# The Morse equation in the Conley index theory for discrete multivalued
dynamical systems

Bogdan Batko  Bogdan Batko, Department of Mathematics, Rutgers University,
Piscataway, NJ 08854, USA<EMAIL_ADDRESS>On leave from Division of
Computational Mathematics, Faculty of Mathematics and Computer Science,
Jagiellonian University, ul. St. Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS>
###### Abstract.

A recent generalization of the Conley index to discrete multivalued dynamical
systems without a continuous selector is motivated by applications to
data–driven dynamics. In the present paper we continue the program by studying
attractor–repeller pairs and Morse decompositions in this setting. In
particular, we prove Morse equation and Morse inequalities.

###### 2010 Mathematics Subject Classification:

primary 54H20, secondary 37B30, 37B35, 37M10, 54C60

The author was partially supported by the Polish National Science Center under
Maestro Grant No. 2014/14/A/ST1/00453, Opus Grant No. 2019/35/B/ST1/00874, and
the DARPA contract HR0011-16-2-0033.

††footnotetext: Keywords and phrases. Morse equation, Morse inequalities,
Morse decomposition, Discrete multivalued dynamical system, Conley index,
Attractor, Repeller, Invariant set, Isolating neighborhood, Index pair, Weak
index pair.

##  1\. Introduction

The present data–driven world sets new challenges to contemporary science.
There is a growing interest in coarse theories capable to extract robust
information hidden in noisy experimental data. A good example is the rapid
development in persistent homology (cf. [15]) invented to investigate
homological features of topological spaces known only from a cloud of points
or samples. Conley theory (cf. [12]) provides a robust topological invariant
for studying dynamical systems. It has been used to prove the existence of
stationary and periodic solutions, heteroclinic connections and chaotic
invariant sets. The potential of Conley theory in the context of data became
apparent via a generalization to multivalued dynamical systems [21] proposed
as a tool in computer assisted proofs in dynamics. Multivalued dynamics,
important on its own right (cf. e.g. [1]), becomes in such proofs a tool in
the study of single valued dynamics (cf. e.g. [28, 30, 40, 35, 22]). However,
the practical use of the generalization [21] has been severely restricted by
some strong assumptions difficult to be fulfilled in practice. Fortunately, a
recent revision of the theory (cf. [5, 3]) removes these limitations opening
the way to applications in data–driven dynamics.

Assume time series data have been collected as a result of a measurement, an
experiment, or an observation of an unknown dynamical system. Using similar
techniques as in [29, 39, 7, 4] one can construct a multivalued map that
represents an unknown generator of the underlying system. Both, toy examples
(cf. [5, 3]) and constructions for real applications (cf. [7, 8]) show that
usually such a map does not admit a continuous selector. Nevertheless, we can
identify isolated invariant sets, and hire the Conley index theory for
multivalued maps, because the construction of the index works under minimal
assumptions on a multivalued map; in particular, it does not require single
valued selectors. Eventually, we extend the obtained results to the underlying
unknown single valued dynamical system using continuation properties of the
index.

In the present paper we continue the program that we started in [5, 3].
Heading towards the comprehensive description of the dynamics we need to gain
more insight into the internal structure of isolated invariant sets. One of
the relevant descriptors is a Morse decomposition of an isolated invariant set
and the associated Morse equation. Leaving mathematical sophistication aside,
one can think that the Morse decomposition is a decomposition of a given
isolated invariant set into a finite union of pairwise disjoint isolated
invariant sets, called Morse sets, and connecting orbits between them, such
that outside the Morse sets the dynamics is gradient–like. There is a rich
variety of approaches to global dynamics that involve Morse decompositions;
see for instance [17, 19, 18, 2] or [11, 27] for continuous-time dynamical
systems without uniqueness. For recent results that combine classical and
combinatorial dynamics we refer to [34, 14, 6]. An approach to the poset
structure of Morse decomposition with the emphasis on lattice structures of
attractors, for both an underlying dynamical system and its
combinatorialization - a combinatorial multivalued map, is demonstrated in the
series of papers [23, 24, 25, 26]. The presented list of papers is not
complete and does not pretend to be complete in any sense.

The Morse equation describes the relationship between the Conley index of an
isolated invariant set and the Conley indices of its Morse sets. In
particular, information on (co)homologically nontrivial connections between
Morse sets can be derived. The classical Morse inequalities concern
nondegenerate critical points of a gradient flow on a compact manifold, and
show the correspondence between the $k$-th Betti number of the manifold and
the number of critical points with Morse index $k$, that is critical points
with the $k$-dimensional unstable invariant manifold (cf. e.g. [9, 10]). One
of the possible generalizations of the classical inequalities due to Morse and
to Smale (cf. [37]) in the Conley index theory was developed by Conley and
Zehnder to flows (cf. [13]). Afterwords, the Morse equation in the Conley
index theory was proved by Rybakowski in [36] for semiflows, and in the
discrete time case by Franks [16] and Mrozek [33].

Assume that we are given a (continuous– or discrete–time) dynamical system on
a locally compact metric space. A set $S$ is said to be isolated invariant if
it is the maximal invariant set contained in some compact neighborhood of
itself. In the cohomological Conley index theory with such a set one
associates a special pair of sets, called an index pair. Then, the
cohomological Conley index of $S$ is defined to be the Alexander–Spanier
cohomology of the index pair. One proves that, in line with the need, this is
an invariant of $S$. Thus, we can associate with $S$ the Poincaré series
$p(t,S)$, the power series in $t$ with the ranks of the cohomology modules as
coefficients. Now, assume that $\text{$\mathcal{M}$}:=\\{M_{i}\ |\
i\in\\{1,2,\dots,n\\}\\}$ is a Morse decomposition of $S$. Hence, in
particular, the sets $M_{i}$ are pairwise disjoint isolated and invariant
subsets of $S$. Then, the Morse equation takes the form

$\sum_{i=1}^{n}p(t,M_{i})=p(t,S)+(1+t)Q(t),$

where $Q$ is a formal power series with nonnegative integer coefficients (cf.
[13]). The terms in $Q$ provide information about nontrivial connections
between pairs of Morse sets. One can observe that the above equation
generalizes the classical Morse inequalities (cf. e.g. [36, 38]).

The aim of this paper is to prove the Morse equation in the Conley index
theory for discrete multivalued dynamical systems.

The organization of the paper is as follows. In Section 2 we provide basic
definitions related to the Conley index theory for multivalued maps. Section 3
is devoted to Morse decompositions. We define repeller–attractor pairs and
provide a characterization of Morse sets via the associated sequence of
attractors. The key step for the prove of the Morse equation is the
construction of the, so called, index triple for a given repeller–attractor
pair. This is presented in Section 4. Finally, we prove the Morse equation in
Section 5.

##  2\. Preliminaries

Throughout the paper $\mathbb{Z}$ and $\mathbb{N}$ will stand for the sets of
all integers and all positive integers, respectively. By an interval in
$\mathbb{Z}$ we mean the trace of a real interval in $\mathbb{Z}$.

Given a topological space $X$ and a subset $A\subset X$, by
$\operatorname{int}_{X}A$, $\operatorname{cl}_{X}A$ we denote the interior of
$A$ in $X$ and the closure of $A$ in $X$ respectively. We omit the subscript
$X$ if the space is clear from the context.

Let $X$, $Y$ be topological spaces, and let $F\colon X\multimap Y$ denote a
multivalued map, that is a map $F\colon X\ni x\mapsto
F(x)\in\text{$\mathcal{P}$}(Y)$, where $\text{$\mathcal{P}$}(Y)$ is the power
set of $Y$. We define an effective domain of $F$ by
$\operatorname{dom}(F):=\\{x\in X\ |\ F(x)\neq\emptyset\\}.$ A multivalued map
$F$ is upper semicontinuous if for any closed $B\subset Y$ its large counter
image under $F$, that is the set $F^{-1}(B):=\\{x\in X\ |\ F(x)\cap
B\neq\emptyset\\}$, is closed.

Assume that a multivalued self-map $F\colon X\multimap X$ is given.

Let $I$ be an interval in $\mathbb{Z}$ with $0\in I$. We say that a single
valued mapping $\sigma\colon I\to X$ is a _solution for $F$ through $x\in X$_
if $\sigma(n+1)\in F(\sigma(n))$ for all $n,n+1\in I$ and $\sigma(0)=x$. Given
a subset $N\subset X$, the set

$\operatorname{Inv}(N,F):=\\{x\in N\ |\
\exists\sigma\colon\mbox{$\mathbb{Z}$}\to N\text{ a solution for }F\text{
through }x\\}$

is called the invariant part of $N$. A compact subset $N\subset X$ is an
isolating neighborhood for $F$ if
$\operatorname{Inv}(N,F)\subset\operatorname{int}N$. A compact set $S\subset
X$ is said to be invariant with respect to $F$ if $S=\operatorname{Inv}(S,F)$.
It is called an isolated invariant set if it admits an isolating neighborhood
$N$ for $F$ such that $S=\operatorname{Inv}(N,F)$ (cf. [5, Definition 4.1,
Definition 4.3]).

######  Definition 2.1 (cf. [5, Definition 4.7]).

Let $N\subset X$ be an isolating neighborhood for $F$. A pair
$P=(P_{1},P_{2})$ of compact sets $P_{2}\subset P_{1}\subset N$ is called a
weak index pair in $N$ if

  * (a)

$F(P_{i})\cap N\subset P_{i}$ for $i\in\\{1,2\\}$,

  * (b)

$\operatorname{bd}_{F}P_{1}:=\operatorname{cl}P_{1}\cap\operatorname{cl}(F(P_{1})\setminus
P_{1})\subset P_{2}$,

  * (c)

$\operatorname{Inv}(N,F)\subset\operatorname{int}(P_{1}\setminus P_{2})$,

  * (d)

$P_{1}\setminus P_{2}\subset\operatorname{int}N$.

Given a weak index pair $P$ in an isolating neighborhood $N\subset X$ for $F$
we set

$T_{N}(P):=(T_{N,1}(P),T_{N,2}(P)):=(P_{1}\cup(X\setminus\operatorname{int}N),P_{2}\cup(X\setminus\operatorname{int}N)).$

Assume $F$ is determined by a morphism, which in particular holds true if $F$
has acyclic values (cf. [20]). Following [5] recall that $F_{P}$, the
restriction of $F$ to the domain $P$, is a multivalued map of pairs,
$F_{P}\colon P\multimap T_{N}(P)$; the inclusion $i_{P}:P\to T_{N}(P)$ induces
an isomorphism in the Alexander–Spanier cohomology; and the index map
$I_{F_{P}}$ is defined as an endomorphism of $H^{*}(P)$ given by

$I_{F_{P}}=F_{P}^{*}\circ(i_{P}^{*})^{-1}.$

Applying the Leray functor $L$ (cf. [31]) to the pair $(H^{*}(P),I_{F_{P}})$
we obtain a graded module $L(H^{*}(P),I_{F_{P}})$ with its endomorphism,
called the Leray reduction of the Alexander–Spanier cohomology of a weak index
pair $P$. By [5, Definition 6.3], this is the cohomological Conley index of
$\operatorname{Inv}(N,F)$, which we denote by $C(\operatorname{Inv}(N,F),F)$.

We introduce the following notation for future use. Given $N\subset X$, $x\in
N$ and $n\in\mbox{$\mathbb{Z}$}^{+}$ we put:

$\displaystyle F_{N,n}(x)$ $\displaystyle:=$ $\displaystyle\\{y\in
N\mid\exists\,\sigma:[0,n]\to N\mbox{ a solution for }F$
$\displaystyle\mbox{\hskip 120.92421ptwith }\sigma(0)=x,\,\sigma(n)=y\\},$
$\displaystyle F_{N,-n}(x)$ $\displaystyle:=$ $\displaystyle\\{y\in
N\mid\exists\,\sigma:[-n,0]\to N\mbox{ a solution for }F$
$\displaystyle\mbox{\hskip 129.46011ptwith }\sigma(-n)=y,\,\sigma(0)=x\\},$
$\displaystyle F_{N}^{+}(x)$ $\displaystyle:=$
$\displaystyle\bigcup\mbox{$\\{\,F_{N,n}(x)\,|\,n\in\mbox{$\mathbb{Z}$}^{+}\,\\}$},$
$\displaystyle F_{N}^{-}(x)$ $\displaystyle:=$
$\displaystyle\bigcup\mbox{$\\{\,F_{N,-n}(x)\,|\,n\in\mbox{$\mathbb{Z}$}^{+}\,\\}$}.$

##  3\. Morse decompositions and repeller–attractor pairs

Given a solution $\sigma:\mbox{$\mathbb{Z}$}\to X$ of a multivalued upper
semicontinuous map $F:X\multimap X$, we define its $\alpha$– and
$\omega$–limit sets respectively by

$\alpha(\sigma):=\bigcap_{k\in\mbox{$\mathbb{Z}$}}\operatorname{cl}\sigma((-\infty,k]),\quad\quad\omega(\sigma):=\bigcap_{k\in\mbox{$\mathbb{Z}$}}\operatorname{cl}\sigma([k,+\infty)).$

Let us point out that unlike in the single valued case, we define $\alpha$–
and $\omega$–limit sets for a given solution $\sigma$ through an $x\in X$, not
for an $x$ itself, because backward nor forward solutions through $x$ need not
be unique.

######  Definition 3.1 (cf. [6, Definition 3.9]).

Let $S$ be an isolated invariant set of $F:X\multimap X$. We say that the
family $\text{$\mathcal{M}$}:=\\{M_{r}\ |\ r\in\mbox{$\mathbb{P}$}\\}$ indexed
by a poset $(\mbox{$\mathbb{P}$},\leq)$ is a Morse decomposition of $S$ if the
following conditions are satisfied:

  1. (1)

the elements of $\mathcal{M}$ are mutually disjoint isolated invariant subsets
of $S$,

  2. (2)

for every full solution $\sigma$ in $X$ there exist
$r,r^{\prime}\in\mbox{$\mathbb{P}$}$, $r\leq r^{\prime}$, such that
$\alpha(\sigma)\subset M_{r^{\prime}}$ and $\omega(\sigma)\subset M_{r}$,

  3. (3)

if for a full solution $\sigma$ in $X$ and $r\in\mbox{$\mathbb{P}$}$ we have
$\alpha(\sigma)\cup\omega(\sigma)\subset M_{r}$, then
$\operatorname{im}\sigma\subset M_{r}$.

(a) Morse decomposition $\text{$\mathcal{M}$}=\\{M_{1},M_{2},M_{3}\\}$, with
$M_{3}=[0,a]$, $M_{2}=[g,h]$, and $M_{1}=[d,e]\cup[j,k]$, of an isolated
invariant set $S=[0,k]$ of $F$. Morse sets depicted as thick lines in orange,
red and green, respectively, and the associated isolating neighborhoods as
thin lines in the same colors.

(b) Morse graph of $F$ \- a directed graph with the set of vertices
$\mathcal{M}$ and arrows presenting the existence of connecting orbits.

(c) Repeller–attractor pair $(A_{1}^{*},A_{1})$ in $S$. Attractor
$A_{1}=M_{1}=[d,e]\cup[j,k]$ and its trapping region $[c,f]\cup[i,l]$ depicted
as green lines. The dual repeller $A_{1}^{*}=[0,m]\cup[n,o]\cup[p,q]\cup[g,h]$
and its isolating neighborhood in yellow.

(d) Repeller–attractor pair $(A_{2}^{*},A_{2})$ in $S$. Attractor
$A_{2}=[d,k]$ and its trapping region $[c,l]$ in magenta. The dual repeller
$A_{2}^{*}=[0,a]=M_{3}$ and its isolating neighborhood depicted as orange
lines. Note that $[0,c]$ is an another isolating neighborhood of $A_{2}^{*}$
(cf. Lemma 3.6).

Figure 1. Graph of a multivalued map $F:[0,1]\multimap[0,1]$ in blue. Panel
(A) - Morse decomposition of an isolated invariant set $S=[0,k]$ for $F$,
panel (B) - Morse graph, panel (C) and panel (D) - repeller–attractor pairs in
$S$.

The partial order $\leq$ on $\mathbb{P}$ will be called an admissible ordering
of the Morse decomposition $\mathcal{M}$. Note that it is not uniquely
determined. Moreover, there is an ”extremal” admissible ordering $\leq_{F}$,
given by $p\leq_{F}q$ if and only if there exists a sequence of distinct
elements $p=r_{0},r_{1},\dots,r_{k}=q$ of $\mathbb{P}$ such that, for any
$j\in\\{1,2,\dots,k\\}$, there exists a solution $\sigma$ for $F$ with
$\alpha(\sigma)\subset M_{r_{j}}$ and $\omega(\sigma)\subset M_{r_{j-1}}$. The
ordering $\leq_{F}$ is extremal in the sense that any admissible ordering of
$\mathcal{M}$ is an extension of $\leq_{F}$. One can observe that for any
admissible ordering $\leq$ there exists its linear extension which is also
admissible. In the latter case, that is whenever the admissible ordering is
linear, for simplicity we write
$\text{$\mathcal{M}$}=\\{M_{1},M_{2},\dots,M_{n}\\}$.

######  Example 3.2.

Consider multivalued map $F:[0,1]\multimap[0,1]$, graph of which is presented
in Figure 1. Note that $S=[0,k]$ is an isolated invariant set for $F$ and
$\text{$\mathcal{M}$}=\\{M_{1},M_{2},M_{3}\\}$, with $M_{1}=[0,a]$,
$M_{2}=[g,h]$, and $M_{3}=[d,e]\cup[j,k]$, is a Morse decomposition of $S$
(see Figure 1(A) and (B)).

######  Definition 3.3.

We say that an isolating neighborhood $T$ for $F$ is a trapping region if
$F(T)\subset T$. A subset $A$ of an isolated invariant set $S$ is called an
attractor (in $S$) if it admits a trapping region $T$ which isolates $A$
(relative to $S$). Given an attractor $A$ in $S$, the set $A^{*}:=\\{x\in S\
|\mbox{ there exists a solution }\sigma:\mbox{$\mathbb{Z}$}\to S\mbox{ through
}x\mbox{ with }\omega(\sigma)\cap A=\emptyset\\}$ will be called the repeller
dual to $A$, and the pair $(A^{*},A)$ will be called the repeller–attractor
pair in $S$.

######  Example 3.4.

Consider Morse decomposition of an isolated invariant set $S=[0,k]$ with
respect to a multivalued map presented in Example 3.2 (see Figure 1). Note
that the pairs of sets $(A_{0}^{*},A_{0})=(S,\emptyset)$,
$(A_{1}^{*},A_{1})=([0,m]\cup[n,o]\cup[p,q]\cup[g,h],[d,e]\cup[j,k])$ (Figure
1(C)), $(A_{2}^{*},A_{2})=([0,a],[d,k])$ (Figure 1(D)) and
$(A_{3}^{*},A_{3})=(\emptyset,S)$ are repeller-attractor pairs in $S$.

Observe that $\mathcal{M}$, the sequence of attractors $\emptyset=A_{1}\subset
A_{2}\subset A_{3}=S$, and the associated sequence of dual repellers
$S=A_{0}^{*}\supset A_{1}^{*}\supset A_{2}^{*}\supset A_{3}^{*}=\emptyset$,
satisfy all the assertions of upcoming Theorem 3.9, Proposition 3.10 and
Theorem 3.11.

Assume that $A$ is an attractor and $T$ is a given trapping region of $A$. It
is easily seen that if $\sigma:\mbox{$\mathbb{Z}$}\to S$ is a solution for $F$
through an $x\in T$, say $\sigma(0)=x$, then
$\sigma(\mbox{$\mathbb{Z}$}^{+})\subset T$. Consequently,
$\omega(\sigma)\subset T$ according to the compactness of $T$. It follows that
$\omega(\sigma)\subset A$, because $\omega(\sigma)$ is invariant and
$A=\operatorname{Inv}(T,F)$. We state this simple observation as a lemma for
further references.

######  Lemma 3.5.

Let $A$ be an attractor in $S$ with a trapping region $T$. If
$\sigma:\mbox{$\mathbb{Z}$}\to S$ is a solution for $F$ with
$\operatorname{im}\sigma\cap T\neq\emptyset$ then $\omega(\sigma)\subset A$.

######  Lemma 3.6.

Let $(A^{*},A)$ be a repeller–attractor pair in $S$. Then $A^{*}$ and $A$ are
disjoint isolated invariant sets. Moreover, if $T$ is a trapping region for
$A$ then $N:=S\setminus\operatorname{int}T$ is an isolating neighborhood for
$A^{*}$.

###### Proof.

Directly by the definition, $A$ is an isolated invariant set, and $A^{*}\cap
A=\emptyset$. It is easy to see that $A^{*}\cap T=\emptyset$. Indeed,
otherwise we would have an $x\in A^{*}\cap T$, and a solution
$\sigma:\mbox{$\mathbb{Z}$}\to S$ with $\sigma(0)=x$ with $\omega(\sigma)\cap
A=\emptyset$, as $x\in A^{*}$. However, $x\in\operatorname{im}\sigma\cap T$,
which according to Lemma 3.5 means that $\omega(\sigma)\subset A$, a
contradiction.

We shall prove that $N$ is an isolating neighborhood for $A^{*}$. Clearly, $N$
is compact. Let $x\in A^{*}$. There exists a solution
$\sigma:\mbox{$\mathbb{Z}$}\to S$ through $x$ with $\omega(\sigma)\cap
A=\emptyset$. Note that $\operatorname{im}\sigma\subset\operatorname{int}N$,
because otherwise we would have $\operatorname{im}\sigma\cap T\neq\emptyset$
and in turn, by Lemma 3.5, $\omega(\sigma)\subset A$, a contradiction. This
shows the inclusions $A^{*}\subset\operatorname{Inv}N$ and
$A^{*}\subset\operatorname{int}N$. Now consider an $x\in\operatorname{Inv}N$
and a solution $\sigma:\mbox{$\mathbb{Z}$}\to N$ through $x$. By compactness
of $N$ we get $\omega(\sigma)\subset N$, showing that $\omega(\sigma)\cap
A=\emptyset$. Thus, $x\in A^{*}$, showing the inclusion
$\operatorname{Inv}N\subset A^{*}$. This completes the proof. ∎

######  Lemma 3.7.

Let $(A^{*},A)$ be a repeller–attractor pair in $S$, let
$\sigma:\mbox{$\mathbb{Z}$}\to S$ be a solution for $F$ through an $x\in S$,
and let $T$ be a trapping region for $A$. The following holds true:

  * (i)

if $x\notin T$ then $\alpha(\sigma)\subset A^{*}$,

  * (ii)

if $x\notin A^{*}\cup A$ then $\alpha(\sigma)\subset A^{*}$ and
$\omega(\sigma)\subset A$,

  * (iii)

if $\omega(\sigma)\cap A^{*}\neq\emptyset$ then
$\operatorname{im}\sigma\subset A^{*}$,

  * (iv)

if $\alpha(\sigma)\cap A\neq\emptyset$ then $\operatorname{im}\sigma\subset
A$.

###### Proof.

Set $N:=S\setminus\operatorname{int}T$ and, without loss of generality, assume
that $\sigma(0)=x$.

For the proof of (i), observe that $\sigma(\mbox{$\mathbb{Z}$}^{-})\subset N$.
By the compactness of $N$ we infer that $\alpha(\sigma)\subset N$.
Consequently, $\alpha(\sigma)\subset A^{*}$, because $\alpha(\sigma)$ is
invariant and, according to Lemma 3.6, $\operatorname{Inv}(N,F)=A^{*}$.

We shall verify (ii). Since $x\notin A^{*}$, we get $\omega(\sigma)\cap
A\neq\emptyset$. Then $\operatorname{im}\sigma\cap T\neq\emptyset$, and by
Lemma 3.5, $\omega(\sigma)\subset A$. If $\operatorname{im}\sigma\not\subset
T$ then the inclusion $\alpha(\sigma)\subset A^{*}$ follows from (i). We have
left the case $\operatorname{im}\sigma\subset T$. However, in that case we
would have $\operatorname{im}\sigma\subset A$, and in turn $x\in A$, leading
to a contradiction.

We verify (iii). For contradiction suppose that
$\operatorname{im}\sigma\not\subset A^{*}$. By Lemma 3.6,
$N:=S\setminus\operatorname{int}T$ is an isolating neighborhood for $A^{*}$,
therefore we have $\operatorname{im}\sigma\not\subset N$. This means that
$\operatorname{im}\sigma\cap T\neq\emptyset$. However, by Lemma 3.5 we get
$\omega(\sigma)\subset A$, a contradiction.

For the proof of (iv) observe that the assertion holds whenever
$\operatorname{im}\sigma\subset T$. Thus, assume the contrary. But then, by
(i), we get $\alpha(\sigma)\subset A^{*}$, a contradiction. ∎

######  Lemma 3.8.

Let $K$ be an isolating neighborhood of $S$, let $(A^{*},A)$ be a
repeller–attractor pair in $S$, and let $N\subset K$ be a compact neighborhood
of $A^{*}$ disjoint from $A$. Then $N$ is an isolating neighborhood for
$A^{*}$.

###### Proof.

Since $A^{*}\subset\operatorname{int}N$ and $N$ is compact, it suffices to
verify that $A^{*}=\operatorname{Inv}(N,F)$.

Inclusion $A^{*}\subset\operatorname{Inv}(N,F)$ is straightforward, as $A^{*}$
is invariant and $A^{*}\subset N$. To see the opposite inclusion consider an
$x\in\operatorname{Inv}(N,F)$ and a solution $\sigma$ through $x$ in $N\subset
K$. Then, by compactness of $N$ we have $\omega(\sigma)\subset N$. In
particular, $\omega(\sigma)\cap A=\emptyset$, as $N$ and $A$ are disjoint.
This along with Lemma 3.7 yields $x\in A^{*}$, and completes the proof. ∎

######  Theorem 3.9.

Let $\text{$\mathcal{M}$}:=\\{M_{1},M_{2},\dots,M_{n}\\}$ be a Morse
decomposition of an isolated invariant set $S$ with respect to an upper
semicontinuous $F:X\multimap X$. Then there exists a family
$\emptyset=A_{0}\subset A_{1}\subset A_{2}\subset\dots\subset A_{n}=S$ of
attractors in $S$ such that $M_{j}=A_{j}\cap A^{*}_{j-1}$ for
$j\in\\{1,2,\dots,n\\}$, where $A^{*}_{j}$ is a repeller dual to $A_{j}$.

###### Proof.

We set $A_{0}:=\emptyset$ and $A_{k}:=\\{x\in S\ |\ \mbox{there exists a
solution }\sigma:\mbox{$\mathbb{Z}$}\to S\mbox{ through }x\mbox{ with
}\alpha(\sigma)\subset M_{1}\cup M_{2}\cup\dots\cup M_{k}\\}$, for
$k\in\\{1,2,\dots,n\\}$.

The inclusions $\emptyset=A_{0}\subset A_{1}\subset A_{2}\subset\dots\subset
A_{n}=S$ are straightforward.

We shall prove that the sets $A_{k}$ are attractors. We proceed inductively.
First note that directly by the definition we have $A_{n}=S$; hence, our claim
holds true for $k=n$. Now suppose that, for given $k\in\\{0,1,\dots,n-1\\}$,
the set $A_{k+1}$ is an attractor. We need to verify that $A_{k}$ is an
attractor.

We begin by showing that $\operatorname{cl}A_{k}$ and $M_{k+1}$ are disjoint.
For contradiction, suppose that there exists $y\in\operatorname{cl}A_{k}\cap
M_{k+1}$, and consider a sequence $\\{y_{t}\\}$ in $A_{k}$ convergent to $y$.
For each $t\in\mbox{$\mathbb{N}$}$ let $\sigma_{t}:\mbox{$\mathbb{Z}$}\to S$
be a solution for $F$ through $y_{t}$ with $\alpha(\sigma_{t})\subset
M_{1}\cup\dots\cup M_{k}$. Since $A_{k+1}$ is an attractor, $\sigma_{t}$ is a
solution in $A_{k+1}$. We construct a solution $\sigma:\mbox{$\mathbb{Z}$}\to
A_{k+1}$ through $y$. Fix $m\in\mbox{$\mathbb{N}$}$, choose an increasing
sequence $\\{t_{p}\\}\subset\mbox{$\mathbb{N}$}$ such that $\sigma_{t_{p}}(l)$
is convergent in $A_{k+1}$ for each $l\in[-m,m]$, and set
$\sigma^{m}(l):=\lim_{p\to\infty}\sigma_{t_{p}}(l)$. Clearly,
$\sigma^{m}(0)=y$. Moreover, $\sigma_{t_{p}}(l+1)\in F(\sigma_{t_{p}}(l))$ for
any $t_{p}\in\mbox{$\mathbb{N}$}$ and $l,l+1\in[-m,m]$. Therefore, by the
closed graph property of $F$ we obtain $\sigma^{m}(l+1)\in F(\sigma(l))$. This
shows that $\sigma^{m}:[-m,m]\to A_{k+1}$ is a solution for $F$ through $y$.
Proceeding inductively and using pointwise limits of solutions $\sigma_{t}$,
we can extend $\sigma^{m}$ to a full solution $\sigma:\mbox{$\mathbb{Z}$}\to
A_{k+1}$ for $F$ through $y$ (cf. [7, Lemma 3.4]).

Now, fix an arbitrary $l\in\\{1,2,\dots,k\\}$. Note that $F(M_{l})\cap
M_{k+1}=\emptyset$, because otherwise by the invariance of $M_{l}$ and
$M_{k+1}$ we would have a solution $\tau:\mbox{$\mathbb{Z}$}\to S$ with
$\alpha(\tau)\subset M_{l}$ and $\omega(\tau)\subset M_{k+1}$, which is in
contradiction with the ordering of the Morse sets in $\mathcal{M}$. Clearly,
$M_{l}$ and $M_{k+1}$ are compact. The set $F(M_{l})$ is compact as an image
of a compact set under an upper semicontinuous map. Therefore, by the upper
semicontinuity of $F$ we can take compact and disjoint neighborhoods $N_{l}$
and $N^{\prime}_{k+1}$ of $M_{l}$ and $M_{k+1}$, respectively, such that
$F(N_{l})\cap N^{\prime}_{k+1}=\emptyset$. It follows that there exists a
compact neighborhood $N_{k+1}$ of $M_{k+1}$ such that

$N_{l}\cap N_{k+1}=\emptyset\mbox{ and }F(N_{l})\cap N_{k+1}=\emptyset$

for all $l\in\\{1,2,\dots,k\\}$. We set

$V_{k+1}:=A_{k+1}\setminus\bigcup_{i=1}^{k+1}\operatorname{int}N_{i}$

and

$V_{k}:=A_{k+1}\setminus\bigcup_{i=1}^{k}\operatorname{int}N_{i}.$

Since $\sigma_{n}(0)$ converges to $y\in
M_{k+1}\subset\operatorname{int}N_{k+1}$, we may assume that $\sigma_{n}(0)\in
N_{k+1}$ for $n\in\mbox{$\mathbb{N}$}$. Fix an $n\in\mbox{$\mathbb{N}$}$.
Observe that there exists a smallest $k_{n}\in\mbox{$\mathbb{N}$}$ such that
$\sigma_{n}(-k_{n})\in V_{k+1}$, because $\alpha(\sigma_{n})\subset
M_{1}\cup\cdots\cup M_{k}$ and $F(N_{1}\cup\cdots\cup N_{k})\cap
N_{k+1}=\emptyset$.

Case 1. Sequence $\\{k_{n}\\}$ is bounded. Then we can find an
$l\in\mbox{$\mathbb{N}$}$ such that $\sigma_{n}(-l)\in V_{k+1}$ for all but
finitely many $n\in\mbox{$\mathbb{N}$}$. Passing in $\sigma_{n}(-l)$ with $n$
to infinity we obtain $\sigma(-l)\in V_{k+1}$, as $V_{k+1}$ is compact. Since
$\sigma(0)\in M_{k+1}$ and $M_{k+1}$ is invariant, there exists a solution
$\tau:\mbox{$\mathbb{Z}$}\to M_{k+1}$ through $\sigma(0)$. Now, the
concatenation

$\sigma^{\prime}(m):=\left\\{\begin{array}[]{rl}\tau(m),&m\geq 0\\\
\sigma(m),&m\leq 0\end{array}\right.$

is a solution for $F$ in $A_{k+1}$ with $\omega(\sigma^{\prime})\subset
M_{k+1}$ and $\alpha(\sigma^{\prime})\subset M_{1}\cup\cdots\cup M_{k+1}$.
However, $\sigma^{\prime}(-l)=\sigma(-l)\notin M_{k+1}$, thus we cannot have
$\alpha(\sigma^{\prime})\subset M_{k+1}$. It follows that
$\alpha(\sigma^{\prime})\subset M_{j}$ for some $j\in\\{1,2,\dots,k\\}$, a
contradiction.

Case 2. Sequence $\\{k_{n}\\}$ is unbounded. Without loss of generality we may
assume that $k_{n}\geq n$. Observe that

$\operatorname{dom}(F_{{V_{k}},{k_{n}}})\cap V_{k+1}\cap A_{k+1}\neq\emptyset$

for all $n\in\mbox{$\mathbb{N}$}$. Taking into account that
$\\{\operatorname{dom}(F_{{V_{k}},{n}})\\}$ is a decreasing sequence of
compact sets, and $V_{k+1}$ and $A_{k+1}$ are compact, we get

$\bigcap\left\\{\operatorname{dom}(F_{{V_{k}},{k_{n}}})\cap V_{k+1}\cap
A_{k+1}\,|\,n\in\mbox{$\mathbb{N}$}\right\\}\neq\emptyset.$

By the identity
$\operatorname{Inv}^{+}(V_{k},F)=\bigcap\left\\{\operatorname{dom}(F_{{V_{k}},{n}})\,|\,n\in\mbox{$\mathbb{Z}$}^{+}\right\\}$
(cf. e.g. [5, Lemma 4.9], [21, Lemma 2.8]) we obtain

$\operatorname{Inv}^{+}(V_{k},F)\cap V_{k+1}\cap A_{k+1}\neq\emptyset.$

Take an $x\in\operatorname{Inv}^{+}(V_{k},F)\cap V_{k+1}\cap A_{k+1}$ and a
solution $\tau^{\prime}:\mbox{$\mathbb{Z}$}^{+}\to V_{k}$ for $F$ through $x$.
Then $\omega(\tau^{\prime})\subset V_{k}$, because $V_{k}$ is compact. This,
in turn, means that $\omega(\tau^{\prime})\subset M_{k+1}$, as
$V_{k}\cap(M_{1}\cup M_{2}\cup\cdots\cup M_{k})=\emptyset$ and $V_{k}\subset
A_{k+1}$. Now, by the inclusion $x\in A_{k+1}$ and the invariance of
$A_{k+1}$, we can extend $\tau^{\prime}$ to the full solution
$\tau:\mbox{$\mathbb{Z}$}\to A_{k+1}$. We have $\omega(\tau)\subset M_{k+1}$
and $\alpha(\tau)\subset M_{1}\cup M_{2}\cup\cdots\cup M_{k+1}$. Moreover,
$\operatorname{im}\tau\not\subset M_{k+1}$, as $x\in V_{k+1}$. Therefore, we
cannot have $\alpha(\tau)\subset M_{k+1}$, showing that $\alpha(\tau)\subset
M_{j}$ for some $j\in\\{1,2\dots,k\\}$, a contradiction.

This completes the proof that $\operatorname{cl}A_{k}$ and $M_{k+1}$ are
disjoint.

Take a compact neighborhood $W_{k}$ of $A_{k}$ such that $W_{k}\cap
M_{k+1}=\emptyset$. We claim that $W_{k}$ is an isolating neighborhood of
$A_{k}$. Indeed, it suffices to verify the identity
$A_{k}=\operatorname{Inv}(W_{k},F)$. To this end consider
$x\in\operatorname{Inv}(W_{k},F)$ and $\sigma:\mbox{$\mathbb{Z}$}\to W_{k}$, a
solution through $x$. By compactness of $W_{k}$ we have $\alpha(\sigma)\subset
W_{k}$, and consequently $\alpha(\sigma)\subset M_{1}\cup\cdots\cup M_{k}$,
showing that $x\in A_{k}$. We need to verify the other inclusion. It is
straightforward to observe that $A_{k}$ is invariant. Thus,
$A_{k}=\operatorname{Inv}(A_{k},F)\subset\operatorname{Inv}(W_{k},F)$. This
completes the proof that $W_{k}$ is an isolating neighborhood for $A_{k}$. In
particular, it follows that $A_{k}$ is compact. Moreover, using the same
reasoning, one can easily see that $\operatorname{Inv}^{-}(W_{k},F)=A_{k}$.

Observe that $F(A_{k})=A_{k}\subset\operatorname{int}W_{k}$ and consider an
open neighborhood $V$ of $A_{k}$ such that

(1) $F(V)\subset\operatorname{int}W_{k}.$

Using [21, Lemma 2.11] we can pick a compact neighborhood $A$ of $A_{k}$ such
that $T_{k}:=F^{+}_{W_{k}}(A)\subset V$. We shall show that $T_{k}$ is a
trapping region for $A_{k}$. First note that the set $T_{k}$ is compact,
because $\operatorname{Inv}^{-}(W_{k},F)=A_{k}\subset A\subset W_{k}$ (cf.
[21, Lemma 2.9]). Moreover,
$A_{k}\subset\operatorname{int}A\subset\operatorname{int}T_{k}$. The identity
$\operatorname{Inv}(T_{k},F)=A_{k}$ is easily seen. Thus, $T_{k}$ is an
isolating neighborhood for $A_{k}$. There remains to verify that
$F(T_{k})\subset T_{k}$. Note that, directly by the definition, $T_{k}$ is
positively invariant in $W_{k}$. Moreover, by (1) we have
$F(T_{k})=F(F^{+}_{W_{k}}(A))\subset F(V)\subset W_{k}$. Consequently,
$F(T_{k})=F(T_{k})\cap W_{k}\subset T_{k}$.

This completes the proof that $A_{k}$ is an attractor.

We will show the identity $M_{j}=A_{j}\cap A^{*}_{j-1}$ for
$j\in\\{1,\dots,n\\}$. Fix $j\in\\{1,\dots,n\\}$. Let $x\in M_{j}$ and let
$\sigma:\mbox{$\mathbb{Z}$}\to M_{j}$ be a solution through $x$. The set
$M_{j}$ is compact, therefore $\alpha(\sigma)\subset M_{j}$. This means that
$x\in A_{j}$. We shall verify that $x\in A^{*}_{j-1}$. For contradiction
suppose that $x\notin A^{*}_{j-1}$. Then, for any solution
$\tau:\mbox{$\mathbb{Z}$}\to S$ we have $\omega(\tau)\cap
A_{j-1}\neq\emptyset$. In particular, $\omega(\sigma)\cap
A_{j-1}\neq\emptyset$, which in turn implies $\omega(\sigma)\subset M_{k}$ for
some $k\in\\{1,2,\dots,j-1\\}$. However, $\sigma$ is a solution in $M_{j}$;
hence, $\omega(\sigma)\subset M_{k}\cap M_{j}$, a contradiction. For the proof
of the other inclusion take an $x\in A_{j}\cap A^{*}_{j-1}$. Then there exists
a solution $\sigma^{\prime}:\mbox{$\mathbb{Z}$}\to S$ through $x$ with
$\alpha(\sigma^{\prime})\subset M_{1}\cup\cdots\cup M_{j}$. We also have a
solution $\sigma^{\prime\prime}:\mbox{$\mathbb{Z}$}\to S$ such that
$\omega(\sigma^{\prime\prime})\cap(M_{1}\cup\cdots\cup M_{j-1})=\emptyset$.
Consequently, there is $k\geq j$ such that
$\omega(\sigma^{\prime\prime})\subset M_{k}$. Since both $\sigma^{\prime}$ and
$\sigma^{\prime\prime}$ are solutions through $x$, we can take their
concatenation $\sigma:\mbox{$\mathbb{Z}$}\to S$ so that $\alpha(\sigma)\subset
M_{1}\cup\cdots\cup M_{j}$ and $\omega(\sigma)\subset M_{k}$. Thus, by the
ordering of the Morse sets $\alpha(\sigma)\subset M_{j}$ and
$\omega(\sigma)\subset M_{j}$. It follows that $\operatorname{im}\sigma\subset
M_{j}$, in particular $x\in M_{j}$.

This completes the proof. ∎

######  Proposition 3.10.

Let $\text{$\mathcal{M}$}:=\\{M_{1},M_{2},\dots,M_{n}\\}$ be a Morse
decomposition of an isolated invariant set $S$ with respect to an upper
semicontinuous $F:X\multimap X$, and let $\emptyset=A_{0}\subset A_{1}\subset
A_{2}\subset\dots\subset A_{n}=S$ be the sequence of associated attractors.
Then $(M_{j},A_{j-1})$ is a repeller–attractor pair in $A_{j}$ for
$j\in\\{1,2,\dots,n\\}$.

###### Proof.

Let $j\in\\{1,2,\dots,n\\}$ and let $T_{j-1}$ be a trapping region for
$A_{j-1}$ in $S$. One can observe that $T_{j-1}\cap A_{j}$ is a trapping
region for $A_{j-1}$ in $A_{j}$, showing that $A_{j-1}$ is an attractor in
$A_{j}$. Now, the assertion of the proposition is straightforward according to
the identity $M_{j}=A_{j}\cap A^{*}_{j-1}$. ∎

The following theorem may be viewed as a converse of Theorem 3.9.

######  Theorem 3.11.

Let $S$ be an isolated invariant set with respect to an upper semicontinuous
map $F:X\multimap X$, and let $\emptyset=A_{0}\subset A_{1}\subset
A_{2}\subset\dots\subset A_{n}=S$ be a sequence of attractors in $S$. Then
$\text{$\mathcal{M}$}:=\\{M_{1},M_{2},\dots,M_{n}\\}$, where $M_{j}=A_{j}\cap
A^{*}_{j-1}$ and $A^{*}_{j}$ is a repeller dual to $A_{j}$, for
$j\in\\{1,2,\dots,n\\}$, is a Morse decomposition of $S$. Moreover,
$A_{k}=\\{x\in S\ |\ \mbox{there exists a solution
}\sigma:\mbox{$\mathbb{Z}$}\to S\mbox{ through }x\mbox{ with
}\alpha(\sigma)\subset M_{1}\cup M_{2}\cup\dots\cup M_{k}\\}$,
$k\in\\{1,2,\dots,n\\}$.

###### Proof.

The idea of the proof is similar to its single valued counterpart (cf. e.g.
[36]), but the details differ. We present the entire proof for the sake of
completeness.

We shall verify all the assertions of Definition 3.1. Fix $i<j$. We have

$\begin{array}[]{rcl}M_{i}\cap M_{j}&=&A_{i}\cap A^{*}_{i-1}\cap A_{j}\cap
A^{*}_{j-1}\\\ &\subset&A_{i}\cap A^{*}_{j-1}\subset A_{j-1}\cap
A^{*}_{j-1}=\emptyset.\end{array}$

Let $j\in\\{1,2,\dots,n\\}$ be fixed. We shall verify that $M_{j}$ is isolated
and invariant with respect to $F$. Let $T_{j}$ be a trapping region for
$A_{j}$. Recall that $N_{A^{*}_{j-1}}:=S\setminus\operatorname{int}T_{j}$ is
an isolating neighborhood for $A_{j-1}^{*}$ (cf. Lemma 3.6). We claim that
$N_{j}:=T_{j}\cap N_{A^{*}_{j-1}}$ is an isolating neighborhood for $M_{j}$.
We have $M_{j}=A_{j}\cap A^{*}_{j-1}\subset
A_{j}\subset\operatorname{int}T_{j}$ and $M_{j}=A_{j}\cap A^{*}_{j-1}\subset
A^{*}_{j-1}\subset\operatorname{int}N_{A^{*}_{j-1}}$. Thus,
$M_{j}\subset\operatorname{int}T_{j}\cap\operatorname{int}N_{A^{*}_{j-1}}=\operatorname{int}(T_{j}\cap
N_{A^{*}_{j-1}})=\operatorname{int}N_{j}$. There remains to show that
$M_{j}=\operatorname{Inv}(N_{j},F)$. To this end, consider an $x\in M_{j}$.
Since $x\in A_{j}$, we can take a solution $\eta:\mbox{$\mathbb{Z}$}\to A_{j}$
with $\eta(0)=x$. Clearly, $\alpha(\eta)\subset A_{j}$. On the other hand
$x\in A^{*}_{j-1}$, therefore there exists a solution
$\tau:\mbox{$\mathbb{Z}$}\to S$ with $\tau(0)=x$ such that $\omega(\tau)\cap
A_{j-1}=\emptyset$. Define $\sigma:\mbox{$\mathbb{Z}$}\to S$ by

$\sigma(k):=\left\\{\begin{array}[]{rl}\eta(k),&k\leq 0,\\\ \tau(k),&k\geq
0.\end{array}\right.$

One easily sees that $\sigma$ is a solution with respect to $F$ through $x$.
Note that $\omega(\sigma)=\omega(\tau)$; hence, $\omega(\sigma)\cap
A_{j-1}=\emptyset$. Then, by Lemma 3.5, $\operatorname{im}\sigma\cap
T_{A_{j-1}}=\emptyset$ and by Lemma 3.6 the inclusion
$\operatorname{im}\sigma\subset A^{*}_{j-1}$ follows. Moreover
$\alpha(\sigma)=\alpha(\eta)\subset A_{j}$, and by Lemma 3.7(iv),
$\operatorname{im}\sigma\subset A_{j}$. Consequently,
$\operatorname{im}\sigma\subset A_{j}\cap A^{*}_{j-1}=M_{j}$ showing that
$M_{j}$ is invariant with respect to $F$. Therefore,
$M_{j}=\operatorname{Inv}(M_{j},F)\subset\operatorname{Inv}(N_{j},F)$. For the
proof of the other inclusion consider an $x\in\operatorname{Inv}(N_{j},F)$ and
a solution $\sigma:\mbox{$\mathbb{Z}$}\to N_{j}$ through $x$. We have
$\operatorname{im}\sigma\subset N_{j}\subset T_{j}$, and in turn
$\operatorname{im}\sigma\subset A_{j}$, as $T_{j}$ is a trapping region for
$A_{j}$. Similarly, $\operatorname{im}\sigma\subset N_{j}\subset
N_{A^{*}_{j-1}}$ and we have the inclusion $\operatorname{im}\sigma\subset
A^{*}_{j-1}$, because $N_{A^{*}_{j-1}}$ is an isolating neighborhood for
$A^{*}_{j-1}$. As a consequence, $\operatorname{im}\sigma\subset A_{j}\cap
A^{*}_{j-1}=M_{j}$. In particular $x\in M_{j}$. We have proved that $M_{j}$ is
an isolated invariant set, which also justifies its compactness. The proof of
Definition 3.1(1) is complete.

In order to prove Definition 3.1(2) consider a solution
$\sigma:\mbox{$\mathbb{Z}$}\to S$. Since the sequence of attractors is
increasing and $A_{n}=S$, there exists a smallest positive integer $i$ such
that $\omega(\sigma)\subset A_{i}$. Similarly, there is a largest
$j\in\mbox{$\mathbb{N}$}$, $j<n$, with $\alpha(\sigma)\subset A^{*}_{j}$, as
the sequence of the dual repellers is decreasing and $A^{*}_{0}=S$. Then,
$\omega(\sigma)\not\subset A_{i-1}$, which according to Lemma 3.5 implies
$\operatorname{im}\sigma\cap T_{i-1}=\emptyset$. Hence, by Lemma 3.6 we get

(2) $\operatorname{im}\sigma\subset A^{*}_{i-1}.$

In particular, $\omega(\sigma)\subset A^{*}_{i-1}$. This, along with the
inclusion $\omega(\sigma)\subset A_{i}$ implies

(3) $\omega(\sigma)\subset M_{i}.$

By the choice of $j$ we have $\alpha(\sigma)\subset A^{*}_{j}$ and
$\alpha(\sigma)\not\subset A^{*}_{j+1}$. We claim that

(4) $\operatorname{im}\sigma\subset A_{j+1}.$

Indeed, otherwise we would have $\operatorname{im}\sigma\not\subset
T_{A_{j+1}}$ and, by Lemma 3.7(i), $\alpha(\sigma)\subset A^{*}_{j+1}$, a
contradiction.

Observe that $j+1\geq i$. If not, then by (4) $\operatorname{im}\sigma\subset
A_{j+1}\subset A_{i-1}$, which along with (2) yields
$\operatorname{im}\sigma\subset A_{i-1}\cap A^{*}_{i-1}=\emptyset$, a
contradiction.

If $j+1=i$ then by (4) we have $\operatorname{im}\sigma\subset A_{j+1}=A_{i}$.
Now, using (2) we obtain $\operatorname{im}\sigma\subset A_{i}\cap
A^{*}_{i-1}=M_{i}$.

If $j+1>i$ then we have $\alpha(\sigma)\subset A^{*}_{j}$ and, by (4),
$\alpha(\sigma)\subset A_{j+1}$. Thus, $\alpha(\sigma)\subset A_{j+1}\cap
A^{*}_{j}=M_{j+1}$. This along with (3) completes the proof of Definition
3.1(2).

For the proof of Definition 3.1(3) fix a $j\in\\{1,2,\dots,n\\}$ and take a
solution $\sigma:\mbox{$\mathbb{Z}$}\to S$ with
$\alpha(\sigma)\cup\omega(\sigma)\subset M_{j}$. Then the inclusion
$\operatorname{im}\sigma\subset A_{j}$ follows from Lemma 3.7(iv), as
$\alpha(\sigma)\subset A_{j}$. Similarly, by Lemma 3.7(iii) and the inclusion
$\omega(\sigma)\subset A^{*}_{j-1}$ we get $\operatorname{im}\sigma\subset
A^{*}_{j-1}$. Consequently, $\operatorname{im}\sigma\subset M_{j}$.

We have proved that the family $M$ is a Morse decomposition of $S$.

There remains to justify the last assertion of the theorem. Fix a
$k\in\\{1,2,\dots,n\\}$. Take an $x\in A_{k}$ and a solution
$\sigma:\mbox{$\mathbb{Z}$}\to A_{k}$ through $x$. Clearly,
$\alpha(\sigma)\subset A_{k}$. Let $i\leq k$ be the smallest integer such that
$\alpha(\sigma)\subset A_{i}$. Then $\alpha(\sigma)\not\subset A_{i-1}$. This
means that $\operatorname{im}\sigma\not\subset T_{i-1}$, which along with
Lemma 3.7(i) yields $\alpha(\sigma)\subset A^{*}_{i-1}$. Thus,
$\alpha(\sigma)\subset A_{i}\cap A^{*}_{i-1}=M_{i}$. For the proof of the
other inclusion consider an $x\in S$ and a solution
$\sigma:\mbox{$\mathbb{Z}$}\to S$ through $x$ with $\alpha(\sigma)\subset
M_{1}\cup M_{2}\cup\dots\cup M_{k}$. There is $i\leq k$ such that
$\alpha(\sigma)\subset M_{i}$. Then $\alpha(\sigma)\subset A_{i}$, which along
with Lemma 3.7(iv) shows that $\operatorname{im}\sigma\subset A_{i}$. In
particular, $x\in A_{i}\subset A_{k}$.

This completes the proof. ∎

##  4\. Index triples

Throughout this section assume that $X$ is a locally compact metric space and
$F:X\multimap X$ is an upper semicontinuous multivalued map determined by a
morphism.

######  Lemma 4.1.

Let $N$ be an isolating neighborhood for $F$ and let a pair $P=(P_{1},P_{2})$
of compact sets $P_{2}\subset P_{1}\subset N$ satisfy conditions (a) and (d)
of Definition 2.1. Then $P$ satisfies condition (b).

###### Proof.

First we show that $\operatorname{bd}_{F}P_{1}\subset\operatorname{bd}N$. For
contradiction assume that there exists a
$y\in\operatorname{bd}_{F}P_{1}\setminus\operatorname{bd}N$. Then
$y\in\operatorname{cl}(F(P_{1})\setminus P_{1})$, and we can consider a
sequence $\\{y_{n}\\}\subset F(P_{1})\setminus P_{1}$ convergent to $y$.
Observe that $y\in\operatorname{int}N$, because $y\in P_{1}\subset N$ and
$y\notin\operatorname{bd}N$. Therefore, $y_{n}\in\operatorname{int}N$ for
large enough $n\in\mbox{$\mathbb{N}$}$. This along with $y_{n}\in F(P_{1})$
and (a) implies $y_{n}\in P_{1}$, a contradiction.

To prove inclusion (b) assume the contrary and consider an
$x\in\operatorname{bd}_{F}P_{1}\setminus P_{2}$. Then $x\in P_{1}\setminus
P_{2}\subset\operatorname{int}N$, by property (d). On the other hand,
$x\in\operatorname{bd}_{F}P_{1}\subset\operatorname{bd}N$, a contradiction. ∎

The next lemma is straightforward.

######  Lemma 4.2.

Let $A,B,N,M$ be subsets of $X$. The following holds true:

  * (i)

if $A,B$ are positively invariant with respect to $F$ in $N$ then so are
$A\cup B$ and $A\cap B$,

  * (ii)

if $A$ and $B$ are positively invariant with respect to $F$ in $N$ and $M$,
respectively, then $A\cap B$ is positively invariant with respect to $F$ in
$N\cap M$,

  * (iii)

if $A$ is positively invariant with respect to $F$ in $N$ and $B$ is
positively invariant with respect to $F$, then $A\cap B$ is positively
invariant with respect to $F$ in $N$,

  * (iv)

if $A$ is positively invariant with respect to $F$ in $M$ and $N\subset M$
then $A$ is positively invariant with respect to $F$ in $N$.

######  Definition 4.3.

We will say that a pair $R=(R_{1},R_{2})$ of compact sets is an $F$–pair if
there is a compact set $M$ with $R_{2}\subset R_{1}\subset M$ and

  * (Fp1)

$R_{1}$, $R_{2}$ are positively invariant with respect to $F$ in $M$,

  * (Fp2)

$\operatorname{cl}(R_{1}\setminus R_{2})$ is an isolating neighborhood,

  * (Fp3)

$R_{1}\setminus R_{2}\subset\operatorname{int}M$.

Note that a weak index pair is a special $F$–pair.

Given an $F$–pair $R$ in $M$ we set

$T_{M}(R):=(T_{M,1}(R),T_{M,2}(R)):=(R_{1}\cup(X\setminus\operatorname{int}M),R_{2}\cup(X\setminus\operatorname{int}M)).$

######  Lemma 4.4.

If $R$ is an $F$–pair in $M$, then

  * (i)

$F(R)\subset T_{M}(R)$,

  * (ii)

the inclusion $i_{R}:=i_{R,T_{M}(R)}:R\to T_{M}(R)$ induces an isomorphism in
the Alexander-Spanier cohomology.

###### Proof.

Property (i) follows from the positive invariance of $R_{1}$ and $R_{2}$ with
respect to $F$ in $M$. Since $T_{M,1}(R)\setminus T_{M,2}(R)=R_{1}\setminus
R_{2}$, inclusion $i_{R}$ is an excision, and property (ii) follows. ∎

By Lemma 4.4 we can define an endomorphism $I_{R}:H^{*}(R)\to H^{*}(R)$ by

$I_{R}:=F_{R}^{*}\circ(i_{R}^{*})^{-1},$

where $F_{R}$ stands for the restriction of $F$ to the domain $R$ and the
codomain $T_{M}(R)$. In the following endomorphism $I_{R}$ will be called an
index map associated with an $F$–pair $R$.

Note that if an $F$–pair $R$ is a weak index pair then the above notion of the
index map coincides with that for a weak index pair.

######  Proposition 4.5.

Let $R$ be an $F$–pair in $M$, and let
$S:=\operatorname{Inv}(\operatorname{cl}(R_{1}\setminus R_{2}),F)$. If
$N\subset M$ is an isolating neighborhood of $S$ such that $R_{1}\setminus
R_{2}\subset\operatorname{int}N$ then

  * (i)

$P:=R\cap N$ is a weak index pair for $F$ in $N$,

  * (ii)

the inclusion $i_{PR}:P\to R$ induces an isomorphism in the Alexander-Spanier
cohomology,

  * (iii)

index maps $I_{F_{P}}$ and $I_{R}$ are conjugate.

As a consequence, $C(S,F)=L(H^{*}(R),I_{R})$.

###### Proof.

We shall verify that $P$ is a weak index pair. First observe that, by the
positive invariance of $R$ in $M$, the inclusion $N\subset M$, and Lemma
4.2(iv), $P$ is positively invariant in $N$, i.e. $P$ satisfies property (a)
of Definition 2.1. Properties (c) and (d) are straightforward. Now, property
(b) is justified by Lemma 4.1.

Since $R_{1}\setminus R_{2}\subset\operatorname{int}N$, we have
$P_{1}\setminus P_{2}=(R_{1}\setminus R_{2})\cap N=R_{1}\setminus R_{2}$.
Therefore, inclusion $i_{PR}$ induces an isomorphism, as an excision.

There remains to verify (iii). Consider the commutative diagram

${(P_{1},P_{2})}$${(T_{N,1}(P),T_{N,2}(P))}$${(P_{1},P_{2})}$${(R_{1},R_{2})}$${(T_{M,1}(R),T_{M,2}(R))}$${(R_{1},R_{2})}$$\scriptstyle{i_{PR}}$$\scriptstyle{F_{P}}$$\scriptstyle{i_{PR}}$$\scriptstyle{i_{P}}$$\scriptstyle{F_{R}}$$\scriptstyle{j}$$\scriptstyle{i_{R}}$

in which $i_{P}$, $i_{R}$, $i_{PR}$ and $j$ are inclusions. Recall that
$i_{P}$ and $i_{R}$ and $i_{PR}$ induce isomorphisms in cohomology by the
strong excision property. And, so does $j$. The commutativity of the diagram
shows that the index maps $I_{F_{P}}$ and $I_{R}$ are conjugate. This
completes the proof. ∎

######  Lemma 4.6.

Assume $S_{1}\subset S_{2}$ are isolated invariant sets with respect to $F$,
with isolating neighborhoods $N_{1}$ and $N_{2}$, respectively. Then,
$N_{1}\cap N_{2}$ is an isolating neighborhood for $S_{1}$.

###### Proof.

We have $S_{1}=\operatorname{Inv}(S_{1},F)\subset\operatorname{Inv}(N_{1}\cap
N_{2},F)$, because $S_{1}$ is invariant and $S_{1}\subset N_{1}\cap N_{2}$. On
the other hand, $\operatorname{Inv}(N_{1}\cap
N_{2},F)\subset\operatorname{Inv}(N_{1},F)=S_{1}$. Thus,
$S_{1}=\operatorname{Inv}(N_{1}\cap N_{2},F)$. There remains to verify that
$S_{1}\subset\operatorname{int}(N_{1}\cap N_{2})$. To this end observe that
$S_{1}\subset\operatorname{int}N_{1}$ and $S_{1}\subset
S_{2}\subset\operatorname{int}N_{2}$. Therefore, $S_{1}=S_{1}\cap
S_{2}\subset\operatorname{int}N_{1}\cap\operatorname{int}N_{2}=\operatorname{int}(N_{1}\cap
N_{2})$. ∎

######  Theorem 4.7.

Let $S$ be an isolated invariant set with respect to $F$ and let $N$ be its
isolating neighborhood. Assume that $(A^{*},A)$ is a repeller–attractor pair
in $S$. Then, there exist a triple $(P_{0},P_{1},P_{2})$ of compact subsets
$P_{2}\subset P_{1}\subset P_{0}$ of $N$ such that

  * (i)

$(P_{0},P_{2})$ is a weak index pair for $F$ and
$C(S,F)=L(H^{*}(P_{0},P_{2}),I_{(P_{0},P_{2})})$,

  * (ii)

$(P_{1},P_{2})$ is an $F$–pair and
$C(A,F)=L(H^{*}(P_{1},P_{2}),I_{(P_{1},P_{2})})$,

  * (iii)

$(P_{0},P_{1})$ is an $F$–pair and
$C(A^{*},F)=L(H^{*}(P_{0},P_{1}),I_{(P_{0},P_{1})})$.

###### Proof.

Let $(P_{0},P_{2}^{\prime})$ be a weak index pair for $S$ and $F$ in an
isolating neighborhood $N$. Consider a trapping region $M^{\prime}$ for
attractor $A$. By Lemma 4.6 the set $M:=M^{\prime}\cap N$ is an isolating
neighborhood of $A$. So, we can take a weak index pair
$Q^{\prime}=(Q_{1}^{\prime},Q_{2}^{\prime})$ for $A$ and $F$ in $M$. Set
$Q_{i}:=Q_{i}^{\prime}\cap P_{0}$, $i\in\\{1,2\\}$, and $Q:=(Q_{1},Q_{2})$.

First we prove that $Q$ is a weak index pair for $A$ in $M$. Fix
$i\in\\{1,2\\}$. Since $Q_{i}^{\prime}$ is positively invariant in $M$ and
$P_{0}$ is positively invariant in $N$, by Lemma 4.2(ii)
$Q_{i}=Q_{i}^{\prime}\cap P_{0}$ is positively invariant in $M\cap N=M$. This
shows that $Q$ satisfies property (a) of Definition 2.1. Observe that
$A\subset S\subset\operatorname{int}P_{0}$ and
$A\subset\operatorname{int}(Q_{1}^{\prime}\setminus Q_{2}^{\prime})$, as
$(P_{0},P_{2}^{\prime})$ and $(Q_{1},Q_{2}^{\prime})$ are weak index pairs for
$S$ and $A$, respectively. Therefore,
$A\subset\operatorname{int}(Q_{1}^{\prime}\setminus
Q_{2}^{\prime})\cap\operatorname{int}P_{0}=\operatorname{int}((Q_{1}^{\prime}\setminus
Q_{2}^{\prime})\cap P_{0})=\operatorname{int}(Q_{1}\setminus Q_{2})$, showing
(c). Property (d) follows from the straightforward inclusions $Q_{1}\setminus
Q_{2}=(Q_{1}^{\prime}\cap P_{0})\setminus(Q_{2}^{\prime}\cap
P_{0})=(Q_{1}^{\prime}\setminus Q_{2}^{\prime})\cap P_{0}\subset
Q_{1}^{\prime}\setminus Q_{2}^{\prime}\subset\operatorname{int}M$. Now,
property (b) is a consequence of (a), (d), and Lemma 4.1.

Set

(5) $P_{1}:=Q_{1}\cup P_{2}^{\prime}\mbox{ and }P_{2}:=Q_{2}\cup
P_{2}^{\prime}.$

We shall show that $(P_{1},P_{2})$ is an $F$–pair in $N$. Clearly,
$P_{2}\subset P_{1}$ are compact subsets of $N$. Property (Fp3) in Definition
4.3 is straightforward, because $P_{1}\setminus P_{2}\subset Q_{1}\setminus
Q_{2}$ and $Q$ is a weak index pair for $F$ in $M\subset N$. For the proof of
(Fp2) observe that $\operatorname{cl}(P_{1}\setminus
P_{2})\subset\operatorname{cl}(Q_{1}\setminus Q_{2})\subset M$, therefore
$\operatorname{Inv}(\operatorname{cl}(P_{1}\setminus
P_{2}),F)\subset\operatorname{Inv}(M,F)=A$. Moreover,
$A\subset\operatorname{int}(Q_{1}\setminus Q_{2})$ and
$A\subset\operatorname{int}(P_{0}\setminus P^{\prime}_{2})$, as $Q$ is a weak
index pair for $A$, $(P_{0},P^{\prime}_{2})$ is a weak index pair for $S$, and
$A\subset S$. Thus, $\operatorname{Inv}(\operatorname{cl}(P_{1}\setminus
P_{2}),F)\subset\operatorname{int}(Q_{1}\setminus
Q_{2})\cap\operatorname{int}(P_{0}\setminus
P^{\prime}_{2})=\operatorname{int}(P_{1}\setminus P_{2})$. There remains to
verify (Fp1). For $i\in\\{1,2\\}$ we have $F(P_{i})\cap N=F(Q_{i}\cup
P_{2}^{\prime})\cap N=((F(Q_{i})\cap N)\cup(F(P_{2}^{\prime}))\cap N)$. By the
positive invariance of $P_{2}^{\prime}$ in $N$ we have the inclusion
$F(P_{2}^{\prime})\cap N\subset P_{2}^{\prime}$. Recall that $M^{\prime}$ is
an attracting neighborhood and $Q_{i}\subset M^{\prime}$; hence,
$F(Q_{i})\subset M^{\prime}$. Taking this into account we obtain $F(Q_{i})\cap
N\subset F(Q_{i})\cap N\cap M^{\prime}=F(Q_{i})\cap M\subset Q_{i}$, where the
last inclusion is a consequence of the positive invariance of $Q_{i}$ in $M$.
Finally, we have $F(P_{i})\cap N\subset Q_{i}\cup P_{2}^{\prime}=P_{i}$, i.e.
$P$ satisfies (Fp1). This shows that $(P_{1},P_{2})$ is an $F$–pair in $N$.

Notice that $\operatorname{Inv}(P_{1}\setminus P_{2},F)=A$ and $M$ is an
isolating neighborhood of $A$ with $P_{1}\setminus
P_{2}\subset\operatorname{int}M$. Therefore, by Proposition 4.5 we infer that
$C(A,F)=L(H^{*}(P_{1},P_{2}),I_{(P_{1},P_{2})})$. This completes the proof of
(ii).

We shall show that $P:=(P_{0},P_{2})$ is a weak index pair for $S$ and $F$ in
$N$. Recall that $(P_{0},P_{2}^{\prime})$ is a weak index pair for $F$ in $N$.
Thus, in order to verify that $P$ satisfies property (a) in Definition 2.1 it
suffices to show that $P_{2}=Q_{2}\cup P_{2}^{\prime}$ is positively invariant
in $N$. Since the sets $Q_{2}$ and $P_{2}^{\prime}$ are positively invariant
in $N$, our claim follows from Lemma 4.2(i). We have $P_{1}\setminus
P_{2}=P_{1}\setminus(Q_{2}\cup P_{2}^{\prime})\subset P_{1}\setminus
P_{2}^{\prime}\subset\operatorname{int}N$, showing property (d). Property (b)
is a consequence of (a), (d) and Lemma 4.1. There remains to verify property
(c), i.e. the inclusion $S\subset\operatorname{int}(P_{1}\setminus P_{2})$.
Since $S\subset\operatorname{int}(P_{1}\setminus P_{2}^{\prime})$, for
contradiction suppose that there is an $x\in S\cap Q_{2}$. Consider a solution
$\sigma$ through $x$ in $S$. Since $Q_{2}$ is compact and positively invariant
in $N$, we infer that $\emptyset\neq\omega(\sigma)\subset Q_{2}$. Moreover,
$Q_{2}\subset M$ and $\omega(\sigma)$ is invariant with respect to $F$, hence
we have
$\omega(\sigma)=\operatorname{Inv}(\omega(\sigma),F)\subset\operatorname{Inv}(M,F)=A$,
showing that $A\cap Q_{2}\neq\emptyset$. However, $(Q_{1},Q_{2})$ is a weak
index pair for $A$, a contradiction.

Since $(P_{0},P_{2})$ is a weak index pair for $S$ and $F$ in $N$, the
identity $C(S,F)=L(H^{*}(P_{0},P_{2}),I_{(P_{0},P_{2})})$ is straightforward.
This completes the proof of (i).

Now we focus on a repeller $A^{*}$ dual to $A$ in $S$. We shall verify that
$(P_{0},P_{1})$ is an $F$–pair in $N$. Clearly, the sets $P_{1}\subset P_{0}$
are compact subsets of $N$. Recall that we have already justified the positive
invariance of $P_{0}$ and $P_{1}$ in $N$, that is property (Fp1). Moreover, by
the inclusions $P_{0}\setminus P_{1}\subset P_{0}\setminus
P_{2}\subset\operatorname{int}N$, (Fp3) follows. We claim that
$\operatorname{cl}(P_{0}\setminus P_{1})$ is an isolating neighborhood for
$A^{*}$. According to (ii) we have $A\subset\operatorname{int}P_{1}$, showing
that $\operatorname{cl}(P_{0}\setminus P_{1})\cap A=\emptyset$. Thus,
according to Lemma 3.8, it suffices to verify that

(6) $A^{*}\subset\operatorname{int}(P_{0}\setminus P_{1}).$

To this end consider an $x\in A^{*}$ and a solution $\sigma$ through $x$ such
that $\omega(\sigma)\cap A=\emptyset$. Observe that $x\notin Q_{1}$, because
otherwise we would have $\emptyset\neq\omega(\sigma)\subset Q_{1}$, and
consequently $\omega(\sigma)\subset A$, leading a contradiction. Thus, by the
inclusion $A^{*}\subset S\subset\operatorname{int}(P_{0}\setminus P_{2})$ and
(5), that is the definition of $P_{1}$, inclusion (6) follows.

Now, using (6), (5) and Lemma 3.8, one can easily prove that
$K:=N\setminus\operatorname{int}Q_{1}$ is an isolating neighborhood of
$A^{*}$. We claim that $(P_{0}\cap K,P_{1}\cap K)$ is a weak index pair for
$F$ in $K$. Indeed, in view of Proposition 4.5(i) it suffices to justify that
$P_{0}\setminus P_{1}\subset\operatorname{int}K$, which follows from the
inclusions $P_{0}\setminus P_{1}\subset\operatorname{int}N$ and
$P_{0}\setminus P_{1}\subset P_{0}\setminus Q_{1}$.

Eventually, according to Proposition 4.5 we have the
identity$C(A^{*},F)=L(H^{*}(P_{0},P_{1}),I_{(P_{0},P_{1})})$, which proves
(iii).

This completes the proof. ∎

######  Example 4.8.

Consider multivalued map given in Example 3.2 (see Figure 1). We present some
examples of weak index triples.

  * (a)

Consider a repeller–attractor pair
$(A_{1}^{*},A_{1})=([0,m]\cup[n,o]\cup[p,q]\cup[g,h],[d,e]\cup[j,k])$ in
$S=[0,k]$ (Figure 1(C)). Note that $([0,l],\emptyset)$ is a weak index pair
for $S$ in $[0,l]$ and $([c,f]\cup[i,l],\emptyset)$ is a weak index pair for
$A_{1}$ in $[c,f]\cup[i,l]$. Observe that $P_{0}:=[0,l]$,
$P_{1}:=[c,f]\cup[i,l]$ and $P_{2}:=\emptyset$ is a triple satisfying the
assertions of Theorem 4.7.

  * (b)

Observe that for the repeller–attractor pair $(A_{2}^{*},A_{2})=([0,a],[d,k])$
in $S$ (Figure 1(D)), the weak index triple is given by $P_{0}:=[0,l]$,
$P_{1}:=[c,l]$ and $P_{2}:=\emptyset$.

  * (c)

Note that $(A_{2}^{*},M_{2})$ is a repeller–attractor pair in $A_{1}^{*}$ (see
Figure 1(C) and (D). Then, $([0,c]\cup[f,i],\\{c\\}\cup\\{f\\}\cup\\{i\\})$ is
a weak index pair for $A_{1}^{*}$, and $([f,i],\\{f\\}\cup\\{i\\})$ is a weak
index pair for $M_{2}$. One can verify that the triple
$([0,c]\cup[f,i],[f,i]\cup\\{c\\},\\{c\\}\cup\\{f\\}\cup\\{i\\})$ satisfies
the assertions.

##  5\. Morse equation

The aim of this section is to prove the Morse equation.

As in the preceding section, $X$ is a locally compact metric space and
$F:X\multimap X$ is an upper semicontinuous multivalued map determined by a
morphism.

######  Proposition 5.1.

Assume $(A^{*},A)$ is a repeller–attractor pair in an isolated invariant set
$S$ with respect to an upper semicontinuous map $F:X\multimap X$. Then

$p(t,A^{*})+p(t,A)=p(t,S)+(1+t)Q(t)$

for some power series $Q$ with nonnegative integer coefficients.

###### Proof.

The proof runs along the lines of the proof of [33, Theorem 5.5] with the use
of Theorem 4.7 instead of [33, Theorem 4.3]. ∎

######  Theorem 5.2.

Let $\text{$\mathcal{M}$}:=\\{M_{1},M_{2},\dots,M_{n}\\}$ be a Morse
decomposition of an isolated invariant set $S$ with respect to an upper
semicontinuous $F:X\multimap X$. Then

(7) $\sum_{i=1}^{n}p(t,M_{i})=p(t,S)+(1+t)Q(t),$

where $Q$ is a formal power series with nonnegative integer coefficients.
Moreover,

$Q(t)=\sum_{i=1}^{n}Q_{i}(t),$

where

(8) $(1+t)Q_{i}(t)=p(t,M_{i})+p(t,A_{i-1})-p(t,A_{i}).$

If $Q_{i}\neq 0$ then there exists a solution $\sigma:\mbox{$\mathbb{Z}$}\to
S$ with $\alpha(\sigma)\subset M_{i}$ and $\omega(\sigma)\subset M_{j}$ for
some $j<i$.

###### Proof.

The proof runs similarly as the proof of [33, Theorem 5.6]. By Theorem 3.9 and
Proposition 3.10 we can consider the associated sequence of attractors such
that $(M_{i},A_{i-1})$ is a repeller–attractor pair in $A_{i}$. Applying
Proposition 5.1 to this pair we find a formal power series $Q_{i}$ with
nonnegative integer coefficients satisfying

(9) $p(t,M_{i})+p(t,A_{i-1})=p(t,A_{i})+(1+t)Q_{i}(t).$

Summing these equations over $i\in\\{1,2,\dots,n\\}$ and taking into account
that $A_{n}=S$ and $A_{0}=\emptyset$ we obtain the first assertion.

In order to prove the second assertion, for contradictions suppose that there
is no solution $\sigma:\mbox{$\mathbb{Z}$}\to S$ with $\alpha(\sigma)\subset
M_{i}$ and $\omega(\sigma)\subset M_{j}$, where $j<i$. This means that $A_{i}$
is a disjoint union of $M_{i}$ and $A_{i-1}$, $F(M_{i})\cap
A_{i-1}=\emptyset$, and $F(A_{i-1})\cap M_{i}=\emptyset$. By the additivity
property of the Conley index [3, Theorem 5.3] we obtain

$C(A_{i},F)=C(M_{i},F)\times C(A_{i-1},F).$

In particular,

$\dim C^{q}(A_{i},F)=\dim C^{q}(M_{i},F)+\dim C^{q}(A_{i-1},F).$

Now, there remains to multiply the above equalities by $t^{q}$ and sum over
$q\in\mbox{$\mathbb{N}$}$ in order to get

$p(t,A_{i})=p(t,M_{i})+p(t,A_{i-1}),$

which, along with (9), implies $Q_{i}=0$, a contradiction. ∎

######  Example 5.3.

Consider Morse decomposition of an isolated invariant set $S=[0,k]$ with
respect to a multivalued map presented in Example 3.2 and Figure 1. One can
verify that $p(t,M_{1})=2$, $p(t,M_{2})=t$, $p(t,M_{3})=0$, and $p(t,S)=1$, so
that the Morse equation has the form

$2+t+0=1+(1+t)Q(t),$

showing that $Q(t)=1$. Moreover, $p(t,A_{0})=0$, $p(t,A_{1})=2$,
$p(t,A_{2})=1$ and $p(t,A_{3})=1$. Therefore, by (8) we obtain that
$Q_{1}(t)=Q_{3}(t)=0$ and $Q_{2}(t)=1$. Thus, the Morse equation shows that
there exists a solution $\sigma:\mbox{$\mathbb{Z}$}\to S$ with
$\alpha(\sigma)\subset M_{2}$ and $\omega(\sigma)\subset M_{1}$.

Assume, we are given a Morse decomposition with respect to a flow on $X$.
Recall that, according to [32, Theorem 1], any isolated invariant set with
respect to a flow is isolated and invariant with respect to time-one-map. Note
that this extends over the Morse decomposition. Thus, by Theorem 5.2 we get
Morse equation (7) for the flow. Therefore, arguing as in [36] or [38], as a
corollary we obtain the classical Morse inequalities.

######  Corollary 5.4.

Let $M$ be a $d$-dimensional compact manifold with $k$-th Betti number
$\beta_{k}$, $k\in\\{0,1,2\dots,d\\}$. Let nondegenerate critical points
$\\{x_{1},x_{2},\dots,x_{n}\\}$ with respect to a given gradient flow on $M$
constitute a Morse decomposition of $M$, and let $\lambda(x_{j})$ stand for
the Morse index of $x_{j}$. Then

$\sum_{j=1}^{n}t^{\lambda(x_{j})}=\sum_{j=0}^{d}\beta_{j}t^{j}+(1+t)Q(t),$

where $Q$ is a formal power series with nonnegative integer coefficients.

## Acknowledgements

The paper was finalized during author’s visit to the Rutgers University, USA.
It is my pleasure to thank the staff of the Department of Mathematics,
especially Professor Konstantin Mischaikow for the invitation and for
inspiring scientific atmosphere.

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