Lefschetz index of arbitrary planar
homeomorphisms at isolated invariant
continua
Francisco R. Ruiz del Portal and José M. Salazar
∗
October 30, 2014
Abstract
Let f : U ⊂ R2 → f (U ) ⊂ R2 be an arbitrary homeomorphism,
with U an open set, and let K ⊂ U be an isolated invariant continuum. If K does not decompose the plane, the xed point indices of
the iterates of f at K , iR2 (f n , K), follow the same behavior than the
indices at a locally maximal xed point. This problem was solved by
P. Le Calvez and J.C. Yoccoz in the orientation preserving case. In
this paper we compute the indices for arbitrary, not only orientation
preserving, homeomorphisms and arbitrary isolated continua and show
some dynamical implications. In particular we show that proper invariant continua of the 2-sphere containing the set of periodic orbits
of a homeomorphism are not isolated.
1. Introduction
The existence of a minimal homeomorphism of Rm is an open problem suggested by Ulam in the Scottish Book [17]. In dimension 2, one can
consider the problem of the existence or not of minimal homeomorphisms
f : R2 \ K → R2 \ K with K a compact set. If K = ∅, the result follows from the Brouwer's translation arcs theorem. Handel, in [10], proved
that, if K has at least two points, there are not minimal homeomorphisms
of R2 \ K . Le Calvez and Yoccoz [15] solved completely the problem in
the multi-punctured plane. Later Franks [6], gave an alternative proof using Conley index methods. The general result for K a compact set is given
in [23]. Recently has been proved that there are not minimal orientation
reversing homeomorphisms in R3 ([11]).
The key of the proof in [15] is the study of the xed point indices of
the iterations of an orientation preserving local homeomorphism f : U ⊂
∗
The authors have been supported by MINECO, MTM 2012-30719.
Mathematics Subject Classication: 37C25, 37B30, 54H20, 54H25.
keywords and phrases. Fixed point index, Conley index, semidynamical systems.
2000
1
R2 → R2 in a neighborhood of a point p which is not an attractor nor a
repeller isolated invariant set. This result was extended in [21] for arbitrary
homeomorphisms with a shorter proof, using the following ideas:
If f : U ⊂ R2 → R2 is a local homeomorphism and p ∈ U is a non-repeller
xed point of f such that {p} is an isolated invariant set, then there are an
AR (absolute retract), D, containing a neighborhood V ⊂ R2 of p, a nite
subset {q1 , . . . , qm } ⊂ D and a map f 0 : D → D such that f 0 |V = f |V and
for every k ∈ N, F ix((f 0 )k ) ⊂ {p, q1 , . . . , qm }. Moreover,
a) (Le Calvez-Yoccoz) If f preserves the orientation, then
k
iR2 (f , p) =
1 − rq
1
if k ∈ rN
if k ∈/ rN
where k ∈ N, q is the number of periodic orbits of f 0 (excluding p) and r is
their period.
b) Assume that f reverses the orientation (see [21]). Then
k
iR2 (f , p) =
1−δ
1 − (2q + δ)
if k odd
if k even
where q and δ are the number of orbits of period 2 and 1 of f 0 in {q1 , . . . , qm }.
More results about the computation of the xed point index for a xed
point p can be obtained in [24], [19], [5], [14], [1], [20], [22], [11], [8], [9] and
[16].
The aim of this paper is to extend the above result by computing the
xed point index iR2 (f k , K) for K an isolated invariant continuum. The
techniques we employ are based on Conley index ideas. From the results of
[16] it follows that this sequence es periodic.
We will prove, in a constructive way, the existence of certain special isolating blocks and index pairs, which we call strong ltration pairs, associated
to an isolated invariant continuum K of a homeomorphism f : U ⊂ R2 →
f (U ) ⊂ R2 , with U an open set. The isolating block, N , of K is a connected
manifold. N and K will decompose the plane in the same number of components. This construction permit us to prove the Main Theorems which
compute the xed point indices of the iterations of a homeomorphism f at
K.
Main Theorem 1. Let f
: U ⊂ R2 → f (U ) ⊂ R2 be a homeomorphism and
let K be an isolated invariant continuum. Then,
iR2 (f k , K) =
if f k is orientation preserving.
if f k is orientation reversing.
2 − C(k) − P (k)
−C 0 (k) − P (k)
where C(k) ≥ 0 is the number of components C of R2 \ K such that f k (U ∩
C) ⊂ C . The integers C 0 (k) (k odd) are also dened only in terms of the
above components as the number of them which are exit regions for f k in a
2
neighborhood of K minus the cardinal of the rest. The integers P (k) ≥ 0
depend on the behavior of f k in the exit set of N .
Let us observe that, if k is even or f is orientation preserving, the index
is iR2 (f k , K) = 2 − C(k) − P (k) ≤ 2. On the other hand, if k is odd and f
is orientation reversing, iR2 (f k , K) = −C 0 (k) − P (k).
Corollary 1. Under the above conditions, if f is orientation preserving,
iR2 (f k , K) ≤ 2. Moreover, there exists k0 ∈ N such that iR2 (f nk0 , K) ≤ 1
for every n ∈ N.
Corollary 2. Given any homeomorphism f , if K decomposes the plain into
three or more components, there exists k0 ∈ N such that iR2 (f k0 , K) < 0.
Consequently f has a periodic orbit in K for the orientation preserving and
for the orientation reversing cases.
Note that a stronger version of the above result of existence of periodic
orbits is well known to be true in the orientation reversing case (see [13] for
example). However, we nd periodic points of non-zero index.
The knowledge of the structure of the set of periodic orbits of homeomorphisms of the sphere is an interesting problem that was stated explicitly by
Le Calvez in his lecture in the ICM 2006. The question of whether cl(P er(f ))
is isolated has been studied in [23] with restrictions on the homeomorphism
(area-preserving). The next corollary is valid for arbitrary homeomorphisms
but we consider just invariant continua containing P er(f ).
Corollary 3. Given a homeomorphism
f : S 2 → S 2 , if K $ S 2 is an
invariant continuum which contains P er(f ), K is not isolated. S
On the other
hand, if K has a nite amount of connected components, K = ni=1 Ki , and
for each Ki , S 2 \Ki has no invariant components Ui,j for some f ni,j which are
locally attracted to K or repelled from K for f ni,j (f ni,j (Ui,j ∩UK ) * Ui,j ∩UK
and f ni,j (Ui,j ∩ UK ) + Ui,j ∩ UK for every neighborhood UK of K ), then K
is not isolated.
When K is not locally maximal but R2 \K has a nite number of components, the computation of the index could be made by using Carathéodory's
prime ends techniques but we're not treating this situation here.
The structure of the article is the following. In section 2 we introduce
the Conley index techniques we will need for our particular setting. Section
3 and 4 are devoted to prove the Main Theorem.
In section 5 we see a theorem which relates the xed point indices with
the local dynamics of f in a neighborhood of K .
The reader is referred to the text of [3], [4], [12] and [18] for information
about the xed point index theory.
3
2. Preliminaries. Conley index and construction of adequate ltration pairs
Let U ⊂ X be an open set. By a local semidynamical system we mean
a locally dened continuous map f : U → X . We say that a function
σ : Z → X is a solution to f through x in N ⊂ X if f (σ(i)) = σ(i + 1)
for all i ∈ Z, σ(0) = x and σ(i) ∈ N for all i ∈ Z. The invariant part of
N , Inv(N, f ), is dened as the set of all x ∈ N that admit a solution to f
through x in N , i. e. the set of all x ∈ N sucht that there is a full orbit γ
such that x ∈ γ ⊂ N . Inv + (N, f ) (respectively Inv − (N, f )) denotes the set
of all x ∈ N such that f n (x) ∈ N for every n ∈ N (respectively f −n (x) is
well dened and belongs to N for every n ∈ N).
Given A ⊂ B ⊂ N , cl(A), clB (A), int(A), intB (A), ∂(A) and ∂B (A) will
denote the closure of A, the closure of A in B , the interior of A, the interior
of A in B , the boundary of A and the boundary of A in B .
A compact set K ⊂ X is invariant if f (K) = K . An invariant compact
set K is isolated with respect to f if there exists a compact neighborhood N
of K such that Inv(N, f ) = K . The neighborhood N is called an isolating
neighborhood of K .
A compact set N is called isolating block if f (N ) ∩ N ∩ f −1 (N ) ⊂ int(N ).
If K = Inv(N, f ) then N is an isolating neighborhood of K and we will say
that N is an isolating block of K .
We consider the exit set of N to be dened as
N − = {x ∈ N : f (x) ∈
/ int(N )}.
Let f : U ⊂ R2 → f (U ) ⊂ R2 be a homeomorphism, with U open, and
let K ⊂ U be an isolated invariant continuum. Given k ∈ N we want to
compute the xed point index of f k in K , iR2 (f k , K), i.e. the xed point
index of each f k in a small enough neighborhood of K .
We begin this study with the choice of an adequate isolating block N of
K with topological properties which give us information about the dynamical
behavior of f in a neighborhood of K .
Among all compact connected smooth manifolds (discs with a nite
amount of holes), isolating blocks for K , we take N to be one of them
which decomposes the plane in the minimum possible number of bounded
components.
Let {D1 , . . . , Dp } be the components of int(N ) and let D(N ) =
S
N ∪ ( pi=1 Di ) be the disc obtained by lling the holes of N .
If we consider K ⊂ N ⊂ S 2 , the set I(K) = S 2 \K is an open subset of S 2
and S 2 \ int(N ) has p + 1 components {D0 , D1 , . . . , Dp } with D0 unbounded.
Let us denote {I(K)j } the connected components of I(K) with I(K)0 the
component which contains D0 .
4
I(K)0
N
K
I(K) \ I(K)0
Figure 1
Take
f (N ).
f∗
:
S2
→
S2
to be a homeomorphism which extends f |N : N →
Lemma 1. If
I(K)j ⊂ I(K), then I(K)j contains a component Dj ∈
{D0 , . . . , Dp }. As a consequence, I(K) decomposes into a nite number of
components on which f ∗ acts as a permutation.
Proof. Let us suppose j 6= 0. Since N is an isolating block, if I(K)j ⊂ N
then each image f n (I(K)j ) ⊂ I(K) is contained in N .
In fact, if there exists Dj 0 ⊂ I(K)j 0 = f ∗ (I(K)j ), since N is an isolating block, f (∂(Dj 0 )) ⊂ int(Dj 00 ) ⊂ I(K)j 00 = f ∗ (I(K)j 0 ). On the other
hand, it is immediate to see that f −1 (∂(Dj 00 )) ⊂ N and then f (∂(Dj 00 )) ⊂
int(Dj 000 ) ⊂ I(K)j 000 = f ∗ (I(K)j 00 ). Therefore all the images by f ∗ of I(K)j
have discs of the nite family {D0 , . . . , Dp }. Then, there exists nj such that
(f ∗ )nj (I(K)j ) = I(K)j , and it must contain a component of S 2 \ N , but this
is a contradiction.
S
Then, the set K ∪ cl( n∈Z (f ∗ )n (I(K)j )) 6= K is a invariant continuum
contained in N . This contradicts the fact that Inv(N, f ) = K .
Lemma 2. Let Dj ⊂ I(K)j . Then
f ∗ (Dj ) 6⊂ N and (f ∗ )−1 (Dj ) 6⊂ N.
Proof. Let us prove that f ∗ (Dj ) 6⊂ N (the other statement has an analogous
proof). If f ∗ (Dj ) ⊂ N , since f and f −1 are dened in the isolating block
N , then Dj ⊂ U . On the other hand, (f ∗ )−1 (Dj ) ⊂ int(Di ) ⊂ I(K)i =
(f ∗ )−1 (I(K)j ) with Di 6= Dj . The set N 0 = N ∪ Dj ⊂ U is an isolating
block but N 0 has one hole less than N which is a contradiction.
Lemma 3.
I(K)j contains a unique component Dj .
Proof. Let us S
suppose that there are two components Dj,1 6= Dj,2 in I(K)j
0
and let D = pi=0 Di ⊂ S 2 . Let Bj be a topological disc with Dj,1 ∪ Dj,2 ⊂
5
Bj ⊂ I(K)j . We consider a nite covering of Bj , in I(K)j , formed by closed
balls {B(x)}x∈F , with F a nite subset of Bj , and such that for all B(x)
there exists some nx ∈ Z such that f nx (B(x)) ⊂ int(D0 ). Let n0 be a
natural number such that |nx | ≤ n0 for all x ∈ F .
Our goal is to construct an isolating block of K , Nn0 , with similar properties than N and such that S 2 \ Nn0 has less components than S 2 \ N and
we will arrive to a contradiction.
Dene E 0 = D0 ∪ V1 ∪ V−1 ⊂ S 2 , with Vi compact manifolds homeomorphic to (f ∗ )i (D0 ), Vi ⊂ int((f ∗ )i (D0 )) for i ∈ {1, −1}. Furthermore, Vi
must be transversal to D0 , close enough to (f ∗ )i (D0 ) and such that if x ∈ F
with (f ∗ )i (B(x)) ⊂ int(D0 ), then B(x) ⊂ V−i .
Let D1 be the set obtained by lling all the components of S 2 \ E 0 which
do not contain K . Using the above lemma, one can choose V1 and V−1 such
that D1 has at most p + 1 components.
V1
D0
V−1
K
N
D1
E 0 = D0 ∪ V1 ∪ V.−1
Figure 2
Let us dene
N1 = N \ int(D1 )
Note that K ⊂ int(N1 ) ⊂ int(N ) and D0 ⊂ D1 . It is easy to see that
N1 is an isolating block of K and a compact, connected manifold such that
S 2 \ N1 has at most p + 1 components. The set N1 and the components D1
are in the conditions of Lemma 2.
If we repeat the last construction n0 times, we get the pairs of manifolds
and holes {(N1 , D1 ), . . . , (Nn0 , Dn0 )} with D0 ⊂ D1 ⊂ · · · ⊂ Dn0 and Nn0 ⊂
· · · ⊂ N1 ⊂ N . Each Dn is a nite union of discs and if Dm ⊂ Dn , each
component of Dn contains, at least, a component of Dm . Since D0 has p + 1
6
S
components and Dj,1 ∪ Dj,2 ⊂ Bj ⊂ x∈F B(x) ⊂ Dn0 , then Dn0 has less
than p + 1 components.
Finally, Nn0 is a compact connected manifold, an isolating block of K
and S 2 \ Nn0 has less than p + 1 components.
Denition 1. ([7]) Let K be a compact isolated invariant set and suppose
L ⊂ N is a compact pair contained in the interior of the domain of f . The
pair (N, L) is called a ltration pair for K provided N and L are each the
closure of their interiors and
(1) cl(N \ L) is an isolating neighborhood of K .
(2) f (cl(N \ L) ⊂ int(N ) and
(3) f (L) ∩ cl(N \ L) = ∅.
In the next proposition we prove the existence of strong ltration pairs.
Proposition 1. Let
f : U ⊂ R2 → f (U ) ⊂ R2 be a homeomorphism,
and let K be an isolated invariant continuum. Then there exists a compact pair (N, L), which we call strong ltration pair, where N is a compact
connected two-manifold, isolating block of K , with a minimum set of holes
{D1 , . . . , Dp }, and L = L1 ∪ · · · ∪ Lm ⊂ N is a nite union of compact
connected two-manifolds with, at most, one hole. The pair (N, L) is such
that:
1) ∂N (Li ) is an arc with ∂N (Li ) ∩ ∂(N ) = {ai , bi } two points, or ∂N (Li )
is a Jordan curve.
2) cl(N \ L) is an isolating neighborhood of K .
3) f (cl(N \ L)) ⊂ int(N ).
4) For all Li there exists a set Bi , which is a disc or an annulus, such
that ∂N (Li ) ⊂ Bi ⊂ Li , Bi ∩ N − 6= ∅ and f (Bi ) ∩ cl(N \ L) = ∅.
Proof. Let N be a compact connected manifold, isolating block of K , such
that S 2 \ N has the minimum possible number of components {D1 , . . . , Dp }.
Following the steps of the proof of Theorema 3.7 in [7], we obtain a pair
(N, J), J = J1 ∪ · · · ∪ Jn is a nite union of two-dimensional compact connected manifolds such that the pair (N, J) is a ltration pair for K , with J
a submanifold of N , and Ji ∩ N − 6= ∅ for all i ∈ {1, . . . , n}.
If we add to J the holes of J which are in N and not intersect K ,
we obtain a compact manifold denoted by D(J). Let us consider the pair
(N, D(J)). Each D(Ji ) (dened as D(J)) has at most one hole. There are
only three possible situations:
a) Ji ⊂ I(K)j , j 6= 0, a connected component of I(K). In this case D(Ji )
has, at most, one hole. If this hole exists, it contains Dj , the hole of N in
I(K)j .
b) Ji ⊂ I(K)0 . Then D(Ji ) has only one hole which contains K .
c) In the remaining cases D(Ji ) has no holes.
7
K
D(Ji )
K
D(Ji )
K
D(Ji )
c)
b)
a)
Figure 3
We can suppose that D(J) = D(J1 ) ∪ · · · ∪ D(Jn ) are disjoints and left
to the reader to verify that the pair (N, D(J)) is a ltration pair for K .
Let us dene Li = N \ c.c.(N \ D(Ji ), K), where c.c.(N \ D(Ji ),SK) is the
n
connected component of N \ D(J
i ) which contains K . Let L =
i=1 Li =
S
S
m
m
L
with
m
≤
n
,
such
that
L
is
a
disjoint
union
of
discs
with, at
i
i
i=1
i=1
most, one hole.
K
K
Li
Li
a)
K
Li
b)
c)
Figure 4
The pair (N, L) is the pair of the proposition and the proof of its properties is easy to check.
Let f : U ⊂ R2 → f (U ) ⊂ R2 be a homeomorphism, and let K be a
compact connected isolated invariant set. Let us consider a strong ltration
pair (N, L) as in the last proposition.
Remark 1. If ∂N (Lj ) is homeomorphic to S 1 , then Lj has a hole. Further-
more, Lj ⊂ I(K)0 contains K in its hole and ∂(Lj ) = ∂N (Lj ) ∪ ∂(D(N )),
or Lj ⊂ I(K)i for some i 6= 0 and the hole of Lj is Di . On the other hand,
if ∂N (Lj ) is an arc, then Lj is a disc without holes. Therefore, cl(N \ L) is
a disc with the same amount of holes than N .
Let cl(N \ L)/ ∼ = NL be the quotient space obtained by identifying
each ∂N (Li ) ⊂ cl(N \L) to a point qi . Take the projection π : cl(N \L) → NL
and a retraction r : N → cl(N \ L) with r(x) = x if x ∈ cl(N \ L) and r
retracts each Li to ∂N (Li ).
Now dene f 0 = π◦r◦f ◦π −1 , f 0 : NL \{q1 , . . . , qm } → NL . The map f 0 is
continuous and, in a neighborhood of K , f 0 ≡ f . Since f (∂N (Li )) ⊂ int(Lj ),
f 0 admits a unique continuous extension
8
f 0 : NL → NL
such that f 0 (U 0 (qi )) = qj for a neighborhood U 0 (qi ) of qi .
We have that f 0 ({q1 , . . . , qm }) ⊂ {q1 , . . . , qm } (f 0 (qi ) = qj if and only
if f (∂N (Li )) ⊂ int(Lj )). Obviously F ix(f 0 ) ⊂ K ∪ {q1 , . . . , qm } and, since
cl(N \ L) is an isolating neighborhood of K , F ix((f 0 )k ) ⊂ K ∪ {q1 , . . . , qm }.
By applying f ∗ to the family {I(K)0 , . . . , I(K)p } we obtain a nite union
of cycles. We have p + 1 = t1 + · · · + tl , with {t1 , . . . , tl } the lengths of the
p
cycles. Let n
us change the notation of the open
o setsn{I(K)i }i=0 and the holeso
1
l
1
l
{Di }pi=0 by {I(K)1,j }tj=1
, . . . , {I(K)l,j }tj=1
and {D1,j }tj=1
, . . . , {Dl,j }tj=1
i
where Di,j ⊂ I(K)i,j and each family {I(K)i,j }tj=1
, with i ∈ {1, . . . , l}, is
∗
t
i
a cycle of length ti . Let us observe that (f ) : I(K)i,j → I(K)i,j with
i ∈ {1, . . . , l}, j ∈ {1, . . . , ti }.
Remark 2. a) Given a cycle
{I(K)i,j }j , if I(K)i,1 contains a component
of L, Lh1 , with Di,1 the hole of Lh1 , then there exists Lhj ⊂ I(K)i,j with
Di,j the hole of Lhj . Moreover, there exists only one component of L, Lhj ,
in each I(K)i,j .
b) Given a cycle {I(K)i,j }j , if I(K)i,1 ∩ L = ∅, then I(K)i,j ∩ L = ∅.
Let us observe that ∂(N \ L)/ ∼ is a nite and disjoint union of jordan
curves and points.
Denition 2. Let θ = {p1 , . . . , ps } ⊂ {q1 , . . . , qm } be such that f 0 (θ) = θ.
We say that pi , pj ∈ θ are adjacent in θ if there is an arc γ ⊂ ∂(N \ L)/ ∼
such that γ ∩ θ = {pi , pj }.
Lemma 4. ([21]) Let
θ = {p1 , . . . , ps } ⊂ {q1 , . . . , qm } such that f 0 (θ) = θ.
If pi , pj are adjacent in θ, then their images by f 0 , pi+1 and pj+1 are adjacent
in θ.
For a proof of this result see Proposition 1 of [21].
3. Main Theorem. Orientation preserving case.
In this section we consider an orientation preserving homeomorphism
f : U ⊂ R2 → f (U ) ⊂ R2 .
Proposition 2. If
i
f is orientation preserving and {I(K)i,j }tj=1
is a cycle,
0
the set of periodic orbits of f |(π◦r)(Sti ∂(D )) is such that all its orbits have
j=1
i,j
the same period ri = ni ti for some ni ∈ N.
Proof. Let us x an orientation in I(K)i,1 ∩ ∂(N \ L) ' S 1 . The jordan
curve f (I(K)i,1 ∩ ∂(N \ L)) ⊂ I(K)i,2 bounds Di,2 ⊂ I(K)i,2 preserving
orientation.
In case a) of Remark 2 we only have a periodic orbit of period ti .
9
S
i
In case b) of Remark 2 the result is obvious because (π◦r)( tj=1
∂(Di,1 ))∩
{q1 , . . . , qm } = ∅.
In any other case, given two periodic orbits θ1 = {pi1 , . . . , pir1 } and
θ2 = {p0i1 , . . . , p0ir2 }, by Lemma 4 it easy to see that r1 = r2 = ni ti for some
ni ∈ N.
Corollary 4. In the conditions Sof the last proposition, and given
k ∈ N,
Denition 3. Let us decompose the set of lengths of cycles of
I(K) ⊂
has xed points in (π ◦ r)(
⊂ ∂(N \ L)/ ∼ if and only
if k ∈ ri N. Then, the number S
of xed points is ri q i , with q i the number of
i
0
periodic orbits of f in (π ◦ r)( tj=1
∂(Di,j )).
ti
j=1 ∂(Di,j ))
(f 0 )k
S2,
{t1 , . . . , tl }, into three disjoint sets tA , tR , tS ⊂ {t1 , . . . , tl } such that
tA ∪tR ∪tS = {t1 , . . . , tl }, with tA the set of lengths of cycles corresponding to
case b) of Remark 2 (attracting), tR the set of lengths of cycles corresponding
to case a) of Remark 2 (repelling), and tS the rest of lengths of {t1 , . . . , tl }
(hyperbolicity). It is obvious that rj = q j = 0 for all tj ∈ tA , rj = tj with
q j = 1 for all tj ∈ tR and rj = nj tj for all tj ∈ tS .
The family of periods {r1 , . . . , rl } and the number of periodic orbits of
each period {q 1 , . . . , q l } permit us to compute the number of xed points of
(f 0 )k in {q1 , . . . , qm } for all k ∈ N.
Proposition 3. If f is orientation preserving, then
iNL ((f 0 )k , NL ) = 2 −
X
ti
ti ∈tA ∪tS
k∈ti N
Proof. If K is not a repeller, NL is homeomorphic to a disc with a nite
amount
P of holes. On the other hand, H0 (NL ) = Q, H1 (NL ) = Q0 ⊕k · · · ⊕ Q
with ti ∈tA ∪tS ti − 1 generators and H2 (NL ) = 0. Since iNL ((f ) , NL ) =
Λ((f 0 )k∗ ), from the study of the trace of (f 0 )k∗ : H1 (NL ) → H1 (NL ) it is easy
to obtain the value of the xed point index (see Figure 5).
d
d
e
e
c
a
NL
c
a
b
(f 0 )k∗
b





a→b
b → −a − b − d − e
=
 d→e



e→d
Figure 5
10
If K is a repeller, NL ' S 2 and H0 (NL ) = Q,
Q. We obtain iNL ((f 0 )k , NL ) = 2.
H1 (NL ) = 0,
H2 (NL ) =
Main Theorem 2. (Orientation preserving case) Let
f : U ⊂ R2 →
f (U ) ⊂ R2 be an orientation preserving homeomorphism and let K be an
isolated invariant continuum. Then,
iR2 (f k , K) = 2 −
X
ti ∈tA ∪tR ∪tS
X
ti −
k∈ti N
ti ∈tS
t i ni q i ≤ 2
k∈ni ti N
Proof. By the additivity property of the xed point index,
X
iNL ((f 0 )k , NL ) = iR2 (f k , K) +
iNL ((f 0 )k , U (qi ))
qi ∈{q1 ,...,qm }∩F ix((f 0 )k )
Since the value of each index in the last summation is 1 (f 0 is constant
in a neighborhood of each qi ), we have
iR2 (f k , K) = 2 −
X
ti ∈tA ∪tS
k∈ti N
obtaining the result we are looking for.
ti −
X
ri q i
ri ∈{r1 ,...rl }
k∈ri N
Remark 3. (Proofs of Corollaries 1, 2 and 3) Let us observe that the
sequence {iR2 (f k , K)}k is periodic and, if p is the number of holes of K ,
1−p−
Q
X
ti ∈tS
ti ni q i ≤ iR2 (f k , K) ≤ 2
Q
k
for all k. If k ∈
ti ∈tA ∪tR ti
ti ∈tS ni ti N, then iR2 (f , K) = 1 −
P
p − ti ∈tS ti ni q i . This proves Corollary 1. Consequently, if p ≥ 2, K has
Q
Q
periodic orbits of period which divides
t
n
t
which
i
i
i
ti ∈tA ∪tR
ti ∈tS
2
proves Corollary 2. The proof of Corollary 3 is also easy. If f : S → S 2 is
a homeomorphism with K an invariant continuum which contains P er(f ),
then K is not isolated because, if K were isolated, there should be k0 ∈ N
with iR2 (f nk0 , K) ≤ 1. Then,
2 = iS 2 (f 2k0 , S 2 ) = iR2 (f 2k0 , K) ≤ 1
and we obtain a contradiction. A similar argument permit us to prove
Sn the
same result if K has a nite amount of connected components K = i=1 Ki
with each Ki in the conditions of the Corollary 3. If K were isolated, there
should be k0 ∈ 2N such that each Ki is invariant and isolated for f k0 and
iR2 (f k0 , Ki ) ≤ 0 (tS 6= ∅ for each Ki ). Then
11
2=
n
X
i=1
iR2 (f k0 , Ki ) ≤ 0
which is a contradiction.
Let us suppose that the domain U of f is an open ball. In this case,
there exists an invariant component I(K)0 of I(K) (t0 = 1) which contains
∞. We obtain the next corollary.
Corollary 5. If the domain
U of f is an open ball, then iR2 (f k , K) ≤ 1
k
for all
P k ∈ N. Moreover, if K is an attractor or a repeller, iR2 (f , K) =
1 − ti 6=t0 ti .
k∈ti N
Example. Let us see an example of index 2 with
U an open set which
is not an open ball. Let p = (p1 , p2 ) ∈ U = R2 \ {0}. We dene the
homeomorphisms:
1
S1 : U → U given by S1 (p) = ||p||
2 p.
2
2
S2 : R → R given by S2 (p1 , p2 ) = (−p1 , p2 ).
f : R2 → R2 given by f (p) = ||p||p.
The homeomorphisms S1 and S2 are orientation reversing and f is orientation preserving. The map g = f ◦ S2 ◦ S1 : U → U is an orientation
preserving homeomorphism. The unit circle K = {p : ||p|| = 1} is an isolated
invariant continuum for g and it is easy to see if we consider the extension
ḡ : S 2 → S 2 that iR2 (g, K) = 2.
4. Main theorem. Orientation reversing case
Proposition 4. Let
f : U ⊂ R2 → f (U ) ⊂ R2 be an orientation reversing
homeomorphism and let K be an isolated invariant continuum. Given a cycle
i
{I(K)i,j }tj=1
, there are two possibilities:
a) ti is even. Then f 0 |(π◦r)(Sti ∂(D )) has q i periodic orbits of the same
period ri = ti ni .
b) ti is odd. Then f 0 |(π◦r)(Sti
j=1
j=1
i,j
∂(Di,j ))
has q i,1 ≤ 2 periodic orbits of period
ri,1 = ti and q i,2 periodic orbits of period ri,2 = 2ti .
Proof. The case a) follows from Lemma 4. Let us see the case b). If ti is
odd, the periodic orbits have period ri,1 = ti or ri,2 = 2ti . In fact, take a
periodic orbit θ = {p1 , . . . , pr } of period r. Let p1 < p2 be adjacent in θ
(with the ordering given by the orientation of ∂(N \ L)/ ∼). By Lemma 4,
(f 0 )ti (p1 ) > (f 0 )ti (p2 ) are adjacent in θ. Let us observe that if (f 0 )ti (p1 ) = p2 ,
then (f 0 )ti (p2 ) = p1 and r = 2ti . Let us suppose that r > 2ti , and let p3 ∈ θ,
p3 6= p1 , such that p2 < p3 are adjacent in θ. Then, by an induction argument
we nd a point of θ with period ≤ 2ti , which is a contradiction. Then r ≤ 2ti .
Let us prove that the number of periodic orbits of period ti is ≤ 2. In
fact, let θ1 , θ2 be two periodic orbits of period ti . If there is another periodic
12
orbit θ3 we obtain, by using Lemma 4 and the fact that f is orientation
reversing, that the period of θ3 is > ti .
Proposition 5. If f is orientation reversing, then
iNL ((f 0 )k , NL ) = 1 + (−1)k −
X
(−1)k ti
ti ∈tA ∪tS
k∈ti N
The proof of this result is analogous to the proof of Proposition 3 except
that now f is orientation reversing. We leave it to the reader.
The family of lengths of cycles {t1 , . . . , tl } decomposes into two disjoint
sets: the set of even lengths tP and the set of odd lengths tI .
Main Theorem 3. (Orientation reversing case) Let
f (U ) ⊂
R2
f : U ⊂ R2 →
be an orientation reversing homeomorphism and let K be an
isolated invariant continuum. Then
iR2 (f k , K) = 1 + (−1)k −
−
P
ti ∈tI ∩tS
k∈ti N
P
ti ∈tA ∪tS
k∈ti N
ti q i,1 −
P
(−1)k ti −
ti ∈tI ∩tS
k∈2ti N
P
ti ∈tP ∩tS
k∈ni ti N
2ti q i,2 −
Proof. By the additivity property of the xed point index
X
iNL ((f 0 )k , NL ) = iR2 (f k , K) +
P
ni ti q i
ti ∈tR
k∈ti N
ti
iNL ((f 0 )k , U (qi )),
qi ∈{q1 ,...,qm }∩F ix((f 0 )k )
where each index of the last summation is 1. Then the result follows automatically.
Remark 4. If
f is orientation reversing, the sequence {iR2 (f k , K)}k is
periodic and, if p is the number of holes of K ,
1−p−
X
ti ∈tP ∩tS
ni ti q i −
X
ti ∈tI ∩tS
ti q i,1 −
Q
X
ti ∈tI ∩tS
Q
2ti q i,2 ≤ iR2 (f k , K) ≤ 2
for all k even. If k ∈ 2 ti ∈tA ∪tR ti
ti ∈tS ni ti N, with ni = 1 if
P
P
ti ∈ tI ∩ tS , then iR2 (f k , K) = 1 − p − ti ∈tP ∩tS ni ti q i − ti ∈tI ∩tS ti q i,1 −
P
i,2 . Consequently, if p ≥ 2, K has periodic orbits of period
ti ∈tI ∩tS 2ti q Q
Q
which divides 2 ti ∈tA ∪tR ti
n
t
with ni = 1 if ti ∈ tI ∩ tS .
i
i
ti ∈tS
Remark 5. Let
f : U ⊂ R2 → f (U ) ⊂ R2 be a homeomorphism and let
K be a compact S
isolated invariant
with a nite amount of connected
Smset
r
1
components K = m
K
∪
·
·
·
∪
K
i=1 1,i
i=1 r,i , such that f (Ki,j ) = Ki,j+1 and
f (Ki,mi ) = Ki,1 . Then by the
P additivity property of the xed point index
applied to f k , iR2 (f k , K) = k∈mi N mi iR2 (f k , Ki,1 ).
13
Corollary 6. Let f
: S 2 → S 2 be a homeomorphism and let K be a compact,
invariant and proper set with a nite amount of connected components. Then
f : S 2 \ K → S 2 \ K is not minimal.
Proof. Our proof is analogous to the given in [15] for K a nite amount of
points. If f |S 2 \K is minimal, K is an isolated invariant set which is not an
attractor nor a repeller. For an adequate 2n we have that the components of
K and S 2 \ K are invariant for the orientation preserving homeomorphism
f 2n and, by Remarks 3 and 5 applied to f 2n , iS 2 (f 2n , K) ≤ 0, but it is not
possible because 2 = Λ(f∗2n ) = iS 2 (f 2n , S 2 ) = iS 2 (f 2n , K).
5. Dynamical behavior and xed point index.
In this section we study the relationship between the xed point indices
iR2 (f n , K) and the local dynamics of f in a neighborhood of K . If (N, L)
is a strong ltration pair, dene cl(N \ L)K and R2K as the spaces obtained
from cl(N \ L) and R2 by the identication of K to a point [k] and dene
f : cl(N \ L)K → R2K as the map induced by f (with the same notation).
Let us observe that cl(N \ L)K is the pointed union of a nite family of
p + 1 discs, EiW
, where p + 1 is de number of components of R2 \ K . Denote
cl(N \ L)K = p+1
i=1 Ei with Ei ∩ Ej = [k] for all i 6= j .
Theorem 1. Let
K be an isolated continuum for a homeomorphism f :
U ⊂ R2 → f (U ) ⊂ R2 which decompose the plane in p + 1 components and
let (N, L) be a strong ltration pair. Then there exist
two families of closed
Wp+1
discs, {A1 , . . . , Aa }, {R1 , . . . , Rr }, in cl(N \L)K = i=1 Ei and two families
of continua without holes in cl(N \ L)K , {S1 , . . . , Ss }, {U1 , . . . , Us }, with
s = −iR2 (f d , K) + 1 − p.
P
a = −iR2 (f d , K) + 2 − ti ∈tR ∪tS ti − s.
P
r = −iR2 (f d , K) + 2 − ti ∈tA ∪tS ti − s.
Q
Q
for d = 2
ti ∈tS ti ni
ti ∈tA ∪tR ti , (ni = 1 if f is orientation reversing
and ti ∈ tI ∩ tS ). These sets satisfy the following properties:
S
S
1. si=1 Si ⊂ K + and si=1 Ui ⊂ K − . The set K + is the connected component of Inv + (cl(N \ L)K , f ) which contains [k] and the set K − is
the connected component of Inv − (cl(N \ L)K , f ) which contains [k].
2. Si ∩Sj = Ui ∩Uj = Si ∩Uj = [k], every Si ⊂ Ek for some k = 1, . . . , p+1
and every Uj ⊂ Ek0 for some k0 = 1, . . . , p + 1.
3. f d (Si ) ⊂ Si , f −d (Ui ) ⊂ Ui ,
T
n∈N f
nd (S
i)
=
T
n∈N f
−nd (U )
i
= [k].
4. The sets {Si }i and {Ui }i , alternate in the circles of ∂(cl(N \ L)) ⊂
cl(N \ L)K
14
T
−d (R ) ⊂ int(R ) and
nd
5. f d (A
i ), f
j
j
n∈N f (Ai )
T i ) ⊂ int(A
−nd
= n∈N f
(Ri ) = [k]. for all i = 1, . . . , a and j = 1, . . . , r.
K
cl(N \ L)
L
A1
S4
S1
R1
U2
U1
[k]
[k]
S3
S5
[k]
U3
E1
E2
U5
U4
E3
S2
S7
[k]
U7
U6
S6
E4
Figure 6
Proof. Let us consider the topological space NL,K obtained from the identication in NL of K to a point [k] and let f 0 : NL,K → NL,K be the
0
induced map
L . The space NL,K is the pointed
P obtained from f : NL → NP
union of ti ∈tA ∪tS ti discs, {Cj }j , and ti ∈tR ti = r spheres, {Tj }j , with
Ci ∩ Cj = Ti ∩ Tj = Ci ∩ Tj = [k]. The map (f 0 )d sends each disc to itself
and each sphere to itself.
The rest of the proof is similar to the given in Proposition 7 of [22]. Let
θ = {p1 , . . . , ps } ⊂ {q1 , . . . , qm } ⊂ NL,K be the biggest subset on which f 0
acts as a permutation and such that its points are not isolated points of
πK (∂(N \ L)) ⊂ NL,K with πK : cl(N P
\ L)K → NL,K dened as π but in
cl(N \ L)K . It is clear that θ has s = ti ∈tS ni ti q i elements if f preserves
P
P
P
orientation and s = ti ∈tP ∩tS ni ti q i + ti ∈tI ∩tS ti q i,1 + ti ∈tI ∩tS 2ti q i,2 elements if f reverses orientation. Moreover, (f 0 )d (pi ) = pi for all i = 1, . . . , s.
Dene A = {x ∈ NL,K such that there exists nx with (f 0 )nx (x) ∈ θ} and
A(pi ) the connected component of A which contains pi ∈ θ. It is clear that
A(pi ) ⊂ Cj for some disc Cj of NL,K .
15
T
Let Ki = n∈N (f 0 )nd (cl(A(pi )), i = 1, . . . , s. Since (f 0 )d (cl(A(pi )) ⊂
cl(A(pi )), Ki is a continuum which contains pi and [k] with (f 0 )d (Ki ) =
Ki ⊂ Cj . Dene
−1
Ui = (πK
(Ki ) ∩ K − )
Let us construct the sets Si . If pi−1 , pi ∈ θ are adjacent in πK (∂(cl(N \
L))), let γ ⊂ πK (∂(cl(N \ L))) be an arc joining pi−1 and pi with γ ∩ θ =
{pi−1 , pi } and γ ⊂ Cj for some disc Cj of the pointed union NL,K . There
is a component Kpi of ∂(A(pi )) separating pi from θ \ pi , [k] ∈ Kpi with
limn→∞ (f 0 )nd (x) = [k] for all x ∈ Kpi , and such that Kpi ∩ γ 6= ∅. Let Bi
be the connected component of Cj \ (Ki−1 ∪ Ki ) containing Kpi ∩ γ . Dene
−1
Si = (πK
(Bi ) ∪ [k]) ∩ K +
It is easy to prove that the sets Ui and Si satisfy the properties 1 to 4 of
the theorem. The details are left to the reader.
The equality relating the number s of sets {Ui } and {Si } with the xed
point index, s = −iR2 (f d , K) + 1 − p follows, in the orientation preserving
case, from the
P Main Theorem 1:
s =
ti ∈tS
ni ti q i = −iR2 (f d , K) + 1 − p.
For the orientation
reversing
P
P case we have, from
P the Main Theorem 2:
i
i,1 +
i,2
s =
ti ∈tP ∩tS ni ti q +
ti ∈tI ∩tS ti q
ti ∈tI ∩tS 2ti q
d
= −iR2 (f , K) + 1 − p.
W
For every disc Ei of the pointed union p+1
i=1 Ei such that Ei ∩ L = ∅,
dene Ai = Ei and for every Ej such that Ej ∩ L is a circumference, dene
Rj = Ej . The proof of the property 5 is immediate.
The P
equalities relating a and r withPthe xed point index follow from
a = ti ∈tA ti = −iR2 (f d , K) + 2 − ti ∈tR ∪tS ti − s.
P
P
r = ti ∈tR ti = −iR2 (f d , K) + 2 − ti ∈tA ∪tS ti − s.
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18