Topology and Homoclinic Trajectories of Discrete Dynamical Systems

0.25 MB·English

by  Jacobo Pejsachowicz

TOPOLOGY AND HOMOCLINIC TRAJECTORIES OF
DISCRETE DYNAMICAL SYSTEMS
2
Dedicated to PetrP. Zabrejko
1
0
2 Abstract. Weshowthatnontrivialhomoclinictrajectoriesofafamilyofdis-
n crete, nonautonomous, asymptotically hyperbolic systems parametrized by a
a circle bifurcate from a stationary solution if the asymptotic stable bundles
J Es(+∞)andEs(−∞)ofthelinearizationatthestationarybrancharetwisted
indifferentways.
0
3

] Jacobo Pejsachowicz
S
D DipartamentodiMatematica
PolitecnicodiTorino
.
h CorsoDucaDegliAbruzzi24
t 10129Torino,Italy
a
m Robert Skiba
[ FacultyofMathematics andComputerScience

NicolausCopernicusUniversity
4
Chopina12/18, 87-100Torun´,Poland
v
2
0 1. Introduction
4
1 In this paper we will investigate the birth of homoclinic trajectories of discrete
. nonautonomousdynamicalsystemsfromthepointofviewoftopologicalbifurcation
1
1 theory. This means thatinsteadofprovingthe existence ofahomoclinic trajectory
1 for a single dynamical system we will consider a one parameter family of discrete
1 nonautonomous dynamical systems on RN having x≡0 as a stationary trajectory
:
v andshowthat,underappropriateconditions,thedynamicalsystemswithparameter
i values close to a given point, must have trajectories homoclinic to 0. Values of the
X
parameter for which this occurs are called bifurcation points.
r
Bifurcation theory for various types of bounded solutions of discrete nonauton-
a
omousdynamicalsystemshavebeenstudiedin[10,21]andmorerecentlyin[18,19].
However our approach is different. We will not look for homoclinics bifurcating at
avalueofthe parametergivenaprioributinsteadwe willdiscussthe appearanceof
homoclinic solutions forced by the asymptotic behavior of the family of linearized
equations at 0.
When the family is asymptotically hyperbolic, the asymptotic stable and unsta-
ble subspaces of the linearized equations form vector bundles over the parameter
space which might be nontrivial when the parameter space carries some nontrivial
topology. We will show that homoclinic trajectories bifurcate from a stationary
1991 Mathematics Subject Classification. Primary: 39A28; 34C23, 58E07; Secondary: 37G20
47A53.
Key words and phrases. Discretedynamical systems,Homoclinics,Bifurcation,Indexbundle,
Fredholmmaps.
ThefirstauthorissupportedbyMIUR-PRIN2009-Metodivariazionalietopologicineifenomeni
nonlineari. Thesecondauthor issupportedinpartbyPolishscientificgrantNN201395137.
1
2 JACOBOPEJSACHOWICZANDROBERTSKIBA
solutionifthe asymptoticstablebundles Es(+∞)andEs(−∞)ofthe linearization
along the stationary branch are ”twisted” in different ways.
Our results require methods going beyond the classical Lyapunov-Schmidt re-
ductionandspectralanalysisata potentialbifurcationpoint. While similarresults
can be proved for general parameter spaces using more sophisticated technology
from algebraic topology, here we will concentrate to on the simplest topologically
nontrivial parameter space, the circle. This is equivalent to considering families of
dynamical systems parametrized by an interval [a,b] with the assumption that the
systems at a and b are the same.
Roughly speaking, we will first translate the problem of bifurcation of homo-
clinic trajectoriesinto a problemofbifurcationfroma trivialbranchof zeroesfor a
parametrizedfamily ofC1-Fredholmmaps. Thenwe willconsider the index bundle
ofthe familyoflinearizationsatpointsofthe trivialbranchgivenbythestationary
solutions of the equation. The index bundle of a family of Fredholm operators is a
refinement of the ordinary index of a Fredholm operator which takes into account
the topology of the parameter space. A special ”homotopy variance” property
of the topological degree for C1-Fredholm maps, constructed in [17] relates the
nonorientability of the index bundle to bifurcation of zeroes. On the other hand,
anelementary index theorem, Theorem4.1, allows us to compute the index bundle
interms ofthe asymptoticstable bundles ofthe linearizedproblem,relatinginthis
waythe appearanceofhomoclinics to the asymptotic behaviorofcoefficients ofthe
linearized equations. The precise result is stated in Theorem 2.3 of Section 2. An
analogous approach applied to nonautonomous differential equations can be found
in [14, 15].
The paper is organizedas follows. In the next section we introduce the problem
and state our main result. In Section 3 we recall the concept of the index bundle
and discuss its orientability. In the fourth section, we compute the index bun-
dle of the family of operators associated to a family of asymptotically hyperbolic
nonautonomousdynamicalsystems. InSection 5,we discussthe parity ofa pathof
Fredholmoperatorsofindex0,andwerecalltheconstructionin[17]ofatopological
degreetheoryforC1-Fredholmmapsofindex0extendingtoproperFredholmmaps
the well known Leray-Schauder degree. In Section 6, using the computation of the
index bundle and the homotopy property of the topological degree constructed in
[17] we prove Theorem 2.3. In the seventh section, we illustrate Theorem 2.3 with
a non-trivialexample. Section 8 is devotedto comments and possible extensions of
our results.
2. The main result
A nonautonomous discrete dynamical system on RN is defined by a doubly in-
finite sequence of maps f = {f : RN → RN | n ∈ Z}. A trajectory of the system
n
f: Z×RN →RN is a sequence x=(x ) such that
n
(1) x =f (x ).
n+1 n n
In the terminology of [18] (1) is a nonautonomous difference equation whose
solutions are trajectories of the corresponding dynamical system.
In what follows we will always assume that the f are C1 and that f (0) = 0.
n n
Under this assumption the system has a stationary trajectory x = 0, where 0 is a
sequence of zeroes. A trajectory x = (x ) of f is called homoclinic to 0, or simply
n
a homoclinic trajectory, if lim x = 0. Under our assumptions the system f has
n
n→±∞
TOPOLOGY AND HOMOCLINIC TRAJECTORIES OF DISCRETE DYNAMICAL SYSTEMS 3
alwaysa trivialhomoclinictrajectory. Namely, the stationarytrajectory0.We will
look for nontrivial homoclinic trajectories.
A natural function space for the study of homoclinic trajectories is the Banach
space
c(RN):={x: Z→RN | lim x =0}
n
|n|→∞
equipped with the norm kxk:= supk∈Z|xn|. Any homoclinic trajectory of f is
naturally an element of this space. Moreover, each dynamical system f induces a
nonlinear Nemytskii (substitution) operator
(2) F: c(RN)→c(RN)
defined by F(x) = (f (x )). Under some natural assumptions (see below) F be-
n n
comes C1-map such that F(0) = 0. In this way nontrivial homoclinic trajectories
become the nontrivial solutions of the equation Sx−F(x)=0, where
S: c(RN)→c(RN)
is the shift operator S(x)=(x ).
n+1
The linearization of the system f at the stationary solution 0 is the nonau-
tonomous linear dynamical system a: Z×RN → RN defined by the sequence of
matrices (a ) ∈ RN×N, with a = Df (0). The corresponding linear difference
n n n
equation is
(3) x =a x .
n+1 n n
If f =f, for all n∈Z, the systemis calledautonomous. We will dealonly with
n
discrete nonautonomous dynamical systems whose linearization at 0 is asymptotic
for n→±∞ to an autonomous linear dynamical system associatedto a hyperbolic
matrix. We will call systems with this property asymptotically hyperbolic.
Letusrecallthataninvertiblematrixaiscalledhyperbolicifahasnoeigenvalues
of norm one, i.e., σ(a)∩{|z|=1}=∅. The spectrum σ(a) of an hyperbolic matrix
a consists of two disjoint closed subsets σ(a) ∩{|z| < 1} and σ(a)∩ {|z| > 1},
so RN has the a-invariant spectral decomposition RN = Es(a) ⊕ Eu(a), where
Es(a) (respectively Eu(a)) is the real part of sum of the generalized eigenspaces
corresponding to the part of the spectrum of a inside the unit disk (respectively
outsideoftheunitdisk). Itiseasytoseethatζ ∈Es(a)ifandonlyif lim anζ =0.
n→∞
The unstable subspace Eu(a) has a similar characterization, i.e., ζ ∈ Eu(a) if and
only if lim a−nζ =0.
n→∞
When the linearized system is asymptotically hyperbolic, the map G = S −F
becomes a Fredholm map (at least in a neighborhood of 0) which will allow us to
apply the results of general bifurcation theory for Fredholm maps to our problem
by relating the corresponding bifurcation invariants to the asymptotic behavior of
the linearization at ±∞.
Let us describe precisely our setting and assumptions.
A continuous family of C1-dynamical systems parametrized by the unit circle S1
is a sequence of maps
(4) f={f : S1×RN →RN |n∈Z}
n
such that f is differentiable with respect to the second variable and, for all n ∈
n
∂jf
Z, 0≤j ≤1, the map (λ,x)7→ n(λ,x) is continuous.
∂xj
4 JACOBOPEJSACHOWICZANDROBERTSKIBA
To put it shortly, a continuous family of C1-dynamical systems is a continuous
map
f: Z×S1×RN →RN,
differentiableinthethirdvariableandsuchthatallthepartialsdependcontinuously
on (λ,x). We will use f to denote the dynamical system corresponding to the
λ
parameter value λ.
Remark2.1. Alternativelyonecanthinkoffasadoubleinfinitesequenceofmaps
f : [a,b]×RN →RN, such that f (a,x)=f (b,x) for all n∈Z.
n n n
Pairs (λ,x) which solve the parameter-dependent difference equation:
(5) x =f (λ,x ), for all n∈Z,
n+1 n n
will be called homoclinic solutions. Equivalently, (λ,x) is a homoclinic solution of
(5) if x=(x ) is a homoclinic trajectory of the dynamical system f .
n λ
Homoclinicsolutionsof(5)oftheform(λ,0)arecalledtrivialandthesetS1×{0}
iscalledthetrivial or stationary branch. We areinterestedinnontrivialhomoclinic
solutions.
We will assume that the family f: Z×S1 ×RN → RN of dynamical systems
satisfies the following conditions:
(A0) For all λ∈S1 and n∈Z, f (λ,0)=0.
n
(A1) ForanyM >0andε>0thereexistsaδ >0suchthatforall(λ,x),(µ,y)∈
S1×B¯(0,M) (1) with d (λ,x),(µ,y) <δ and all j, 0≤j ≤1,
(cid:0) (cid:1)
∂jf ∂jf
n n
sup (λ,x)− (µ,y) <ε.
n∈Z(cid:13) ∂xj ∂xj (cid:13)
(cid:13) (cid:13)
Here d is the pro(cid:13)duct distance in the metr(cid:13)ic space S1×RN.
(cid:13) (cid:13)
(A2) For all bounded Ω⊂(cid:13)S1×RN one has (cid:13)
∂f
n
sup (λ,x) <∞.
(n,λ,x)∈Z×Ω(cid:13) ∂x (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
∂fn (cid:13) (cid:13)
(A3) Letan(λ):= (λ,0).Asn→(cid:13) ±∞thef(cid:13)amilyofmatricesan(λ)converges
∂x
uniformlytoafamilyofhyperbolicmatricesa(λ,±∞). Moreover,forsome,
andhence forall λ∈S1, a(λ,+∞)and a(λ,−∞)havethe same number of
eigenvalues (counting algebraic multiplicities) inside of the unit disk.
(A4) There exists λ ∈S1 such that
0
(6) x =a (λ )x
n+1 n 0 n
admits only the trivial solution (xn ≡0)n∈Z.
By (A3) the map λ → a(λ,±∞) is a continuous family of hyperbolic matrices.
Since there are no eigenvalues of a(λ,±∞) on the unit circle, the projectors to the
spectral subspaces corresponding to the spectrum inside and outside the unit disk
dependcontinuouslyontheparameterλ(see[12]). Itiswellknownthattheimages
of a continuous family of projectorsform a vector bundle over the parameterspace
[13]. Therefore, the vector spaces Es(λ,±∞) and Eu(λ,±∞) whose elements are
the generalizedrealeigenvectorsofa(λ,±∞)correspondingto the eigenvalueswith
1Given anormedspaceE, B¯(x,r)andB(x,r), wherex∈Eandr>0,denote the closedand
openballaroundxofradiusr inE,respectively.
TOPOLOGY AND HOMOCLINIC TRAJECTORIES OF DISCRETE DYNAMICAL SYSTEMS 5
absolute value smaller (respectively greater) than 1 are fibers of a pair of vector
bundles Es(±∞)andEu(±∞) overS1 which decomposethe trivialbundle Θ(RN)
with fiber RN into a direct sum:
(7) Es(±∞)⊕Eu(±∞)=Θ(RN).
In what follows Es(±∞) and Eu(±∞) will be calledstable and unstable asymp-
totic bundles at ±∞.
Ourmaintheoremrelatestheappearanceofhomoclinicsolutionstothetopology
of the asymptotic stable bundles Es(±∞). Due to relation(7) the considerationof
the unstable bundles would give the same result.
In what follows, for notational reasons,it will be convenient for us to work with
the multiplicative groupZ ={1,−1}insteadof the standardadditive Z ={0,1}.
2 2
A vector bundle over S1 is orientable if and only if it is trivial, i.e., isomorphic
to a product S1×Rk. Moreover,whether a givenvectorbundle E overS1 is trivial
or is not is determined by a topological invariant w (E)∈Z .
1 2
In order to define w (E) let us identify S1 with the quotient of an interval
1
I =[a,b]byitsboundary∂I ={a,b}.Ifp: [a,b]→S1 =I/∂I istheprojection,the
pullback bundle p∗E =E′ is the vector bundle over I with fibers E′ =E . Since
t p(t)
I is contractible to a point, E′ is trivial and the choice of an isomorphism between
E′ and the product bundle provides E′ with a frame, i.e., a basis {e (t),...,e (t)}
1 k
of E′ continuously depending on t. Since E′ = E = E = E′, {e (a) | 1 ≤
t a p(a) p(b) b i
i ≤ k} and {e (b) | 1 ≤ i ≤ k} are two bases of the same vector space. We define
i
w (E)∈Z by
1 2
(8) w (E):=signdetC,
1
where C is the matrix expressing the basis {e (b)|1≤i≤k} in terms of the basis
i
{e (a)|1≤i≤k}.
i
Itiseasytoseethatw (E)isindependentfromthechoiceoftheframe. Weclaim
1
that w (E)=1 if and only if E is trivial. The if part is an immediate consequence
1
of the definition of w (E). On the other hand, if w (E) = 1, then detC > 0 and
1 1
there exists a path C(t) with C(a)=C and C(b)=Id. Now, f (t)=C(t)e (t) is a
i i
frame suchthatf (a)=f (b) andhence Φ(t,x ,...,x )=(t, x f (t)) induces an
i i 1 k i i
isomorphismoverS1 betweenthe productbundleS1×Rk andE. ThusE istrivial.
P
Remark 2.2. Under the isomorphism H1(S1;Z ) ∼= Z , w (E) can be identified
2 2 1
with the first Stiefel-Whitney class of E.
Our main result is:
Theorem 2.3. If the system (5) verifies (A0)–(A4) and if
(9) w (Es(+∞))6=w (Es(−∞)),
1 1
thenfor all εsmall enoughthereis ahomoclinic solution (λ,x)of (5)withkxk=ε.
The proof will be presented in Section 6.
A point λ ∈ S1 is a bifurcation point for homoclinic solutions of (5) from the
∗
stationary branch (λ,0) if in every neighborhood of (λ ,0) there is a point of a
∗
nontrivial homoclinic solution (λ,x) of x =f (λ,x ).
n+1 n n
ByTheorem2.3wecanfindasequenceofnontrivialhomoclinicsolutions(λ ,x )
n n
of (5) such that kx k → 0. Since S1 is compact λ possesses a subsequence con-
n n
verging to some λ ∈S1. Hence we obtain:
∗
6 JACOBOPEJSACHOWICZANDROBERTSKIBA
Corollary 2.4. Under the assumptions of Theorem 2.3 there exists at least one
bifurcation point λ ∈ S1 of nontrivial homoclinic solutions from the branch of
∗
stationary solutions. In other words there exists a λ ∈S1 and a sequence (λ ,x )
∗ k k
such that λ →λ and x 6=0 is a nontrivial homoclinic trajectory of f.
k ∗ k
Letusobservethatiff: Z×S1×RN →RN verifiesthe assumptionsofTheorem
2.3 and˜f: Z×S1×RN →RN is defined by˜f=f+h, where
h=(h ): Z×S1×RN →RN,
n
verifies (A0)–(A2) and moreover
∂h
(A3′) n(λ,0)→0 as n→±∞ uniformly on λ,
∂x
∂h
(A4′) sup n(λ ,0) is small enough,
0
n∈Z(cid:13) ∂x (cid:13)
(cid:13) (cid:13)
then also˜f v(cid:13)(cid:13)erifies Assu(cid:13)(cid:13)mptions (A0)–(A4). Indeed, (A3′) for h implies (A3) for˜f.
(cid:13) (cid:13)
On the other hand, it is shown in the proof of Theorem 2.3 that (A3) and (A4)
together imply that the operator L : c(RN)→c(RN) defined by
λ0
L x=(x −a (λ )x )
λ0 n+1 n 0 n
is invertible. Now, that˜f verifies (A4) follows from (A4′) and the fact that the set
of all invertible operators is open.
Summing up we have:
Corollary 2.5. If f verifies the assumptions of Theorem 2.3 then any perturbation
˜f = f+h as above must have nontrivial homoclinic solutions bifurcating from the
stationary branch at some point of the parameter space.
3. The index bundle
LetusrecallthataboundedoperatorT ∈L(X,Y)(2)isFredholmifithasfinite
dimensionalkerneland cokernel. The index of a Fredholmoperatoris by definition
indT := dimKerT − dimCokerT. The Fredholm operators will be denoted by
Φ(X,Y) and those of index 0 by Φ (X,Y).
0
The index bundle generalizes to the case of families of Fredholm operators the
conceptofindexofasingleFredholmoperator. IfafamilyL ofFredholmoperators
λ
depends continuously on a parameter λ belonging to some topologicalspace Λ and
if the kernels KerL and cokernels CokerL form two vector bundles KerL and
λ λ
CokerL overΛ, then, roughly speaking,the index bundle is KerL−CokerL where
one has to give a meaning to the difference by working in an appropriate group
generalizing Z. We will first define such a group and then will see how to handle
the case where the kernels do not form a vector bundle.
IfΛ isa compacttopologicalspace,the GrothendieckgroupKO(Λ)is the group
completion of the abelian semigroup Vect(Λ) of all isomorphisms classes of real
vector bundles over Λ. In other words, KO(Λ) is the quotient of the semigroup
Vect(Λ) ×Vect(Λ) by the diagonal sub-semigroup. The elements of KO(Λ) are
called virtual bundles. Eachvirtual bundle can be written as a difference [E]−[F]
whereE,F arevectorbundlesoverΛand[E]denotestheequivalenceclassof(E,0).
Moreover,one can show that [E]−[F]=0 in KO(Λ) if and only if the two vector
2ByL(X,Y)wewilldenotethespaceofboundedlinearoperatorsbetweentwoBanachspaces
XandY.
TOPOLOGY AND HOMOCLINIC TRAJECTORIES OF DISCRETE DYNAMICAL SYSTEMS 7
bundles become isomorphic after the addition of a trivial vector bundle to both
sides. Taking complex vector bundles instead of the real ones leads to the complex
Grothendieckgroupdenotedby K(Λ). In what followsthe trivialbundle with fiber
Λ×V will be denoted by Θ(V). The trivial bundle, Θ(RN), will be simplified to
ΘN.
LetX, Y be realBanachspacesandletL: Λ→Φ(X,Y)be acontinuousfamily
of Fredholm operators. As before L ∈ Φ(X,Y) will denote the value of L at the
λ
point λ∈Λ. Since CokerL is finite dimensional, using compactness of Λ, one can
λ
find a finite dimensional subspace V of Y such that
(10) ImL +V =Y for all λ∈Λ.
λ
Becauseof the transversalitycondition(10) the family offinite dimensionalsub-
spaces E =L−1(V) defines a vector bundle over Λ with total space
λ λ
E = {λ}×E .
λ
λ∈Λ
[
Indeed, the kernels of a family of surjective Fredholm operators form a finite di-
mensional vector bundle [13]. Denoting with π the canonical projection of Y onto
Y/V,from(10)itfollowsthatthe operatorsπL aresurjectivewithKerπL =E ,
λ λ λ
which shows that E ∈Vect(Λ).
We define the index bundle IndL by:
(11) IndL=[E]−[Θ(V)]∈KO(Λ).
NoticethattheindexbundleofafamilyofFredholmoperatorsofindex0belongs
tothereducedGrothendieckgroupKO(Λ)which,bydefinition,isthe kernelofthe
rank homomorphism rk: KO(Λ)→Z given by
g
rk([E]−[F])=dimE −dimF .
λ λ
We will mainly, but not always, work with families of Fredholm operators of index
0. If Λ = pt consists of just one point, then the rank homomorphism rk is an
isomorphism and the index bundle coincides with the ordinary numerical index
indL = dimKerL − dimCokerL of a Fredholm operator L. The index bundle
enjoysthesamenicepropertiesoftheordinaryindex. Namely,homotopyinvariance,
additivity with respect to directs sums, logarithmic property under composition of
operators. Clearly it vanishes if L is a family of isomorphisms. We will use these
properties in the sequel. The precise statements and proofs can be found in [16,
Appendix A].
It can be shown that any element η ∈ KO(Λ) can be written as [E]−[ΘN].
Moreover,[E]−[ΘN]=[E′]−[ΘM] in KO(Λ) if and only if there exist two trivial
bundlesΘandΘ′ suchthatE⊕ΘisisomorpghictoE′⊕Θ′,(see[11,Theorem3.8]).
The obstruction w (E) to the trivialgity of vector bundle E over S1 defined in
1
Section 2 induces a well defined homomorphism w : KO(S1)→Z by putting
1 2
(12) w ([E]−[F])=w (E)w (F).
1 1 1 g
Indeed,takingΛ=S1weobservethatw (E)remainsunmodifiedunderaddition
1
of a trivial vector bundle which, on the basis of the above discussion, proves that
(12) is well defined.
Proposition 3.1. The homomorphism w : KO(S1)→Z is an isomorphism.
1 2
g
8 JACOBOPEJSACHOWICZANDROBERTSKIBA
Proof. This follows again from the above discussion and the fact that w (E) = 1
1
implies that E is a trivial vector bundle over S1. (cid:3)
4. The index bundle of the family of operators associated to linear
asymptotically hyperbolic systems
In this section we will deal only with linear asymptotically hyperbolic systems
a: Z×S1 → GL(N), where GL(N) is the set of all invertible matrices in RN×N.
This means:
(a) As n→±∞ the sequence a(λ)=(a (λ)) convergesuniformly with respect
n
to λ∈S1 to a family of matrices a(λ,±∞).
(b) a(λ,±∞)∈GL(N) is hyperbolic for all λ∈S1.
GivenafamilyaofasymptoticallyhyperbolicsystemsparametrizedbyS1 letus
consider the family of linear operators
L={L : c(RN)→c(RN)|λ∈S1}
λ
defined by L =S−A , where S is the shift operator and
λ λ
A : c(RN)→c(RN)
λ
is defined by A x:=(a (λ)x ).
λ n n
Sincethesequence(a (λ))convergesuniformly,itisbounded,fromwhichfollows
n
immediately that A and L are well defined bounded operators. Moreover it is
λ λ
easy to see that the map A: S1 → L(c(RN),c(RN)) defined by A(λ) := A is
λ
continuouswithrespectto thenormtopologyofL(c(RN),c(RN)). Hence thesame
holds for the family L.
Clearly, x = (x ) ∈ c(RN) verifies a linear difference equation x = a (λ)x
n n+1 n n
if and only if L x=0.
λ
By the discussion in the previous section the families a(λ,±∞)∈GL(N) define
two vectorbundles Es(±∞) overS1. The next theorem relatesthe index bundle of
the family L to Es(±∞).
Theorem 4.1. Let a: Z×S1 →GL(N) be a continuous map verifying (a) and (b).
Then the family L: S1 →L(c(RN),c(RN)) verifies:
(i) L is a Fredholm operator for all λ∈S1.
λ
(ii) IndL=[Es(+∞)]−[Es(−∞)]∈KO(S1).
Remark 4.2. Inthe proofofTheorem4.1we will alsocompute the index ofL in
λ
terms of dimensions of the stable spaces at ±∞. This is far from being new, and
similar computations using exponential dichotomies can be found in many places,
e.g., [5, 22]. Here we are not interested in the index but rather in the index bundle
and our theorem can be considered an extension to the case of families of the
computations quoted above.
Proof. Let a¯: S1×Z→GL(N) be defined by
a(λ,+∞) if n≥0,
(13) a¯(λ,n)=(a¯ (λ))=
n
(a(λ,−∞) if n<0.
Put X := c(RN). Fix λ ∈ S1 and denote by A¯ ∈ L(X,X) the operator associ-
λ
ated to a¯ . We claim that the operator K = A −A¯ is a compact operator. To
λ λ λ λ
this end, we will show that K is the limit (in the norm topology of L(X,X)) of
λ
TOPOLOGY AND HOMOCLINIC TRAJECTORIES OF DISCRETE DYNAMICAL SYSTEMS 9
a sequence of operators K˜m with finite dimensional range. We observe that K is
λ λ
defined by K x=(k (λ)x ), where k (λ)=a (λ)−a¯ (λ) and define
λ n n n n n
k (λ)x if |n|≤m,
(14) K˜mx= n n
λ
(0 if |n|>m.
Clearly ImK˜m is finite dimensional. We are to prove that
λ
(15) sup k(K −K˜m)xk−−−−→0,
λ λ
kxk=1 m→∞
for x∈X. Observe that
(16)
k(K −K˜m)xk= sup kk (λ)x k≥ sup kk (λ)x k=k(K −K˜m+1)xk,
λ λ n n n n λ λ
|n|>m |n|>m+1
for all m∈N. Since
lim k (λ)=0,
n
|n|→∞
weinfer thatforall ε>0 thereexists n >0suchthatforall |n|>n andkxk=1
0 0
one has
kk (λ)x k<ε.
n n
Consequently, for all ε>0 there exists n >0 such that
0
(17) sup k(K −K˜n0)xk≤ε.
λ λ
kxk=1
Now taking into account (16) and (17), we deduce that for all ε > 0 there exists
n >0 such that for all m≥n one has
0 0
(18) sup ||(K −K˜m)x||≤ε,
λ λ
kxk=1
which proves (15) and the compactness of the operator K .
λ
Let L¯ = S−A¯ . Then L −L¯ =K and hence the family L differs from the
λ λ λ λ λ
family L¯ by a family ofcompactoperators. ThereforeL is Fredholmif andonly if
λ
L¯ is Fredholm and moreover the homotopy invariance of the index bundle applied
to the homotopy H(λ,t) = L¯ +tK shows that IndL¯ = IndL. Hence in order to
λ λ
prove the theorem we can assume without loss of generality that a has already the
special form of (13), which we will do from now on. Let
c+ ={x∈c(RN)|x =0 for i<k},
k i
c− ={x∈c(RN)|x =0 for i>k}.
k i
Bothc±areclosedsubspacesofc(RN). Thespacec+ canbeisometricallyidentified
k k
with
c (RN):={x: [k,∞)∩Z→RN | lim x =0}
k n
n→∞
and similarly for c−.
k
Put X+ = Y+ = c+ and X− = c−, Y− = c− . Let us consider four linear
0 0 −1
operatorsI: Y−⊕Y+ →X,J: X →X−⊕X+,L+: X+ →Y+andL−: X− →Y−
λ λ
10 JACOBOPEJSACHOWICZANDROBERTSKIBA
defined respectively by
I(x,y)=x+y,
(x ,x ) if n=0,
0 0
J(x)(n)= (x ,0) if n<0,
 n
(0,xn) if n>0,
(L+x)(n)=xn+1−a(λ,+∞)xn for n≥0,
λ (0 for n<0,
0 for n>−1,
(L−x)(n)=
λ (xn+1−a(λ,−∞)xn for n≤−1.
We decompose L : X →X via the following commutative diagram:
λ
X−⊕X+ L−λ⊕✲L+λ Y−⊕Y+
✻
(19)
J I
❄
✲
X X.
Lλ
The commutativity of diagram (19) is easy to check.
Indeed, one has
(L+x)(n) if n≥0,
I(L−⊕L+)Jx(n)=L−Jx(n)+L+Jx(n)= λ
λ λ λ λ ((L−λx)(n) if n<0,
which is the same as
x −a(λ,+∞)x if n≥0,
(20) (L x)(n)= n+1 n
λ
(xn+1−a(λ,−∞)xn if n<0.
Next, we will show that L±: X± → Y± are Fredholm and we will compute the
λ
index bundles of L±.
For L+ this is the content of the following Lemma:
λ
Lemma 4.3. ([2, Lemma 2.1]) Let a ∈ GL(N) be an hyperbolic matrix. Then the
operator S−A: c+ →c+, defined by
0 0
x −ax if n≥0,
((S−A)x)(n)= n+1 n
(0 if n<0,
is surjective with
ker(S−A)={x∈c+ |x =anx for all n≥0 and x ∈Es(a)}.
0 n+1 0 0
Thislemmawasprovedin[2,Lemma2.1]byconstructinganexplicitrightinverse
to the operator S−A: c+ →c+. (cid:3)
0 0
By Lemma 4.3
(21) KerL+ ={x∈X+ |x =a(λ,+∞)nx and x ∈Es(λ,+∞)}.
λ n 0 0
Hence the transformation x 7→ x defines an isomorphism between KerL+ and
0
Es(λ,+∞), which is finite dimensional. Being CokerL =0, L+ is Fredholm with
λ λ
indL+ =dimEs(λ,+∞). Clearly the index bundle IndL+ =[Es(+∞)].
λ