Indices of the iterates of
R3-homeomorphisms at fixed points which

are isolated invariant sets

Patrice Le Calvez, Francisco R. Ruiz del Portal and José M. Salazar ∗

March 24, 2010

Dedicated to professor José M. Montesinos on the occasion of his 65th
birthday.

Abstract
Let U ⊂ R3 be an open set and f : U → f(U) ⊂ R3 be a homeo-

morphism. Let p ∈ U be a fixed point. It is known that, if {p} is not
an isolated invariant set, the sequence of the fixed point indices of the
iterates of f at p, (i(fn, p))n≥1, is, in general, unbounded. The main
goal of this paper is to show that when {p} is an isolated invariant set,
the sequence (i(fn, p))n≥1 is periodic. Conversely, we show that for
any periodic sequence of integers (In)n≥1 satisfying Dold’s necessary
congruences, there exists an orientation preserving homeomorphism
such that i(fn, p) = In for every n ≥ 1. Finally we also present an
application to the study of the local structure of the stable/unstable
sets at p.

1. Introduction.

Let U ⊂ Rm be an open set and f : U → f(U) ⊂ Rm be a continuous
map. Let p ∈ U be a fixed point of f that is an isolated fixed point of fn for
every n ≥ 1. Since Dold [4], it is known that the sequence (i(fn, p))n≥1 of Lef-
schetz indices must satisfy some rules, called Dold’s congruences. Shub and
Sullivan proved that for C1-maps the sequence is bounded. Chow, Mallet-
Paret and Yorke [3] gave bounds about the form of the sequence of indices
in terms of the spectrum of the derivative Df(p). Babenko and Bogatyi [1]
proved that these bounds are sharp in dimension 2 and in a more recent
paper [7] Graff and Nowak-Przygodzki have proved that for n = 3 and C1-
maps, the sequence of fixed point indices follows one among exactly seven
different periodic patterns.
∗The authors have been supported by MICINN, MTM 2009-07030, the first author has

also been supported by ANR (Symplexe, ANR-06-BLAN-0030-01)
2000 Mathematics Subject Classification: 37C25, 37B30, 54H25.
Keywords and phrases. Fixed point index, Conley index, entropy, filtration pairs.

1



Suppose now that f is a homeomorphism. For m = 2, see [13], [14], [15]
and [21], it is known that the sequence of indices (i(fn, p))n≥1 is periodic
with a very particular behavior. This sequence contains important dynamical
information. For m ≥ 3, the behavior is completely different from that of
diffeomorphisms. If an isolated fixed point p is not isolated as invariant
set, the sequence of indices of the iterates in general is not bounded, even
if the fixed point is Lyapunov stable. More precisely, in [22] it is proved
that any sequence satisfying Dold’s congruences is realized as the sequence
of fixed point indices of the iterates of a Rm-homeomorphism at an isolated
and stable fixed point (see also [23]).

What happens if p is an isolated invariant set ? There are examples, for
m = 8, where the sequence (i(fn, p))n≥1 is unbounded (see [19]). The main
result of the paper asserts that such a situation cannot occur if m = 3, more
precisely:

Theorem 1. Let U ⊂ R3 be an open set and f : U ⊂ R3 → f(U) ⊂ R3 be
a homeomorphism. Let p be a fixed point of f such that {p} is an isolated
invariant set. Then the sequence (i(fn, p))n≥1 is periodic.

From [1] (see also [11]), one knows that a bounded sequence that sat-
isfies Dold’s congruences is periodic. Therefore it would be sufficient to
prove that the sequence is bounded in the main theorem. In fact we will
prove directly the periodicity. If f : M → M is a continuous map on a
m-dimensional compact manifold M , denote by f∗i the induced map on the
singular homology group Hi(M,Q). A classical result of Manning [16] as-
serts that htop(f) ≥ log(sr(f∗1)), where htop(f) is the topological entropy
of f and sr(f∗1) the spectral radius of f∗1. As explained by Manning, this
implies by Poincaré duality that if f is a homeomorphism of a 3-dimensional
compact manifold, one has htop(f) ≥ sup0≤i≤3 log(sr(f∗i)). The inequality
htop(f) ≥ sup0≤i≤n log(sr(f∗i )) is known to be true if f is C∞ (see [28]) but
fails to be true is f is only continuous : what is constructed in [19] is an ex-
ample of a homeomorphism on a 8-dimensional manifold where the previous
inequality does not occur. As we will see later Manning’s result will be the
key-point of the proof of the main result. We will need to apply it, not on
a manifold but on an absolute neighborhood retract ANR constructed via
a filtration pair and we will need to use Lefschetz-Dold’s formula to get a
duality result. In fact we will prove the following more precise result.

Theorem 2. Let U ⊂ R3 be an open set and f : U ⊂ R3 → f(U) ⊂ R3 be
a homeomorphism. Let S be a connected compact isolated invariant set such
that htop(f |S) = 0. Then the sequence (i(fn, S))n≥1 is periodic.

We will denote by cl(Y ), int(Y ) and ∂Y the closure, the interior and the
frontier of a subset Y of a topological space X.

2



1.1. Lefschetz index

The reader is referred to [2], [4] and [18] for information about the Lef-
schetz index theory. Let us denote by Sm−1 the unit sphere of the euclidian
space Rm. Let U ⊂ Rm be a neighborhood of 0 and f : U → f(U) be a
continuous map having an isolated fixed point at 0. If ε > 0 is small enough,
the map

z 7→ εz − f(εz)
‖εz − f(εz)‖

is well defined on Sm−1 and its degree does not depend on ε, it is the Lef-
schetz index i(f, 0). For example, if f is differentiable at 0 and 1 is not an
eigenvalue of Df(0), this number is equal to (−1)l, where l is the number of
real eigenvalues larger than 1. Recall the following facts in the case where f
is a homeomorphism :

- if f preserves the orientation, then i(f−1, 0) = i(f, 0) if m is even and
i(f−1, 0) = −i(f, 0) if m is odd;

- if f reverses the orientation, then i(f−1, 0) = −i(f, 0) if m is even and
i(f−1, 0) = i(f, 0) if m is odd.

Using charts one can define similarly the Lefschetz index of a continuous
map on a manifold at an isolated fixed point x. Such a definition has been
extended by Dold to the case where f is defined on an ANR. Among the
properties of the index, just recall that if x is an attracting fixed point
(i.e. there exists a compact neighborhood V of x satisfying f(V ) ⊂ V and⋂
n≥0 f

n(V ) = {x}) then one has i(f, x) = 1. If S ⊂ X is a compact
invariant set (i.e. f(S) = S) and if there is a neighborhood V of S whose
fixed points are all contained in S, one may define the index i(f, S) ∈ Z. In
the case where there are finitely many fixed points in S, it is equal to the
sum of all Lefschetz indices of fixed points that are in S. A particular case
is the case where X is compact and S = X. The number i(f,X) is called
the Lefschetz number and denoted by Λ(f). It is related to the action on the
singular homology groups by the Lefschetz-Dold formula

Λ(f) =
n∑
i=0

(−1)itr(f∗i),

where f∗i is the morphism induced by f on Hi(X,Q) and n the biggest
integer such that Hi(X,Q) 6= 0.

1.2. Discrete Conley index, filtration pairs

Let U ⊂ M be an open set of a m-dimensional manifold M and f :
U → f(U) ⊂ M be a homeomorphism. A compact invariant set S is iso-
lated with respect to f if there exists a compact neighborhood N of S such

3



that
⋂
k∈Z f

−k(N) = S. The neighborhood N is called an isolating neigh-
borhood of S. Let us say that S is an attractor (resp. a repeller) if there
exists a compact neighborhood N of S such that

⋂
k∈N f

k(N) = S (resp.⋂
k∈N f

−k(N) = S). Attractors and repellers are special classes of isolated in-
variant sets. An isolating block N is a compact set such that cl(int(N)) = N
and f−1(N) ∩ N ∩ f(N) ⊂ int(N). Isolating blocks are a special class of
isolating neighborhoods.

We consider the exit set of a set N ⊂ U to be defined as

N− = {x ∈ N : f(x) /∈ int(N)}.

Let S be an isolated invariant set and suppose L ⊂ N is a compact
pair contained in U . The pair (N,L) is called a filtration pair for S (see
Franks and Richeson paper [6]) provided N and L are each the closure of
their interiors and

1) cl(N \ L) is an isolating neighborhood of S,

2) L is a neighborhood of N− in N ,

3) f(L) ∩ cl(N \ L) = ∅.

We will recall in the next section how to construct a filtration pair (N,L)
that is simplicial. In that case, the quotient space NL obtained from the pair
(N,L) by an identification of L to a point [L] is an ANR. The properties
of filtration pairs imply that this identification induces a continuous map
f̄ : NL → NL that fixes [L] and that sends each point x 6∈ L to the projection
in NL of f(x). The fixed point [L] is an attractor in the following strong
sense : the map f̄ is locally constant equal to [L] on a neighborhood of [L]. In
fact the dynamics of f̄ is easy to understand : for every point x, either there
exists an integer n ≥ 0 such that f̄n(x) = [L] or its ω-limit set is included in
S. One can apply the Lefschetz-Dold formula to each iterate of f̄ . Using the
fact that [L] is an attracting fixed point and that Hi(NL,Q) ∼ Hi(N,L,Q)
(where ∼ means isomorphic) , one gets

Λ(f̄n) =
m∑
i=0

(−1)itr(fn∗i),

where
Λ(f̄n) = i(f̄n, [L]) + i(f̄n, S) = 1 + i(fn, S)

The second aim of this article is to prove the following result:

Theorem 3. For any periodic sequence of integers (In)n≥1 satisfying Dold’s
necessary congruences, there exists an orientation preserving homeomor-
phism of R3 that fixes {0}, that does not fixed any other compact set, such
that {0} admits an isolating block which is a topological ball and such that
i(fn, 0) = In for every n ≥ 1.

4



The paper is organized as follows: the next section is devoted to the
proof of the main theorem of this paper and to presenting its dynamical
consequences. Section 3 is dedicated to Theorem 3.

2. Proof of the Main Theorem.

2.1. Topological entropy

The topological entropy is a numerical invariant related to the orbit
growth. It represents the exponential growth rate for the number of orbit
segments distinguishable with arbitrary fine precision. Let f : X → X be
a continuous map with X a compact metric space. We define an increasing
sequence of metrics dfn, n = 1, 2, . . . , with df1 = d,

dfn(x, y) = max
0≤i≤n−1

d(f i(x), f i(y)).

We denote the open balls

Bf (x, ε, n) = {y ∈ X : dfn(x, y) < ε}.

A set E ⊂ X is said to be (n, ε)-spanning if X ⊂
⋃
x∈E Bf (x, ε, n). If

Sd(f, ε, n) is the minimal cardinality of an (n, ε)-spanning set then we define
the topological entropy of f as

htop(f) = lim
ε→0

lim sup
n→∞

1
n

logSd(f, ε, n).

This definition does not depend on the metric d but only on the topology
of X.

It is known that the restriction of a map to the non-wandering set Ω(f)
captures the entropy of the system. If Ω(f) is the (closed and invariant) set
of points x ∈ X such that for every open neighborhood U of x there is n > 0
such that fn(U) ∩ U 6= ∅, then we have the following equality

htop(f) = htop(f |Ω(f))

due to Bowen (see [12], [20] or [27] for details).
Recall now the following result of Manning (Theorem 2 in [16]) which

can be applied to spaces with nice local properties like finite CW-complexes
of compact ANR’s.

Theorem 4. If X is a compact metric space with metric d such that:
i) For every ε > 0 there exists δ > 0 such that if d(x, y) < δ then there

exists a path σ : [0, 1]→ X from x to y with diam(σ([0, 1]) < ε.
ii) There exists ε0 > 0 such that any loop of diameter < ε0 is homotopi-

cally trivial in X.
Then for any continuous map f : X → X

htop(f) ≥ log(|λ|)

for every eigenvalue λ of f∗1 : H1(X,Q)→ H1(X,Q).

5



2.2. Some algebraic lemmas

Recall the classical following results :

Proposition 1. Let A and B two finitely generated C-vector spaces and
u : A → A and v : B → B two linear maps. If tr(uk) = tr(vk) for every
k ≥ 1, then u and v have the same non-zero eigenvalues counted with their
multiplicities

Proof. If {λ1, . . . , λr} and {µ1, . . . , µs} are the non-zero eigenvalues of u and
v respectively, we have

∞∑
k=1

tr(uk)
k

Zk =
∞∑
k=1

λk1 + · · ·+ λkr
k

Zk = −
r∑
i=1

log(1− λiZ).

The hypothesis gives us the polynomial equality

r∏
i=1

(1− λiZ) =
s∏
j=1

(1− µjZ),

which implies that r = s and that for every j there exists ij such that
µj = λij .

Proposition 2. Let A be a finitely generated C-vector space and u : A→ A
be a morphism which can be represented by a matrix (aij) such that aij ∈ Z
for all i, j. If all the eigenvalues of u have modulus ≤ 1, then all the non-zero
eigenvalues are roots of unity.

Proof. The characteristic polynomial of u may be written

Pu(X) = Xs
∏

1≤i≤r
(X − λi),

where λ1,. . . , λr are the non-zero eigenvalues of u. The coefficients of Pu
being integers, one deduces that

∏
1≤i≤r λi ∈ Z. By hypothesis, this implies

that each λi satisfies |λi| = 1 and may be written λi = e2iπαi , where αi ∈
R/Z. One knows that every point of Rr/Zr is a recurrent point of the rotation

(t1, . . . , tr) 7→ (t1 + α1, . . . , tr + αr).

One deduces that there exists n > 0 such that every complex number λni =
e2iπnαi is close to 1. This implies that tr(un) = λn1 + · · · + λnr is close to r.
But this number being an integer must be equal to r and this implies that
each λni is equal to 1.

6



2.3. Proof of Theorem 2.

Let us begin with a more detailed construction of filtration pairs (see also
[6]).

Proposition 3. Let U ⊂ Rm be an open set and f : U → f(U) ⊂ Rm be
a homeomorphism. If S is a connected invariant compact set isolated with
respect to f , there exists a filtration pair (N,L) for S that is homeomorphic
to a finite simplicial pair: the sets N , L and cl(N \L) are compact topological
m-dimensional manifolds. More precisely:

1) if S is neither an attractor, nor a repeller, one may suppose that N
is connected, that L is not empty and that no bounded component of
Rm \ L is included in N ;

2) if S is a repeller, one may suppose that N is connected, that L is not
empty and that there exists a unique bounded component of Rm \L that
is included in N and this component contains S;

3) if S is an attractor, one may suppose that N is connected and that L
is empty.

Proof. Suppose first that S is neither an attractor, nor a repeller. One
constructs first an isolating block, that means an isolating neighborhood N
of S satisfying

f(N) ∩N ∩ f−1(N) ⊂ int(N),

(see [6]) for example). Replacing N by a small neighborhood that is a m-
dimensional manifold, one may suppose that N is itself a manifold. The
connected component N ′ of N that contains S satisfies a similar equality
because ∂N ′ ⊂ ∂N . Replacing N by N ′ one may suppose that N is con-
nected. The exit set N− is not empty because S is not an attractor. The
inclusions N− ⊂ N and f(N−) ∩N ⊂ ∂N imply that

f(N−) ∩N ∩ f−1(N) = ∅.

One can find a small neighborhood L of N− in N satisfying

f(L) ∩N ∩ f−1(N) = ∅,

such that (N,L) is a finite simplicial pair. Observe that (N,L) is a filtration
pair because cl(N \ L) ⊂ N ∩ f−1(N).

WriteWi, 1 ≤ i ≤ r for the bounded connected components of Rm\L that
are included in N and define L′ = L ∪ (

⋃
1≤i≤rWi). To get the proposition,

one must prove that (N,L′) is a filtration pair. As one knows that

cl(N \ L′) ⊂ cl(N \ L) ⊂ N ∩ f−1(N),

7



it is sufficient to prove that for every i ∈ {1, . . . , r} one has S ∩ cl(Wi) = ∅
and f(Wi) ∩N ∩ f−1(N) = ∅. From

f(∂Wi) ∩ cl(Wi) ⊂ f(L) ∩ cl(N \ L) = ∅,

one deduces that either cl(Wi) ⊂ f(Wi) or cl(Wi)∩cl(f(Wi)) = ∅. In the first
case the connected S must be included inWi and equal to

⋂
k≥0 f

−k(cl(Wi)).
It would be a repeller. So one knows that cl(Wi)∩cl(f(Wi)) = ∅ and therefore
that S ∩ cl(Wi) = ∅.

The inclusion Wi ⊂ N \N− implies that f(Wi) ⊂ int(N) and therefore
that f(∂Wi) ⊂ N . The inclusion ∂Wi ⊂ L implies that

f(∂Wi) ∩ f−1(N) ⊂ f(L) ∩N ∩ f−1(N) = ∅.

By connectedness of f−1(N), one deduces that either f−1(N) ⊂ f(Wi) or
f−1(N)∩ cl(f(Wi)) = ∅. The first case does not occur because S ⊂ f−1(N)
and S ∩ f(Wi) = ∅. So we are in the second case and we deduce that
f(Wi) ∩N ∩ f−1(N) = ∅.

In the case where S is a repeller, one may find an isolating block N such
that N ⊂ f(int(N)). We can suppose that it is a connected m-dimensional
manifold and construct in a similar way a pair (N,L). Observe that ∂N ⊂
N− ⊂ L. To get the proposition, keeping the same notations as before, we
must prove that one of the Wi contains S. In other words we must prove
that the connected component of Rm \ L that contains S is bounded and
contained in N . It is a consequence of the fact that Wi meets N (it contains
S) but does not meet ∂N .

In the case where S is an attractor one may find an isolating block n such
that f(N) ⊂ int(N). Here again one may suppose that N is a connected
compact m-dimensional manifold. The exit set is empty and (N, ∅) satisfies
the proposition.

Suppose that (N,L) is a filtration pair that satisfies the previous propo-
sition and write f̄ : NL → NL for the induced map.

Proposition 4. In the cases 1) and 2) one has H0(NL,Q) ∼ Q, in the case
3) one has H0(NL,Q) ∼ Q2, in all cases one has f̄∗0 = Id.

In the cases 1) and 3) one has Hm(NL,Q) = {0}, in the case 2) one
has Hm(NL,Q) ∼ Q. In this last case one has f̄∗m = Id if f preserves the
orientation and f̄∗m = −Id if f reverses the orientation.

Proof. In cases 1) and 2), the set N being connected and L being non empty,
one knows that NL is connected. Therefore H0(NL,Q) = Q and obviously
f̄∗0 = Id. In case 3), NL is no more connected because one add an isolated
point. This point is fixed and the other component N is sent into itself,
therefore H0(NL,Q) ∼ Q2 and f̄∗0 = Id.

8



Recall thatHm(NL,Q) is equal to the relative homology groupHm(N,L,Q).
By Alexander’s duality theorem (see [26], page 313) one knows that

Hm(N,L,Q) ∼ H0(Rm \ L, Rm \N, Q).

Therefore one must look at locally constant functions defined on Rm \ L
that vanishes on Rm \N . Equivalently, one must look at bounded connected
components of Rm \L that are included in N . There is none in cases 1) and
3) , there is exactly one in case 2). In this case a generator of Hm(N,L,Q) is
given by (W,∂W ) where W is the connected component of Rm \L that con-
tains S. Observe now that the pair (f(W ), f(∂W )) ⊂ (N,L) is homologous
to (W,∂W ) if f preserves the orientation and to -(W,∂W ) if f reverses the
orientation.

Let us prove now Theorem 2. We suppose thatm = 3 and that htop(f |S) =
0. We choose a filtration pair (N,L) satisfying Proposition 3. Suppose first
that S is neither an attractor, nor a repeller. From the two equalities

Λ(f̄n) = tr(f̄n∗0)− tr(f̄n∗1) + tr(f̄n∗2)− tr(f̄n∗3)

and
Λ(f̄n) = i(f̄n, [L]) + i(f̄n, S) = 1 + i(fn, S),

and from Proposition 4 we deduce that

i(fn, S) = −tr(f̄n∗1) + tr(f̄n∗2).

To prove that the sequence (i(fn, S)n≥1 is bounded, it is sufficient to
prove that the spectral radius of both maps f̄n∗1 and f̄n∗2 is bounded by 1.

Since Hi(NL,Q) = Hi(NL,Z) ⊗ Q, Proposition 2 will imply that the
eigenvalues are roots of unity, which will give us the periodicity condition.
The fact that the spectral radius of the first map is bounded by 1 is a
consequence of Manning’s theorem. Indeed, the set of non wandering points
Ω(f̄) is included in {[L]} ∪S because Ω(f̄) \ {[L]} is a closed set included in
N \ L and invariant by f . Therefore

htop(f̄) = htop(f̄ |Ω(f̄)) = htop(f |S) = 0.

To prove a similar result for f̄∗,2 we will use the following duality argument.
Let (N,L′) be a filtration pair for f−1 satisfying Proposition 3 and let us
consider the quotient space NL′ and the induced map f−1 : NL′ → NL′ .

The pointed union M = NL ∨NL′ of NL and NL′ obtained by the iden-
tification of the points [L] and [L′] to a point ∗ is a compact metric ANR
and ∗ is an attractor of the induced map ḡ = f̄ ∨ f−1 : M → M . Applying
the Lefschetz-Dold formula, one gets

9



Λ(ḡn) = tr(ḡn∗0)− tr(ḡn∗1) + tr(ḡn∗2)− tr(ḡn∗3)

where

Λ(ḡn) = i(ḡn, ∗) + i(fn, S) + i(f−n, S).

We shall assume first that f is orientation preserving. Then, i(fn, S) =
−i(f−n, S) and it follows that Λ(ḡn) = 1. On the other hand, one knows that
tr(ḡn∗0) = 1, that each spaceHi(M,Q), 1 ≤ i ≤ 3 may be writtenHi(M,Q) =
Hi(NL,Q)⊕Hi(N ′L′ ,Q) and that each map ḡ∗i may be decomposed as ḡ∗i =
f̄∗i ⊕ f−1

∗i. One deduces that

−tr(ḡn∗1) + tr(ḡn∗2) = 0,

for every n ≥ 0. By Proposition 1, this implies that the non vanishing
eigenvalues of ḡ∗1 are equal to the non vanishing eigenvalues of ḡ∗2, with the
same multiplicities. The spectral radius of ḡ∗1 being ≤ 1 because htop(ḡ) = 0,
we have a similar result for ḡ∗2 and therefore a similar result for f̄∗2.

In the case where f is orientation reversing, we will get

−tr(ḡn∗1) + tr(ḡn∗2) = 0,

for every even number n. But this will be sufficient to prove that the non
vanishing eigenvalues of ḡ∗1 are equal, up to the sign, to the non vanishing
eigenvalues of ḡ∗2.

Is S is an attractor of f , then it is a repeller of f−1. We construct
filtration pairs (N,L) and (N ′, L′) for f and f−1 that satisfy Proposition 3.
If f is orientation preserving, by looking at Λ(f̄) and Λ(f−1) we get

1 + i(fn, S) = 2− tr(f̄n∗1) + tr(f̄n∗2),
1 + i(f−n, S) = 1− tr((f−1

∗1)n) + tr((f−1
∗2)n)− 1.

and
1 = Λ(ḡn) = 2− tr(ḡn∗1) + tr(ḡn∗2)− 1

and we deduce that
−tr((ḡn∗1) + tr((ḡn∗2) = 0.

We can conclude as in the first case. If f is orientation reversing we will
get the equality for even integers and this permits us to conclude. The case
where S is a repeller is treated in the same way.

Observe that we have proven the following:

Corollary 1. Let U ⊂ R3 be an open set and f : U ⊂ R3 → f(U) ⊂ R3 be
a homeomorphism. Let p be a fixed point of f such that {p} is an isolated
invariant set.

10



- If p is an attracting fixed point, the sequence (i(fn, p))n≥1 is constant
equal to 1;

- If p is a repelling fixed point and f preserves the orientation, the se-
quence (i(fn, p))n≥1 is constant equal to −1;

- If p is a repelling fixed point and f reverses the orientation, one as
i(fn, p) = (−1)n+1;

- if p is neither an attracting fixed point, nor a repelling fixed point, there
exist roots of unity λ1, . . . , λr, µ1, . . . , µs such that

i(fn, p) = −
r∑
i=1

λni +
s∑
j=1

µnj .

Remark 1. Let U ⊂ R3 be an open set, f : U ⊂ R3 → f(U) ⊂ R3 be an
orientation preserving homeomorphism and K ⊂ U be a continuum that is
an isolated invariant set. Using the same ideas as above we have that there
exist λ1, . . . , λr, µ1, . . . , µs such that

i(fn,K) = −
r∑
i=1

λni +
s∑
j=1

µnj .

Taking into account that htop(f |K) = htop(f−1|K) we have that

lim sup
n

log |i(fn,K)|
n

≤ htop(f |K).

2.4. Local Stable/unstable sets.

Let U ⊂ R3 be an open set and let f : U ⊂ R3 → f(U) ⊂ R3 be an
orientation preserving homeomorphism. Let p be a fixed point that is an
isolated invariant set and that is neither an attractor nor a repeller. Let us
consider a filtration pair (N,L) and let us define the local unstable set of p
in cl(N \ L):

Λ−cl(N\L)(p) =
⋂
n≥0

fn(cl(N \ L)).

A point x belongs to Λ−cl(N\L)(p) if and only if the sequence (f−n(x))n≥0 is
well defined and takes its values in cl(N \ L). Obviously the set Λ−cl(N\L)(p)
is backward invariant, which implies that

⋂
n≥0 f

−n(Λ−cl(N\L)(p)) is invariant.

As cl(N\L) is an isolating neighborhood of {p}, the set
⋂
n≥0 f

−n(Λ−cl(N\L)(p))

11



is reduced to the singleton {p}. Therefore, every x ∈ Λ−cl(N\L)(p) satisfies

lim
n→∞

f−n(x) = p.

We will prove the following:

Theorem 5. In the situation of the above paragraph, suppose in addition
that i(f q, p) = r > 0 , where q is the period of the sequence (i(fn, p))n≥1.

a) The second Alexander-Čech cohomology group with compact supports
Ȟ2
c (Λ−cl(N\L)(p) \ L,Q) contains at least r copies of Q.

b) If NL can be embedded in R3 then the image of (Λ−cl(N\L)(p) ∪ L)/L
by this embedding decomposes R3 into at least r + 1 components.

c) The set Λ−cl(N\L)(p) \ L decomposes int(N) \ L into at least r + 1 −
dimH1(cl(N \L),Q) components. In particular, if cl(N \L) is a closed
ball, Λ−cl(N\L)(p)\L decomposes int(N)\L into at least r+1 components.

Proof. The proof will involve shape theory arguments. More precisely, the
homology and cohomology theory that is a shape invariant. It is more appro-
priate, to study spaces that can have bad local behavior, than the singular
homology theory. We recommend the book of Mardešić and Segal ([17]) for
information about the theory of shape.

We keep the same notations as in the previous sections. The set

Inv−(NL, f̄) =
⋂
n≥0

f̄n(NL)

is a continuum, being the intersection of a decreasing sequence of continua
(recall that N is connected and L not empty). Observe that it is the image
of Λ−cl(N\L)(p) by the projection π : N → NL. Indeed if x ∈ Λ−cl(N\L)(p) then
for every n ≥ 0, the point x has a pre-image f−n(x) which is in N and one
has f̄n(π(f−n(x))) = π(x). Conversely, suppose that ξ ∈ (Inv−(NL, f̄) is
not equal to [L] and equal to the projection of x ∈ N \ L. A point η such
that f̄n(η) = ξ is not equal to [L]. Such a point is unique, it must be the
projection of f−n(x). We have proven that

Inv−(NL, f̄) = [L] ∪ π
(

Λ−cl(N\L)(p)
)
.

As we know that the set on the left is connected and both sets on the right are
compact, we deduce that [L] ∈ π

(
Λ−cl(N\L)(p)

)
. In other words, Inv−(NL, f̄)

is Alexandroff’s compactification of Λ−cl(N\L)(p) \ L .

Without loss of generality we can assume that q = 1. In this case we have
that all the eigenvalues of f̄∗1 and f̄∗2 are equal to 1. Since the Lefschetz

12



number can be computed also in terms of the cohomology groups the same
statement holds for

f̄∗1 : H1(NL,Q)→ H1(NL,Q)

and
f̄∗2 : H2(NL,Q)→ H2(NL,Q).

The set Inv−(NL, f̄) is the inverse limit of the sequence

. . . NL
f̄→ NL

f̄→ NL
f̄→ NL . . .

Since i(f, p) = r > 0, there are at least r eigenvalues of f̄∗2 that are equal
to 1. As a consequence, the (co)pro-group associated to the above inverse
system

. . . H2(NL,Q)
f̄∗2→ H2(NL,Q)

f̄∗2→ H2(NL,Q)
f̄∗2→ H2(NL,Q) . . .

is nontrivial and the Alexander-Čech cohomology groups

Ȟ2(Inv−(NL, f̄),Q) = Ȟ2((Λ−cl(N\L)(p)∪L)/L,Q) = Ȟ2(Λ−cl(N\L)(p)∪L,L,Q)

contain at least r copies of Q.
As we know that

Ȟ2(Λ−cl(N\L)(p) ∪ L,L,Q) = Ȟ2
c (Λ−cl(N\L)(p) \ L,Q).

(see [25], page 321, Lemma 11) we deduce that Ȟ2
c (Λ−cl(N\L)(p) \ L,Q) con-

tains at least r copies of Q. This proves a).
In order to check b) let us consider an embedding ι : NL → R3. We know

that

H̃0(R3 \ ι((Λ−cl(N\L)(p) ∪ L)/L),Q) = Ȟ2(ι((Λ−cl(N\L)(p) ∪ L)/L),Q)

(see [25], page 296, Theorem 16). The fact that Ȟ2((Λ−cl(N\L)(p)∪L)/L),Q)
contains r copies of Q implies that ι((Λ−cl(N\L)(p) ∪ L)/L) decomposes R3

into at least r + 1 components.
Now we are going to prove c). Using Theorem 10 in page 342 of [25], we

have that

H1(int(N)\L, (int(N)\L)\ (Λ−cl(N\L)(p)\L),Q) = Ȟ2
c ((Λ−cl(N\L)(p)\L),Q).

Then, from a) and the exactness of the homology sequence of the pair
(int(N) \ L, (int(N) \ L) \ (Λ−cl(N\L)(p) \ L)), if follows that

dimH0(int(N) \ L) \ (Λ−cl(N\L)(p) \ L),Q) ≥ r + 1− dimH1(cl(N \ L),Q).

13



Remark 2. Note that using shape theory arguments it is easy to
check that the covering dimension dim(Inv−(NL, f̄)) ≥ 2. Indeed, pro −
H2(Inv−(NL, f̄),Q) can be also computed in terms of the Čech expansion
of Inv−(NL, f̄), it follows that there is a cofinal family of open coverings of
Inv−(NL, f̄) whose nerves are polyhedra of dimension ≥ 2. Then, there is
a cofinal family of open coverings of Inv−(NL, f̄) of order ≥ 3. Then, the
covering dimension dim(Inv−(NL, f̄)) ≥ 2.

Remark 3. If in the above theorem f is orientation preserving and
i(f q, p) = r < 0 we obtain a similar result for the local unstable set of {p}
associated to f−1.

3. Proof of Theorem 3.

We have proved that the sequence (i(fn, p))n≥1 is periodic. Let us answer
the converse question, i.e., for any periodic sequence of integers I = (In)n≥1,
which satisfies Dold’s necessary congruences, there exists a homeomorphism
with i(fn, p) = In for every n ≥ 1. Moreover, we can construct the home-
omorphism f : R3 → R3 to be orientation preserving with no compact
invariant sets but {p}.

There are different ways to define Dold’s congruences. The simplest one
uses the normalized periodic sequences σk = (σkn)n≥1, where for each k ≥ 1,
one has

σkn =
{
k if n ∈ kN,
0 if n /∈ kN

Any sequence I = (In)n≥1 may be written formally I =
∑

k≥1 akσ
k in an

unique way. It satisfies Dold’s congruences if and only if the ak are integers.
As proved by Dold ([4]), a sequence of integers I = (In)n≥1 satisfies Dold’s
congruences if and only if there exists an ENR X, a continuous map f : X →
X and an open subset U ⊂ X such that for every n ≥ 1, the set Fix(fn)∩Un
is compact and In = iX(fn, Un), the sequence (Un)n≥1 being inductively
defined by U1 = U and Un = f−1(Un−1) ∩ U . As observed by Babenko and
Bogatyi (see [1] or [11]), a bounded sequence that satisfies Dold’s congruences
must be periodic. That means that the sum I =

∑
k≥1 akσ

k is finite.

To prove Theorem 3 we will find a map in the class of radial homeomor-
phisms. The 3-dimensional sphere is nothing but the end compactification of
S2×R where one adds to S2×R the lower end e−, adherent to S2×(−∞, 0],
and the upper end e+, adherent to S2×[0,+∞). Let us fix a homeomorphism
g of S2 and a continuous map ϕ : S2 → R. The skew product

fg,ϕ : (x, y) 7→ (g(x), y + ϕ(x))

induces on S3 a homeomorphism fixing the two ends. If one wants {e−} to
be locally maximal it is sufficient to suppose that the following property (P)
is satisfied:

14



(P) for every x ∈ S2, ϕ(x) ≥ 0⇒ ϕ(g(x)) > 0.
Indeed the property (P) implies that there exists ε > 0 such that for every
x ∈ S2, ϕ(x) ≥ −ε⇒ ϕ(g(x)) ≥ ε. Writing

f−1
g,ϕ : (x, y) 7→ (g−1(x), y − ϕ(g−1(x))),

one deduces that every orbit (xk, yk)k∈Z of fg,ϕ satisfies at least one of the
following properties:
- yk+1 − yk ≥ ε for every k ≥ 0
or
- yk−1 − yk ≥ ε for every k ≤ 0.
This implies that either limk→+∞ yk = +∞ or limk→−∞ yk = +∞. This also
implies that each ball (S2 × (−∞, r]) ∪ {e−} is an isolating block.

Write X0 = ϕ−1([0,+∞)). The following points are easy to state:
- the exit set ofN = (S2×(−∞, 0])∪{e−} isN− = {(x, y) |x ∈ X0, −ϕ(x) ≤
y ≤ 0};
- the unstable set of e− is X− × R ∪ {e−} where X− =

⋂
k≥0 g

k(X0), it is
the infinite cone over X−;
- the stable set of e− is X+ × R where X+ =

⋂
k≤0 g

k(S2 \X0);
- one obtains a filtration pair (N,L) writing L = {(x, y) |x ∈ X, −ψ(x) ≤
y ≤ 0} where X ⊂ S is a subsurface with boundary, neighborhood of X0 sat-
isfying g(X) ⊂ int(X0), and ψ : X → [0,∞) a continuous function vanishing
exactly on ∂X and satisfying ψ(g(x)) > ψ(x)− ϕ(x) > 0 for every x ∈ X.

One can compute the Lefschetz index i(fng,ϕ, e−) by looking at the induced
maps on the spaces Hi(N,L,Q). We will give an alternative way, supposing
for convenience that X0 itself is a subsurface with boundary.

Proposition 5. For every n ≥ 1, one has i(fng,ϕ, e−) = 1 − Λ((g|X0)n),
where

Λ((g|X0)n) = tr((g|X0)∗0)n − tr((g|X0)∗1)n

is the Lefschetz number of the n-th iterate of g|X0 : X0 → X0.

Proof. Write N+ = {(x, y) |x ∈ X0, 0 ≤ y ≤ ϕ(x)} and observe that N ′ =
N ∪N+ is a 3-dimensional sphere. One constructs a continuous map fg,ϕ :
N ′ → N ′ satisfying

fg,ϕ(x, y) =
{

(g(x), 0) if (x, y) ∈ N− ∪N+,
fg,ϕ(x, y) if (x, y) /∈ N− ∪N+.

Observe that both sets N ′ and N− ∪ N+ are attracting sets of fg,ϕ and
that r : (x, y) 7→ (x, 0) is a strong deformation retraction from N− ∪N+ to

15



X0 × {0} that satisfies r ◦ fg,ϕ = g|X0 ◦ r. This implies that

1 = Λ((fg,ϕ)n)

= i(fng,ϕ, e
−) + Λ((fg,ϕ|N∪N ′)n)

= i(fng,ϕ, e
−) + Λ((g|X0)n).

Now we can prove Theorem 3. Let I = (1 + a1)σ1 +
∑

2≤k≤k0 akσ
k be a

periodic sequence that satisfies Dold’s congruences. Write

A− = {k ≥ 1 | ak < 0}, A+ = {k ≥ 1 | ak > 0},

and define
a− =

∑
k∈A−

ak, a
+ =

∑
k∈A+

ak.

Choose−a− closed disks (Zki )k∈A−, 1≤i≤−ak
and a+ closed annuli (Zki )k∈A+, 1≤i≤ak

on S2, all pairwise disjoint. One can easily construct an orientation preserv-
ing homeomorphism g of S2 that satisfies the following properties :
- for each k ∈ A− and each i ∈ {1, . . . ,−ak} there exist k disjoint closed
disks Dk

i,j , j ∈ Z/kZ, in the interior of Zki such that g(Dk
i,j) ⊂ int(Dk

i,j+1)
and such that the maximal invariant set contained in Xk

i =
⋃
j∈Z/kZD

k
i,j is

a periodic orbit of period k;
- for each k ∈ A+ and each i ∈ {1, . . . , ak} there exist k disjoint closed disks
Dk
i,j , j ∈ Z/kZ, in the interior of Zki such that g−1(Dk

i,j) ⊂ int(Dk
i,j+1), such

that Xk
i = Zki \

⋃
j∈Z/(k)Z int(Dk

i,j) is an attracting set of g and such that
the maximal invariant set contained in Xk

i is a wedge of k + 1 circles.
Choose now a continuous function ϕ : S2 → R that is positive in the

interior of X0 =
⋃

1≤k≤k0, 1≤i≤|ak|X
k
i and negative outside X0. Observe that

Λ((g|Xk
i
)n) =

{
k if n ∈ kN,
0 if n /∈ kN

if k ∈ A−, and

Λ((g|Xk
i
)n) =

{
−k if n ∈ kN,
0 if n /∈ kN

if k ∈ A+. By Proposition 5, one deduces that the sequence J = (i(fng,ϕ, e
−))n≥1

satisfies

J = σ1 −
∑
k∈A−

∑
1≤i≤−ak

σk −
∑
k∈A+

∑
1≤i≤ak

(−σk)

= σ1 +
∑
k∈A−

akσ
k +

∑
k∈A+

akσ
k

= I

16



�

Remark 4. Of course, a given sequence I corresponds to infinitely
many such constructions. In the construction above, the sequence I0 = σ1

corresponds to a map fg,ϕ such that ϕ < 0 (the ball N being an attracting
set), the sequence I1 = −σ1 to a map fg,ϕ such that X0 is the union of two
attracting disks (an explicit example being given by the hyperbolic linear
map l : (x1, x2, x3) 7→ (x1/2, x2/2, 2x3)), the sequence I2 = σ1−σ2 to a map
fg,ϕ such that X0 is the union of two permuting disks (an explicit example
being given by l′ : (x1, x2, x3) 7→ (x1/2, x1/2,−2x3)). Observe that I0 also
corresponds to a map fg,ϕ such that X0 is an attracting annulus (an explicit
example being given by l−1) and that I1 also corresponds to a map fg,ϕ such
that ϕ > 0 (the ball N being a repelling set).

Remark 5. Applying the previous proposition to f−1
g,ϕ : (x, y) 7→

(g−1(x), y − ϕ(g−1(x))), one knows that i(f−ng,ϕ , e−) = 1− Λ((g−1|S2\g(X0
)n)

for every n ≥ 1. As we are on a 2-dimensional sphere and as we know that
g(X0) ⊂ X0 we deduce that

Λ((g−1|S2\g(X0))
n) = i(g−n, X+) = i(gn, X+),

which implies that

i(f−ng,ϕ , e
−) + (fng,ϕ, e

−) = 2− i(gn, X−)− i(gn, X+) = 0

by the Poincaré-Hopf formula.

Remark 6. All the constructions above may be done in higher dimen-
sion. Radial homeomorphisms may be constructed similarly and Proposition
5 may be generalized. Replacing all the disks by balls and the annuli by filled
tori in the construction above, one gets a generalization of Theorem 3 in any
dimension. These constructions permit us to understand why Theorem 1
is not true in higher dimension. One can construct a diffeomorphism g on
S3 satisfying the following: there exists an embedded torus T cutting the
sphere in two solid tori T− and T+ such that both maps g : T+ → T+ and
g−1 : T− → T− are solenoidal maps of degree m, that means conjugate to
a mapping

(θ, z) 7→ (θm,
1
2
θ + εzθk−m+1)

defined on the filled torus {θ ∈ C | |θ| = 1}×{z ∈ C | |z| ≤ 1} (see for example
[10]). Let us consider a function ϕ : S3 → R positive (resp. negative) on the
interior of T+ (resp. T−). The unstable manifold of e− is an infinite cone
over the solenoid

⋂
n≥0 g

n(T+) and one knows that for every n ≥ 1

i(fng,ϕ, e
−) = 1− L((g|T+)n) = 1− (1−mn) = mn

which implies that limn→+∞ i(fng,ϕ, e
−) = +∞.

17



The authors would like to thank Federico Rodríguez Hertz, María Ale-
jandra Rodríguez Hertz and Raúl Ures for indicating us a mistake in the
construction of a previous example.

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19



Patrice Le Calvez
Institut de Mathématiques de Jussieu (UMR 7586 CNRS), UPMC 175

rue du Chevaleret 75013 Paris, France.
E-mail: lecalvez@math.jussieu.fr

Francisco R. Ruiz del Portal
Departamento de Geometría y Topología, Facultad de CC.Matemáticas,

Universidad Complutense de Madrid, Madrid 28040, Spain.
E-mail: R−Portal@mat.ucm.es

José Manuel Salazar.
Departamento de Matemáticas. Universidad de Alcalá. Alcalá de Henares.

Madrid 28871, Spain.
E-mail: josem.salazar@uah.es

20