. Let p be a hyperbolic periodic point of period j of f: Mll -* M
then:
1) The stable manifold of p, Ws(p) = (x E M I 3n such that
fni (x) C Wlocs (p) ) ; n C Z+.
2) The unstable manifold Wu(p) - (xC M I 3n and yE Wl0,u(p) such
that fni(y) =x); nCZ+.
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COMPACT DIFFERENTIABLE MANIFOLDS. 177
3) If q -is another hyperbolic periodic point such that W8 (q) is a 1-1
immersed submanifold of M then Wu (p) is transversal to We (q)
[we abbreviate this as Wu (p) 4 Ws (q) ]; if fni I WI.,u (p) is transversal to WS (q) on WI,,u (p) for all n E Z+.
THEOREM (p8). Let M be a compact manifold without boundary and let
KSr (M) be the set of f C Er (M) which satisfy:
I) The periodic poits of f are all hyperbolic.
2) If p is a periodic point of f, W1s8(p) is a 1-1 immnersed submanifold
3) If p and q are periodic points of f, Wu(p) ,fiWs(q).
of constant dimensionz.
Then KSr((M) is a Baire set in Er (M).
Since Diffr (M, M) is an open subset of Er (M) we have as an iirimediate corollary
COROLLARY. The Kupka-Smale automorphisms are a Baire set in
Diffr (M, M)1.
This corollary is the Kupka-Smale approximation theorem for diffeomorphisms.
In III we consider lnon-wandering sets of endomorphisms.
The period during which this paper was written was a period of great
activity in the study of global analysis at Berkeley which gave me the oppor-
tunity to speak to many mathematicians working on similar subjects. I would
particularly like to thank the following mathematicians for valuable conversa-
tions, suggestions, and encouragements: D. Epstein, M. Hirsch, N. EKopell,
I. Kupka, E. Lima, C. Moore, J. Palis, Ml. Peixoto, C. Pugh, J. Wolf, E.
Zeeman, and especially my advisor S. Smale.
I. Expanding endomorphisms. The beginning of this section is
devoted to a description of some of the geometric properties of expanding
endomorphisms on compact manifolds.
THEOREM 1. Let Ml be a compact manifold a2id f: 11- M an expanding
endomorphism. Then:
a) f has a fixed point.
b) The universal covering space of M1 is diffeomorphic to R18, n = dim M.
c) If V C 31 is acn open subset theni U fill(T) =3I.
in1 E Z+
12
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178 MICHAEL SHLUB.
d) The stable manifold of any periodic point of f i's dense in M.
e) The unstable manifold of any periodic point of f is all of M.
f) f has a dense orbit.
g) The periodic points of f are dense in lkl.
LEMMA 1. Let f: N-*N be expanding; then f is a covering map. In
particular, if N is simply connected, f is a diffeomorphism.
Proof. This is a consequence of (26), for instance.
LEMMA 2 (Contraction mapping Lemma). Let X be a complete metric
space and g: X - X a continuous map such that 3k, k C ER u < 1 such that
d(gn (x),gn(y))?k7und(x,y) for all x, yCX and for all n>0. Then g
has a unique fixed point.
A g satisfying the hypotheses of Lemma 2 is called a contraction mapping.
LEMMA 3. Let f: N->AN be an expanding diffeomorphism. Then f has
a unique fixed point.
Proof. f is a diffeomorphism so it has an inverse f-1. Since f is
expanding 3c > 0, X > 1 such that 11 Tf-n (v) 11 _CAn
I jj v 11 for all v C TN and
for all n > 0. Let d (x, y) be the metric on N induced by the Riemannian
metric on TN then d(f-n(x),f-n(y)) ? d(x, y) for all x,yCN and for
all n > 0 by the following argument. Let h: [0, 1] -> M such that h (0) x
and h (1) y be a minimal differentiable arc joining x and y, then
d(x,y) f h'(t) 11 dt and
d (f-n (X) .f-n (y)) 11 Tf-nh' (t) 11 dt_ c ( 1 h' (t) 11 dt.
Thus f-1: N -- N is a contraction mapping and f' has a unique fixed point
which is obviously also a unique fixed point for f.
M. Hirsch pointed out the following lemma to me.
LEMMA 4. Let f: N ->N be an expanding diffeomorphism and let c, X
and d be as is Lemma 3. Let x C N and Br (x) (y C N I d (x, y) C r); then
fn(Br(X)) D BcX-nr(fn(X) ).
Proof. Let h: [0, 1] -*fn (Br(X) ) be a minimal differentiable arc from
fn (x) to a (fn (Br (X)). Then f-nh: [0,1] Br (x) is an arc from x to 3Br (x)
so 1j Tf-nh'(t) 11 dt ? r and 4' h'(t) 11 dt _ cXnl .
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COMPACT DIFFERENTIABLE MANIFOLDS. 179
LEMMA 5. Let f: N-- N be an expanding diffeomorphism. Then N is
diffeomorphic to Rn, n = dim N.
Proof. Let x0 be the unique fixed point for both f and f-1. Then the
eigenvalues of Tf,-;: TxoN -- T0,N are all less than one in absolute value;
see (18, p. 425). Thus there is an embedding of the closed unit disk D-n C Rn,
i: Dn - N, such that i(O) - x0 and f-1 o i(Dn) C interor i(Dn). By Lemma 4
U fn(f-1 o i(Dn) ) is both open and closed in N, hence all of N, and N is the
n?O
expanding union of differentiably embedded disks, thus N is diffeomorphic
to Rn.
Notation. For jC ER+ let mj: R --Rn be the map mj(x) = jx and
mi: Rn-- Rn be the map mj o r where r is an orientation reversing reflection
of Rn.
The proof of the following proposition may essentially be found in [28]
or [22].
PROPOSITION 1. Let f: N-->N be an expanding diffeomorphism and
j > 1. Then f is topologically conjugate to mj if f is orientation preserving,
or to m^j if f is orientation reversing.
Notation. If 1M1 is a Riemannian manifold, its universal covering space
will be denoted by M with covering map P: Mf-- M and the Riemannian
metric which is the pull back of the Riemannian metric on M by P. Thus
the covering transformations of M are isometries. If M is complete so is M.
Given a continuous function a: N -- M a lifting of a is a continuous function a: 17 -> such that
PN
a
PM
N - -> M commutes.
It is well known from covering space theory that any a: N-- iM has a lifting
and if il and a2 are both liftings of a then there exists a unique covering
transformation ( of Mi such that =il ai
Let DN (Di) denote the group of covering transformations of N(M).
Then a continuous function /8: N -M A is a lifting if and only if there is a
group homomorphism h: DjW -- D- such that 380 h (p)13 for all 4 C D;, in
which case h will be denoted by p8#. A compact subset C of X1l will be called
a covering domain if C is the closure of its interior and P (C) Z M.
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180
MICHAEL
SHIUB.
LEMMA 6. Let f: M -->M be expanding and let f: M_-* M be a lifting
of f, then 7 is expanding.
PROPOSITION 2. Let f: M-* M be expanding. Then the universal
covering space of M is diffeomorphic to Rn, n = dim Ml, and f has a fixed point.
Proof. Lemmas 1, 3, 5, and 6, and the fact that if x0 is a fixed point
for a lifting 7 of f then P (x0) is a fixed point for f.
Proof of Theorem 1. (a) and (b) are special cases of Proposition 2.
Since the local stable manifold of a periodic point of an expanding endomorphism is the periodic point itself and the local unstable manifold is a
neighborhood of the point (c) implies (d) and (e). (c) also implies (f)
see Birkhoff [7]. It remains to show (c) and (g). Let C C X be a covering
domain and V C M an open set. Then ' =P-1 (v) n interior of C 7 0.
By Lemmas 4 and 6 for any lifting 7 of f, there exists an n > 0, a 0 C Dm
and a closed ball B C V' such that (1) 7n(B) D ?>(C), which proves (c).
Moreover, by (1) B D 7-p (C) and in particular B D f-r"j (B) which implies
3x C B such that x =f -p (x) or equivalently ?" (x) =cp (x). Thus f" (P (x))
=P (x), which proves (g).
PROPOSITION 3. Let f: Al -* AI be an expandintg endomor phism and
m CE31. Then:
(a) 3l is a K(7r1 (MI, m), 1).
(b) wi(Ml,m) is torsion free.
(c) f#: 7ri(JM,m)-7,(M,f(m)) is injective. Mloreover, if Ml is compact f# (rT (ll., (m) ) is of finite index in 7r, (M, f (m) ).
(d) If 31 is compact then X (A1) the Euler chacracteristic of Al equtals
zero.
Proof. (a) By Proposition 2, 1Ff is diffeomorphic to RB, n =dim ll so
for i > I -i (Ml, m) ~7ri ( it, m) w B(RBn,, =-0.
(b) By a theorem of P. A. Smith Z,, cannot act freely on Rn so 7ri (AI, m)
cannot have a torsion subgroup. An alternate proof is given in [15, p. 103].
(c) Lemma 1 says that f is a covering map.
(d) If M is compact, by Lemma 6 f is a kc sheeted covering map k > 1
and consequently X(MI) -IcX((M) = 0.
LEMMA 7. Let G be a group and c: G -> G a monomorphism. Let
00
Gckqn obJfl(G). Then p (Gm,) ==Gm.
m21=0
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COMPACT DIFFERENTIABLE MIANIFOLDS. 181
Proof. If hC EGQ, then hC E)n?(G) for all m, and hence + (h) C Om+' (G)
for all m ? . As +$(G) C (pi(G) for j < i, (GQ,) C n pm(G)-G(,. Now
m=O
if y C Ge,. there exists x' E (G) such that 0 (xi) = y, but as ( is injective
xi=xi for all iI i O and xi E G(,.
PROPOSITION 4. Let f: 321-11I be expanding and x0 a fixed point of f.
Then 7r,, =q n fe ((r1 (,xQ) ) =1.
in=0
Proof. Let Mr,0 be the covering space of MI corresponding to 7ITC. By
Proposition 3(c) f# is a monomorphism and by above Lemma f#( )=7r,,.
Therefore f lifts to an expanding diffeomorphism of Mr1, and and by Lemma
5, 31n is diffeomorphic to Rn. Thus r, 1.
COROLLARY 1. f#: 7r1 (If, xO) --->1 (211, x0) leaves no proper subgroup
invar-iant and in particular the identity is its unique fixed point.
Conjugacy theorems for expanding endomorphisms. I would like to
thank J. Tate who communicated to me a proof of Theorem (a) of the introduction for the case where the compact manifold is S1, which I had proven
geormetrically. Tate's methods were more amenable to generaization and
the proofs presented below are generalizations of his method. Throughout
this sction N and AM will be maniifolds; N will be compact and 2l will be
complete.
Definition. Let a: Nl-- 21I be a continuous function and oc: N -> f a
lifting of a. Then 'V, will be the set of continuous fulnctions B: N ->
which are liftings of continuous functions from N to M2 such that B# -.
If A,BE VF then D(A,B) ==-sup(d(A(n),B(n)).
nEN
PROPOSITION 5. D is a complete metriic on YV,x.
Proof. (a) D (A, B) <oo: let C be a covering domain in N. Then for
any x C N there exists a EC D13; such that 0 (x) C C. Thus d (A (x), B (x))
- d(A('-l0(x), B(-1%(x)) and as A# =B#= =#,
d (AO-lo (x) BO-10 (x) ) ==d (V (0-1) A (pa) (,~ x(0-1) Bo (X) )
But V#(0-1) C DX and is an1 isometry of M so d(A(x), B(x)) = d(Ao((x), Bo((x))
and sup (A(n),B(n)) sup(A (x) , B(x)). As C is compct, D(A,B) <m.
C1EN
nEC
(b) D is complete: if 1c1 is a Cauchy sequence in TV; then ki converges
to a continuous ftunction k: N9 X> f, and kic-> k- for any (p C DjN,. But
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182
MICHAEL
SHUB.
k(p k,# (c) k7# ((p) k,. Thus V (c) klc--c kc and as k -> k; V# (p) k-k =
and so 7c C V;.
THEOREM 2. Let f : M 31-> be expanding and let g: N-N and
a: N M be continuous. If there exists liftings J: A? -> ., ?: N N , and
d: JV A of f, g, and a respectively, such that fa -= V#P# then there exists
a unique B E VF such that fB Bg.
Proof. For A E Vc, define T(A) = fT'A#. It suffices to show that T is
a contraction mapping on Vii. B is then its unique fixed point.
(a) T(Ty;) C V7g: let AC EV, and p C DzN. Then
?-'A#) f (A5## (i) )AAg.
Since 7# is injective and
A#g# (qp) - #j#(() C Irf#; (7#-lA#g# () ) -'A# =-1(A #g# (,) )Ag
So j#-lA#g# (0)T-1Aflo. f= 7-AAgo. But j#-lA#g# -= -#-li## = #. Thus
7-'A#?=a #(?p)y-lAg and so f-lA# E VI'.
(b) T is a contraction mapping: As in Lemma 3 there exist c, A; 0 < c
and 1 <A such that d(7f-}(x),P-n (y)) _ 1 d(x, y) for all x,y E i and all
n > 0. Since D (Agn, Bgn) < D (A, B) for all A, B E VI, D (f7-AgtL, f-nBgn)
1
?-D (A, B) for all n > 0 and T is a contraction mapping.
THEOREM 3 (Theorem a of the Introduction). Let M1 be a compact
manifold and let f: M3-> M, g: M3 -*3M be two homotopic expanding endomorphisms. Then f and g are topologically conjugate.
Proof. Let m C M then there exists a path ,B from g (m) to f (m) such
that the map 13# 1.r1 (M, f (m) ) ->iri (M, g (m)): a /3 satisfies 13#fs = g#
see [23, p. 52]. Let f be any lifting of f; f: A- A?, and let eO E P-1(m)
and e1-==(eo). Let : i- > A? be the lifting of g which takes eo into the
endpoint of the lifting of the path 1-1 which starts at e,. Then it is easy to
check that f-# = #0. Thus Theorem 2 may be applied with a the identity of
31, and i the identity of A. Consequently there exist unique B1,B2E Vid
such that (1) fB1 =B,# and gB2=B2f. Thus B2fB1,= B2B1?7 and B1,B2
B1B2f. By applying (1) ?B1B2,=B1B2f and gB2B1.= B2B1?7. But B1B2
and B2B1 are elements of Vid and by the uniqueness part of Theorem 2,
B1B= B2BA = the identity of ff. B* is the lifting of a continuous function
hi: MIM->M, i= 12 which satisfy fh1=h1g and gh2=h2 and h1h2=h2h,
identity of M1.
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COMPACT DIFFERENTIABLE MANIFOLDS. 183
THEOREM 4. Let g: N -* N be a continuous function and have a fixed
point no, and let MO be a fixed point for the expanding endomorphism
f: M->M. There is a one to one correspondence between the continuous
functions h: N-> i?, such that h (no) = mo and fh = hg, and the group
homomorphisms A: 7r, (N, no) - 7r1 (M, in0) which make the following diagram
comm.ute:
A
7,(N, no)- > 7r1 , PpMO)-
A# f#
, A 11 ,
7Jl (tN, 110)- -? 7r, pip MO)
The correspondence is given by h -> h#.
Proof. Since 31 is a K(,r-(31,in0o) there exists a continuous function
a: N ->IIY such that a(no) =m- o and a#: 7rl(N,no)-->wr(M,mo) is equal to
A; see [23, p. 427]. Let fnI0EPM-1(mO) and io EPN'1(no). Lift f to
?: Al-> M such that f (ino) =no; lift g to g: 1i->RN such that g (AO) -nO
and lift a to i: A -> if such that a(Ao) =no. Then 7#5#= #g#. Let
T: VaT - Va1 be as in Theorem 2 and let V * (B E Va I B(io) = ffo). Then
VaT is a closed subset of V17 and T (VY*) C VI7* and the unique fixed point B
of Theorem 2 actually lies in Vt. B lifts a continuous function h: N->3M
such that h (no) = mo h# = A and fh = hg. If h1 is another continuous func-
tion such that fhl = hig, h, (no) = mo and hl# = A, then lift h1 to h,: N ->M
such that hi (o) =no; then hi# = i#. Since f,h1 = h,g, 7h, and h1g are
liftings of the same continuous function so there exists a E C D1: such that
h1=-- hi Thus pThl (O) =hig(ibo); ff(fno) =-o and (fno) =-- fO so
is the identity map of i. Consequently 7h, =- hig and h = B so h1 = h.
THEOREM 5. Let 31 and N be compact and let g: N->N and f: M3 - M
be expanding with fixed points nO of g and mo of f. There is a one to one
correspondence between the homeomorphisms h: N--> M, such that h(no) = mO
and fh =hg, and the group isomorphisms A: 7r, (N, no)) ->7 ((M nmo) which
Make the following diagram commute:
A
7T, (NT, n,,) - 7 r (>1",1 MO)
19# A jf#
The (cr, no) on c 7rs n b, hMO) -
The corriespondence is given by h-->h#.
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184
MrICHAEL
SHUB.
Proof. By Theorem 4 there exist unique h1: N - 31M and ht: 71 -l N
which are continuous and which satisfy h1 (no) =mo, h2 (mo) =no,
h1#: 7r1 (N, no) -7r, (M, mo)
is equal to A,
h2#: 7rl (M, mO) -*7r, (N, no)
is equal to A-1, fh, = h,g, and gh, = h,f. Consequently fhlh, = h1h2f,
(h1h2) (iO) =m mo, and (h1h2)#: irl (M1, M0) -i7rl (AI, MO) is the identity map.
By applying Theorem 4 again hlh., -- the identity of ll. Similarly h,hl = the
identity of N. Thus h, and h2 are homeomorphisms.
COROLLARY 2. Let AI be a compact manifold and let f: M-> M1 be
expanding. Then the centralizer of f in the monoid of continuous functions
from 31 to 31 is a countable discr ete subset.
Proof. f has a finite number of fixed points, and for any mo Al,
7r(1(M, mo) is finitely generated. Thus Hom (7r1 (3, no), 7rl (31, iml)) is countable. As any continuous map h: M -*31 such that fh = hf must take a fixed
point of f into a fixed point of f, Theorem 4 may be applied to the case N =1 ,
g = f.
In a similar way Theorem 5 may be used to determine the centralizer
of f in the group of homeomorphisms of l31. It is possible by using Theorem 2
to determine the algebraic structure of the centralizers for some cases.
Remark. Theorem 2 asserts in particular that given a continuous map
g: S1 .> S31 of degree n, I n I > 1, there exists a continuous map a: S1 *
of degree one such that f%a = ag, where f (z) = zn. a is, of course, not a
homeomorphism in general.
Examples. The examples given here are motivated by the exaniples of
Anosov diffeomorphisms given in [20, sec. I-3]. It would be very helpful
to the reader to be familiar with them. Recall that if H is a connected Lie
group arLd E: H -E H is an expanding group automorphism then by Lemma 5,
H is diffeomorphic to Rn. In particular, H is simply connected. And as in
[20, I-3. 6] H is nilpotent. Up to topological conjugacy all examples of
expanding maps on compact manifolds, that I know of, are given by the
following construction:
Let N be a connected, simply connected nilpotent Lie group, with a left
invariant metric. Let C be a compact group of automorphisms of N, and let
G = N* C, be the Lie group obtained by considering N as acting on itself
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COMPACT DIFFERENTIABLE MIANIFOLDS. 185
by left translation and taking the semi-direct product of N and C in Diff (NY).
Let r be a discrete uniform subgroup of G, such that N/r (the orbit space of
N under the action of r) is a compact manifold. Then by a theorem of
L. Auslander [5] r n N is a uniform discrete subgroup of N and r/r n N
is finite. Now let E: N -* N be an expanding group automorphism of N
such that by conjugating r by E in Diff (N) ErE-1 C r. Then E projects
to an expanding map of N/r; E0: N/r -4 N/r.
Remarks. (1) As r/rFn N is finite, C could have been assumed to be
finite to begin with, and as r n N is a uniform discrete subgroup of NV, N/r
is finitely covered by the nilmanifold N/r n N. Since E(N n r)E-C c N n r,
Eo lifts to an expanding map E1: N/r n N -> N/r n N. (Conjugating an
element n of N by E in Diff (N) gives E (N) ).
(2) N/r may be considered as the double coset space r\G/C.
(3) Let 0O-H --rF-> F--0 be an exact sequence of groups where H
is finitely generated, torsion free, discrete nilpotent group, and F is a finite
group such that in the induced action F acts effectively on H by isomorphisms.
Then by the results of Malcev [14] which are summarized in [6] or [20],
H may be embedded as a uniform discrete subgroup of a connected, simply
connected nilpotent Lie group N, and the action of F on H extends uniquely
to an effective action of F on N by Lie group automorphisms. Thus r may
be considered as a subgroup of N o F and if r acts freely on N, N/r is a
compact manifold.
Pr oblemt. Does the general construction above give all examples of
expanding endomorphisms of compact manifolds up to topological conjugacy?
I have tried to use Proposition 4 and other facts about the fun-damental
group of a compact manifold with an expanding endomorphism in conjunction
with Theorem 5 to show that it does. Starting with an entirely different
approach to the problem, M. Hirsch has shown me some very promising ideas
and partial results; but at this writing it remains unsolved.
The toral expanding endomorphisms. The n-Torus, Tn, may be con-
sidered as the quotient of the abelian Lie group Rn by the uniform discrete
subgroup Zn, the subgroup of points all of whose entries are integers. Let A
be an n by n matrix such that all the entries of A are integers and all the
eigenvalues of A are greater than one in absolute value. A may be thought
of as an expanding Lie group automorphism of Rn such that A (Zn) C Zn
(Or equivalently considering Zn as contained in Diff(RI), AZnA-' C Zn).
Then A projects to an expanding endomorphism of Tn; Ao: T --> Tn. Ao is
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186
MICHAEL
SHUB.
called a linear expanding endomorphism. Conversations with Al. Hirsch
were very helpful in proving the following:
PROPOSITION 6. Let f: Tn-* Tn be expanding then f is topologically
conjugate to a linerar expanding endomorphism.
Proof. Let 7: Rn- Rn be a lifting of f. Then '71: DRvn-* DR may be
considered as a group homomorphism of Zn C Rn, in which case 7# (z) = f(z);
for z E Zn. 7-0: Zn -* Zn may be written as an n by n matrix A, all the entries
of which are integers. A may be considered as a Lie group automorphism
(i. e., linear map) of Rn such that A I Zn =f7 Zn. Thus there exist C > 0,
A > 1 such that in the usual metric on Rn, 11 An(z) 11 > CAn 11 z 11 for all z E Zn.
By the linearity of A, 11 Al (y) 11 ? Cn 11 y 11 for all y E Rn and A is an expanding Lie group automorphism of RI, such that A (Zn) C Z". Thus A
projects to a linear expanding endomorphism Ao: T -> Tn. As A is a lifting
of Ao, and AO/= 7 the proof of Theorem 3 shows that Ao and ft are topologically conjugate.
RemafrA. A, is homotopic to f, and A: RA'-*1Rn is f*: H1(T,AR)
-- H1 (Tn, R). It should be noted, however, that two expanding endomorphisms which are topologically conjugate need not be homotopic.
PROPOSITION T. If M is a compact, connected (real analytic) Lie group
and if fi: Al -- AM is expanding, then Al is the n-Torus, T"n where n = dim M.
Proof. Al is of the form-D X where T is a Toroid, G a compact
semisimple group and D a finite central subgroup of T X G such that D n T
and D n G are trivial; see [10, p. 144]. Thus M is homeomorphic to P X G.
P is homeomorphic to Rm, m = dim T. and G is a compact semi-simple group.
But since ft: Ml -I Al is expanding, M is diffeomorphic to RA, n = dim N, thus
C7 is trivial and M== T.
If AM were not compact by using [10, p. 180] the same argument would
prove that Ml is homeomorphic to the direct product of a Torus and a Euclidean
space.
If C is a compact group of Lie group automorphisms of Rn and r is a
uniform discrete subgroup of C Rn, then RnA r is a compact manifold which
has a flat Riemannian metric. See [8] and [23] for general references.
The following is proven in [91.
THEOREM. If A is a compact flat Riemannitan manifold, then thtere
exists an expanding endomorphism of Ml.
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COMPACT DIFFERENTIABLE MANIFOLDS. 187
The situation when the group N is not abelian seems considerably more
complicated. I would like to thank C. Moore for pointing out the following
example to me. Let N be the simply connected non-abelian Lie group of
dimension three; that is, we may suppose that N is represented by the group
of lower triangular matrices
I1 0 O'
x 1 0 x,y,zER.
Let r be the uniform discrete subgroup of matrices of the form
'1 O O'
a: 1 0 -y E Z.
sY A Ij
If a, b, and c are integers which are greater than one in absolute value and
ab c, then the map
x ax
y by
z cz
defines an expanding Lie group automorphism of N which takes r into itself
and hence projects to an expanding endomorphism of N/r.
*Now let C be the compact group of Lie group automorphisms of N
consisting of the identity, and the automorphism A defined by A (x) -x,
1 O O'
A (y)- y, and A (z) = z. Let A1=, 0 1 0 A. Then the group
21 O 1
generated by A1 and r is a uniform discrete subgroup of N C; call it r1.
It is easy to check that rP r U A,r. If E: N -1 N is the expanding Lie
group automorphism defined by
x ax
y by
z cz
where a and b are odd integers greater than one in absolute value and c = ab,
then Er1E-1 C r1 and E projects to an expanding endomorphism of N/PJ.
N/rl is not a nilmanifold, since rP is torsion free. But
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188
0
MICHAEL
1
[l0
l
SHUB.
1
0]
of
102
1OI
0 A, A,.
O
~ 1 =A
0and in particular
1 O f1
2 1 0 Al ,A
l J l= J ll J J
and thus r1 is not nilpotent; see [12, p. 247].
Anosov endomorphisms. We begin by recalling some definitions from
[20]. A bundle map of a Riemannian vector space bundle sb: E -- E is called
contracting if there exists c > 0, 0 <X < 1 such that for all v C E and all
integers m > 0, 11 p02 (v) 1 < cAtm 1j v 11. ( will be called expanding if there
exists d > 0, u > 1 such that for all v C E and all integers m > 0, 11 I,,(v) 11
> du", 11 v 11. If E is a Riemannian vector bundle over a compact space, then
the property of contracting or expanding is independent of the metric. Let
M1 be a Rienmannian manifold; f: 31 - 3M an endomorphism; and A a subset
of M1 such that f(A) C A. Then f is said to be hyperbolic on A if TM I A is
the Whitney sum of two subbundles Es and Eu such that Tf (Es) C Es;
Tf(Eu) C Eu; Tf is contracting on Es; and Tf is expanding on Eu.
Definition. Let 31 be a complete Riemannian manifold without boundary.
f: M1- 3M is an Anosov endomorphism if f is an immersion, a covering map,
and f is hyperbolic on all of M.
Expanding endomorphisms and Anosov diffeomorphisms are, of course,
examples of Anosov endomorphisms. Anosov immersions can be constructed
ill a similar fashion to the Anosov diffeomorphisms constructed in [20].
THEOREM 6. The Anosov endomorphisms of a compact manifold Mi are
open in E(M1) and are structurally stable.
The proof of this Theorem is the same as the proof of the openness of
Anosov diffeomorphisms [20, 1-8] and J. Moser's proof of the stability of
Anosov diffeomorphisms, as exposited by J. Mather in the appendix to [201,
except for the following modification in the proof of the second statemlent.
Let Co (M1, M) denote the continuous functions from Ml to 31 and let OO(Sf-, ilt)
denote the continuous functions from M to M which are liftings of continuous
functions from lkl to M, both spaces with the compact open topology. Then
the natural projection map p: O? (M, M) -->Co (3131) is a covering map
and defines a Banach manifold structure on 00C(M, Vf). If 7 is any lifting
of f, one applies all the arguments of the appendix to the map h -471h,
h E CO (Mft, M) which is defined in a neighborhood of idj; anid then one
projects.
I would like to give two examples of Anosov diffeomorphisms of compact
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COMPACT DIFFERENTIABLE MANIFOLDS. 189
mianifolds which are not nilmianifolds. The examples were motivated by (*)
and are elaborations on [20, 1-3. 8]. Once again the general framework is:
r C N C is a uniform discrete subgroup; A is a hyperbolic Lie group auto-
morphism of N such that ArA-1 = r; and A projects to an Anosov diffeomorphism of N/r.
Let Q [V\3] be the field of rational numbers with the V/3 adjoined;
let Z [ V/ 3] be the algebraic integers in Q [ V/ 3], and let _--. Q [ -\/ 3 Q [V\ 3]
be the non-trivial Galois automorphism (sending V/ 3 into - V 3). For
a C Q [ V/ 3], o (a) will be written i. Let A=2 + V/ 3, theni AA 1. Finally,
let w C Q[V\3] be chosen such that w W Z[\/3], 2w C Z[ /3], and Aw -a + w
where a C Z[V\3]. It is easy to find such a w.
(1) Let N be the abelian Lie group R4, and let C be the compact group
of Lie group automorphisms of R 4 consisting of the identity and the map B,
B (a, b, c, d) = (- a, b, - c, d) for a, b, c, d C R. Let r be the uniform discrete
subgroup of R4 consisting of all elements of the form (Q, fl, i, /), a,: C Z [ \/3].
Let A: R 4-B R4 be defined by A(a,b,c,d) = (Aa,Ab,Xc,Xd). Let rF be the
uniforni discrete subgroup of N( C generated by r and B1, where B1
=- (0, w,0, wv)B. Then r == rU Blr and AF1A-1= r, so A projects to an
Anosov diffeomorphism of N/Fr, which is a Riemannianly flat compact four
manifold. AT/rF is not a nilmanifold, however, by the same argument as in
(e). ((1,,O,O)B,)2=B12.
(2) Uet N be the direct product of two copies of the lower triangular
3 by 3 matrices. For convenience we will write the elements of this group
as (a, b, c, d, e, f) a, b, c, d, e, f C B where the multiplication is:
(a. b, c, d, e, f) (a,, bi, cl, di, el, fl)
= (a+a1,b + b,,c+ bal+ cl,d+ di,e+ elpf+ edi+f1).
Let r be the unifornl discrete subgroup of N consisting of all elements of
the form 8 a, /,yCZ[V/3]. Let C be the compact group of
Lie group automorphisms of N consisting of the identity and B, where
B(a, b, c, d, e,.f) = (-a,-b, c,-d,-e,f). Let A be the hyperbolic Lie
group automorphism of N defined by A(a, b, c, d, e, f) = (Xa, A2b, Ac, Ad, X2e, eXf).
Let r1 be the uniform discrete subgroup of N- C generated by r and B1, where
B1 = (0, 0, w, 0, 0,w iv)B. rF = r U Bir and Ar1A-1 r1, so A projects to an
Anosov diffeomorphism of N/rl which is not a nilmanifold by the argument
in (*).
I would like to thank J. Palis for his help in calculating the above
examples.
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190
MICHAEL
SHUB.
II. Kupka-Smale endomorphisms. The proof of the Kupka-Smale
approximation theorem presented here is in its essentials a reworking of the
proof for automorphisms given in [22]. The stable and unstable manifolds for
endomorphisms are, however, more complicated than those for automorphisms,
and the proof of transversality theorems is correspondingly more difficult.
The Abraham transversality theorem is used to deal with the new difficulties.
In order to apply the theorem, some facts about and examples of Banach
manifolds are necessary. The following material may be found in [1], [2],
and [3].
Throughout this section N and M will be finite dimensional Coo manifolds
such that aM = 0 and N is compact. G will be a Banach Manifold. Let
Cr (N, M) denote the space of Cr functions, r> , from N to M11 with the
topology of uniform convergence of the first r derivatives. (Note that
Cr(M,M) =-Er(M)). If f: N->M is an element of Cr(N,M) then f*TM
is the vector bundle over N which is the pullback of the tangent bundle of Mll
by f. If E is a vector bundle over a compact manifold then rr (E) is the
Banach space of Cr sections of E with the topology of uniform convergence
of the first r derivatives.
THEOREM (1) see [1, 11.1]. Cr(N, M) is a Banach manifold, and the
tangent space at f C Cr(N, M), TfCr(N, M) may be identified with rr(f*TM).
Definition. Ev: Cr(N, M) X N--M is the map defined by Ev(f, n) = f(n).
THEOREM (2) see [1,11.6 and 11.7]. Ev: Cr(N,M) XN->M is Cr.
Moreover, if h C rr (f *TM) and z C TnN, then TEv(f,,,) (h, z) = Tfn (z) + h (n)
where h(n) is considered as an element of Tf(n)M.
Definition. Let G be a Banach manifold and let a: G - >Cr (N, M)
be a function. Then Ev: G X N -- M is the function Ev (a X idm).
Eva(g,n) =c,(g)n for all gC G and all nCN. If Ev is of class Cr, r>1,
x: G -* Cr (N, M) or briefly, (G, a) is called a manifold of mappings.
Examples. (1) By Theorem 3, if G is a submanifold of Cr(N, M) and
j: G -- Cr (N, M) is the inclusion map, then (G, j) is a manifold of mappings.
(2) Let M be compact. Let p: Er (M) -.> Er (M) be the map p (f) fP,
and let G be a submanifold of Er (M). Then (G, p I G) is a manifold of
mappings.
(3) Let Ml be compact. Let graphp: Er(M) --> Cr(M, M X M) be
defined by graphp (fi) = idm X fP. Then if G is a submanifold of Er (M),
(G, graphp I G) is a manifold of mappings.
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COMPACT DIFFERENTIABLE MANIFOLDS. 191
LEMMA 1. Let G be a submanifold of Er (M). Let f C G, and let m C M.
If hC TfG and zC TmM then:
p-l
(a) TEVP(t,n) (h, z) E , TffP-j(m) ih (fP-i-- (m)) + Tfmnp (z).
j=0
(b) TEVgraphp f m) (h, z) = (z, TEvp(f ,m) (h, z)).
Proof. It is sufficient to prove (a). We proceed by induction. For p 1
this is Theorem 3.
TEvp+(f,tm) (h, z)
p-l
TEV ff P(X)) (h, ,. Tffp-i(m) h (fP-j-1 (nm) ) + TfmnP (z) )
j=o
P-1
TffP (n) ( E TffP-J nm) 'h (fP-j-1 (m))) + h (fP (m)) + TmfP+l (z)
j=o
which by reindexing (j goes to j + 1) equals
p
El Tffp+l i-h (fp+l-i-l (m)) + TfmP+1 (z)
j=o
The Abraham transversality theorems. Let K (N) be the topological
space of compact subsets of N with the. weakest topology such that all the
compact subsets of an open set of N form an open set. Let &k(TM) be the
Grassmannian bundle of k planes in TM. Let Wk be the topological space
of closed kc dimensional sumanifolds of M with the weakest topology which
makes the map T: Wk -> k (TM), r (W) =TW for W C Wk continuous.
Recall from [13] that a differentiable map f: G -> M is transversal to WC Wk
at a point gc G if and only if f(g) f W or Tfg(TgG) +Tf(,)W-Tf(,)M.
fi: G -- M is said to be transversal to W C W7V on a subset K of G if f is transversal to W at each point of K.
With slight changes, the proofs of the following two theorems are given
in [1, 13.4 and 14.2] and [2].
OPENNESS THEOREM. Let ca: G -- Cr (NY, M) be a manifold of mappings.
Let A: G -- >K(N) and p: G -- Wk be continuous functions. Then Go
-(gc G I a (g) is transversal to +p(g) on /3(g)) is an open subset of G.
DENSITY THEOREM. Let K C K (N), W C W7, and let c: G -- Cr (N, M)
be a manifold of mappings. Let GK,W-==(gC G I (g) is transversal to W
on K). If Evag is transversal to W on G X K and if Ev, is Cr where
r> max(dimN+k-dimM, 0), then GK,W is an open and dense subset
of G.
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192
MICHAEL
SHUB.
Remark. In applying the transversality theorems to the Kupka-Smale
approximation theorem for endomorphisms the condition in the density
theorem r>max(dimN+k- dimM11,0) will be irrelevant since E,(M) is
dense in EP (M) for q > p and the openness theorem makes no assumption
on r other than r ? 1. In order to apply the transversality theorems to the
approximation theorem some lemmas on the transversality of Eva are needed.
Instead of writing f: G -> M1 is transversal to W on K we will write f 4 W
on K.
LEMMA 1. If a: G-* Cr(N, 91) is a manifold of mappings, and if
a (g) 4 W at m, then Eva + W I at (g, in). From now on let M be compact.
LEMMA 2. Let f C Er (M). Let m be a hyperbolic periodic point of f.
Then graphp (f)4 Amxm at mr, and EVgraphp \ AMXm at (f, m) for all positive
integers p.
LEMMA 3. Let f E Er (M), and let mC EM be a periodic point of f of
period p. Then EVgraphp 4 AmxM at (f, n).
Proof. Since p is the minimal positive integer n such that f(r(m) ~m,
if O<i, j p, then f (m) 54fi(m) when i=/=j. By Lemma 1(b),
p-l
TEVgraphp (h, z) (z, ,TffP-i (m) ih (fP-j-1 (m)) + TfmP (Z).
(f,m)
j=O
We may construct sections h E rr (f*TM) such that h (fP-l (in)) 0 for
j 0, and h (fP-1 (m)) considered as an element of T(mfi)M is any element we
wish. Letting z - 0, this argument shows that TEvgraphp (f ,)T(f,m)Er (31) X 31
contains 0 X T,,M. Thus Evgraphp i AmXm at (f, i).
Definition. Let ON = 0, and let V C N be a compact submanifold,
perhaps with boundary. Then R: Cr (N, lM) > Cr (V,17,1) is the restriction
map, R1(f) ==-f I V for all f C Cr(N,M).
LEMMA 4. (Er (M), R op) is a manifold of mappings. Mor-eover, let
V and W1 be closed subinanifolds of N such that V4 W, and let g C Er(M)5),
then:
(a) Let mC EV such that gi (m) X=/ gk(m) for 0? j < k p -1 and
such that gP-1(m) f W1. Then EvROp 0 W at (g,rm).
(b) Let g have maximal rank ont W and let g(W) C W. Let n C V.
Then, if gi(m) =gk(m) for 0?-< k?p-1 implies that gi(m) C W for
all i j, EvR0p 4 W at (g,m).
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COMPACT DIFFERENTIABLE MANIFOLDS. 193
Proof. That (Er(M),Rop) is a manifold of mappings follows from
Theorem 2 and the fact that EVRO, = Evp I Er (M) X V. The proof of
(a) is the same as the proof of Lemma 3. To prove (b) notice that by the
hypotheses we may assume that there exists an n < p such that g (Mn) , W.
Let s be the maximal such n. Then by (a) EVROS+1 4 W at g (n). By Lemma
1 TEvR0p = TgP-81 (TEvR08+l) +terms which can be made equal to zero
by the proper choice of h's and z's, as in Lemma 3. Since g has maximal
rank on W, EvROV4 W at (g,m).
Definition. Let T.= (f E Er(M)I)graph fP40 Amxm on all of M). Let
Hp = (f E Tv I all periodic points of f of period < p are hyperbolic).
The following proposition is well known.
n
PROPOSITION 1. Let f C n Tk7, n < oo. Then f has a finite number of
periodic points of period _<nn; r1,. - ,,8,. Moreover, there exist neighborhoods Uj of the fj and W of f such that U1 n Uk = 0 for j # k, which satisfy
n
the following condition. If g c W, then g Fn
Tk and g has precisely i
7C=1
periodic points of period ? n, a , * such that aj C Uj and period
cj 1= period /3j.
THEOREM 1. T1p and H. are open and dense subsets of Er (M).
Proof. By Lemma 3 and the transversality theorems, T1, is open and
dense in Er (M). By the proof of [22, 5. 1] H, is open and dense in T1.
We proceed by induction, assuming that H., is open and dense in Er (M).
We will show that T,,+1 nrH is open and dense in Er (M). Then, by the
openness of transversality, T,+, is open and dense in Er (M) and once again
by the proof of [22, 5. 1] H1p+1 is open and dense in T,+1. Let f C H1. If mn
is a periodic point of f of period p + 1, then by Lemma 3 EVgraphP+i 40 AMXM
at (f, in). If ft' (n) = m, and m is not of period p + 1 then m is of period
kc <p + 1 and n is a hyperbolic periodic point of f. By Lemma 2,
EVgraphb+1 40 Amxm. Thus if f E H1p, Evgraphp,j l4 AmXm on all of M, and by the
transversality theorems Tp+il nHp is open and dense in H,,.
Definition. Let f E Er (M)A, and let mn be a hyperbolic periodic point of f.
Then if dim Wlocu (in) = dim MA, n is called a source. If 0 <dim W1 (m)
< dim M, m is called a saddle. And if dim W1e (m) =0, in is called a sink.
Remark. It is well known that the local stable and unstable manifolds
vary continuously in the sense of the transversality theorems.
13
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194
MICHAEL
SHUB.
PROPOSITION 2. There exist open and dense subsets Gp C H, and con-
tinuous functions EjP: G, --1 R+ for 0 < j C p such that:
(a) G.+, C Ga.
(b) EjP -EjP+l 0<j p.
(c) If fE G. and m is a saddle point of period j?p then Wl12 (m),
W1j.' (m) contain closed Ej (f) disks centered at m, called Lfs (in) and
Lf'u(m) respectively, such that Wl0u (M) D f2j (Lfu (m)i).
(d) Iff E G,+1, no periodic point of f of period less than or equal to
p + 1 is contained in fi (Lfu (m) ) where period m = j < p and m is a saddle,
except m itself.
(e) Let f C G,, and let n and m be saddle points of f of period j and k
respectively. Then (Lf8(n) U fi(Lfu(n)) n (Lfs(m) U fk(Lfu(m)) 0 unless
n m, in which case fi(Lfu(n)) nLfLs(n) mn.
Proof. By the proof of the corresponding proposition [22, 6. la] it is
sufficient to verify inductively that the subset of G. n HE+1 which satisfies
(d) is open and dense. It is clearly open. So we must prove density. It is
clear that G1 may be taken equal to H1. Let n be a periodic point of period
p+ 1 of f C Gp nH E+1. Let m be a saddle point of f of period j?p such
that n C fi(Lfu(m) ). Let k be the largest integer O < k ?p such that fk(n)
is not an element of any Lfu (w) where period w ? p. Such a k exists since
if n C Lfu (m) then fui (n) C fi (Lfu (m) ) -Lfu (m) for some positive integer i.
Now by a standard argument there is an arbitrarily small perturbation of f
defined in a compact neighborhood V of fI(n) such that V does not intersect
any Lfu(w) where period w ? p and such that the orbit of the new periodic
point in V does not intersect any Lfu(w). The continuity of E1? finishes the
argument.
LEMMA 5. Let f C G,, and let m be a periodic point of f of period j.
i-l
Let V be a closed submanifold of Mi such that V4+ U ft (Lf8 (n)). Then
J=o
EVRokj 4 Lf,s (m) at (f, q) for any positive integer kc and any q C V.
Proof. By definition of Lf-(m), fi(Lft(m)) C Lfs(m) f has maximal
rank on Lfs(m), and m is the only periodic point in Lfs(m). If fjk(q) - fin(q)
for 7k> n, then fjk-jn(fin(q)) fin(q). Hence either fik(q) , Lf8(m) for
any k, in which case we are finished, or fik(q) = m for all kc n. Then we
j=o
may apply Lemma 4(b) to Ufi(Lfs(m)), and EvpO kj Lfs(m) at (f,q).
i-l
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COMPACT DIFFERENTIABLE MANIFOLDS. 195
PROPOSITION 3. Let Dp' C GP be the set of f C GP such that if m C M
is a periodic point of f of period j ? p then fiki Lfs (m) on M for all jk Cp.
Then Dp' is open and dense in G,.
Proof. Let W be the neighborhood of f given by Proposition 1. By
Lemma 5 for any fixed k EvJk4iLf8(m) at (f, q) for any q C M. Thus the
h C W such that hik Lf 8((m) on M are dense in W and by the openness of
transversality theorem, if h is close enough to f, then hik4Lhs(a) on MI.
Where a is the point corresponding to m, given in Proposition 1. Since f has
a finite number of periodic points of period ? p and since p is finite, Dp'
is dense in Gp. .By the openness of transversality theorem D,P' is also open
in Gp.
PROPOSITION 4. Let D,2 C G. be the set of f C Gp which satisfy the
following condition. If m and q are saddle points of f of period - p, and
period q-=j, then fik I Lfu(q) Lfs8(m) on Lf"(q) for all jklCp. Then
Dp2 is open and dense in Gp.
Proof. Proposition 2 allows us to use Lemma 5 again, and the proof
proceeds as the proof of Proposition 3.
COROLLARY. Dp Dpl n'Dp2 is open and dense in Gp and consequently,
00
D,o n_K Dp is a Baire set in Er(M).
p=1
THEOREM 3. (Theorem A of the Introduction). KSr(M) D Dx.
00
Proof. (1) Since Dc, C n 1p, if f C D,o all the periodic points of t
are hyperbolic.
(2) Let m be a periodic point of f of period j. Let L(i(m) be the open
E?v disk contained in Lfs (m) for any p > j. Then any point q C WS (m) is
contained in an inverse image (fik) -'Ls (m) which by Proposition 3 is a sub-
manifold of M. If q C (fik)-lLs(m) and q C (fin)-lLs(m) then fi(k+n)(q) C Ls(m).
fi(k+Ifn)iLs(m) at q; and thus (fik) lLs(m) and (fin)-'Ls(m) coincide on a
neighborhood of q. So Ws (m) is a 1-1 immersed submanifold of constant
dimension equal to dimension Ls (n).
(3) Let xC Ws(m) n Wu(q).
(a) If q is a sink then Wu'(q) =q and Ws(m) contains a periodic
point. Thus m q and Wu(q)iWs (m).
(b) Let q be a source. Let period q j and let period m = i. Then
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196
MICHAEL
SHUB.
x - Pik (z) for some positive integer kc and some z C Lfu (q). There exists an
n such that ftn(x) C Lf8(m). Since fC D,', finjLft8(m) at x; and since
f C Dftjk', flin k4Lf81(m) at z. As
Tf: TW8 (m) - Tf m(_) Ws (m);
filk cW8(m) at z. Thus Wi(q)4&W9(m).
(c) If m and q are both saddle points then the same argument applies,
since f E Dqfjk2. Thus Wu(q) 4 Ws(m).
(d) If m is a sink, then We (m) has top dimension, so WI (q) 4 W8(m).
(e) If m is a source, then q must be a sink since f C Dp2 for any p.
Thus by (a) Wu (q) 4 Ws (m).
Examples. Expanding endomorphisms are examples of K-S endomorphisms for which the unstable manifold of the periodic points is the entire
manifold M. A contracting endomorphism need not be K-S; for instance let
pCM and f: M->M be the constant map f(m) =p for all mCM. As any
sufficiently small K-St approximation of f is contracting it has a unique periodic
point the stable manifold of which is all of M. More generally any compact
submanifold of M with trivial normal bundle can be an arc component of the
stable manifold of a periodic point of a K-S endomorphism as follows:
Let V be the submanifold and N (1V) its embedded normal bundle. By
the triviality of N (V) there exists a differentiable map g: N (V) >RK
K = dim M -dim V, such that g (V) =0 and 0 is a regular value of g. Now
let U be the domain of a chart of M, : U-* Rm, such that UC N(NV) =0,
=0(p)= for p C U and + is onto. Consider Rm as RK X Rm-K and define
L: RPm>Rm by L(x,y) =- (2x,y/2). Let i: RK RK X Rm-K be the map
i(x) *G (x, 1). Now
f(x) J+-1Lfr(x) ;xC U
1 v-lig (x) ; x E N (V)
defines a differentiable map of N(V) U U into U with p as the unique periodic
point and V on the stable manifold of p. In fact, fim j W10s (p) on V.
fl, perhaps restricted to a slightly smaller set, can be extended to an endomorphism f2 of M. Let f3 be any sufficiently small K-S approximation of f2Then there is an embedding j: V17 M isotopic to the inclusion of V and a
periodic point q of f3 close to p such that J(V) is contained in WqS and
dim Ws(q) = dim V. Let h: M-> M be a diffeomorphisin of M1 such that
h 1 V= j. Then f =- h-Lf3h is a K-S endomorphism with V in the stable
manifold of h-1(q). These examples show, in particular, that while WS(h1l(q))
is an immersed submanifold f I Ws (h1 (q)) is not, in general, an immersion.
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COMPACT DIFFERENTIABLE MANIFOLDS. 197
III. Non-wandering sets of endomorphisms. Recall that a property
of endomorphisms is called generic, if it is true for a Baire set of E' (M).
As above, M will denote a compact manifold without boundary.
Iff C E' 1(M), x C M is called a wandering point for f if there is a neighborhood U of x such that U fm(U) n U 0. A point is called non-wandering
mo>
if it is not wandering. The non-wandering points form a closed subset of M
which will be denoted by 12 (f) or simply by a if no confusion is possible.
It is clear that f (s) C 02, but as f of a wandering point need not be wandering
it is not immediate (as it is for automorphisms) that f(0) D2.
Problem. Does f (Q) = Q, if not always at least generically?
The second part of this problem would follow from the next problem,
which is a generalization of [29] to endomorphisms.
Problem. Let P P(f) be the set of periodic points of f. Is P =
a generic property of endomorphisms?
Example. Let f: Sl -Sl be described by the foHowing diagram:
B
(;~Da
A is the western hemisphere and B, C, and D are each one-third of the eastern
hemisphere. f is an immersion of degree two with one fixed sink x, and two
fixed soucres y and z. f (A) = A and the interior of A is on the stable manifold of x. B is taken linearly by f onto BCD. C is taken linearly onto A
and D is taken linearly onto BCD. It is very easy to see that any point other
than x in W = U hn (Interior A) is wandering, and that the complement of
n2:0
W = WC is non-wandering. WC is the Cantor middle third set of BCD. A
better description of WC and f I We is given by the description of Group 0
in [20; I. 9. 4] Map WO into the Cantor set. aoa1a2' , aj = 0 or 1, by
a (x) = 0 if f (x) C B and a (x) = 1 if f (x) C D. This correspondence is a
homeomorphism and transforms f equivariantly into the map
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198
MICHAEL
SHUB.
aoala2a3 * * -* * aja2a3 *
It follows that P (f) = (f). Moreover, f is hyperbolic on W?. In fact,
Tf is expanding on TS' I We.
It is possible to extend more of the results about automorphisms to
endomorphisms. For example, the rationality of the zeta-function studied in
[4] for the case of a sink [25] applies equally well to endomorphisms. There
is, however, as yet no satisfactory version of the Q-stability or spectral decom-
position theorems of [20].
UNIVERSITY OF CALIFORNIA, BERKELEY.
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COMPACT DIFFERENTIABLE MANIFOLDS. 199
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