Chain transitivity for semigroup actions on flag bundles

Abstract

This paper studies chain recurrence for semigroup actions on fiber bundles with emphasis on flag bundles. The Selgrade theorem is extended from the setting of linear flows to the setting of linear actions of semigroups on projective bundles.

1 Introduction

The Selgrade theorem for linear flows on vector bundles shows that the maximal chain transitive sets of a flow on the projective bundle which covers a chain transitive flow on the base space decompose the vector bundle in a Whitney sum (see Selgrade [18]). The theorem includes the existence of a finest Morse decomposition in the projective bundle whose Morse sets coincide with the maximal chain transitive sets. It has been generalized by Colonius–Kliemann [8] by showing the existence of a finest Morse decomposition in the bundles whose fibers are flag manifolds of subspaces of a vector bundle. Recently, Braga-Barros–San-Martin [4] extended the result to generalized flag manifolds by using the shadowing semigroup methodology. Their method of proof shows that maximal chain transitive sets for the flow are intersections of control sets for the shadowing semigroups (see [4, Theorem 4.7]).

Chain recurrence of semiflows on fiber bundles has been studied by Patrão–San-Martin [15]. They have considered chain transitivity and Morse decomposition of discrete and continuous-time semiflows on fiber bundles with emphasis on generalized flag bundles. Their method of describing the maximal chain transitive sets of a semiflow on a fiber bundle has also been the shadowing semigroup one, although in a topological version (see Theorem 6.2 therein).

Originally, the shadowing semigroup methodology has been developed by Braga-Barros–San-Martin [5]. They have introduced the formal concept of chain recurrence for semigroup actions which depends on a family of subsets of the semigroup. The shadowing semigroups are generated by the neighborhoods of the subsets in the family. The main result characterizes the chain control sets as intersections of control sets for the shadowing semigroups (see [5, Theorem 3.7]). This topological methodology of shadowing has been used in [19] to show how limit behavior can be studied in arbitrary topological spaces and extend Conley’s characterization of chain recurrence in terms of attractors from the setting of flows on compact metric spaces to the setting of semigroup actions on any topological space.

It is well-known that a flow on a topological space defines a group action. Analogously, a semiflow defines an action of a semigroup. In more general setups, a $k$-dimensional flow (or $k$-dimensional semiflow) on a topological space defines a group action (or semigroup action), and a control system on a manifold defines an action of a semigroup that describes the trajectories of the system (see [1, 3, 7]). Many questions on $k$-dimensional semiflows and control systems depend, in fact, only on the action of a semigroup. This procedure has provided a fruitful interrelation between those theories and semigroup actions (as references sources we mention [1–4, 7, 10, 13–15]). These are some of the motivations in working with semigroup actions on topological spaces and to give special attention to fiber bundles.

By following the line of investigation of [5], Braga-Barros–Souza [1, 2] have introduced the formal concepts of attractor and Morse decomposition for semigroup actions on topological spaces and generalized the result of Conley [9] which characterizes the chain recurrence set in terms of attractors (see [1, Theorem 4.1]). Under compactness and broad conditions on the family, they have shown that the existence of a finest Morse decomposition is equivalent to the existence of only finitely many maximal chain transitive sets (see [2, Theorem 5.2]).

In this paper, we extend the results of chain recurrence from the setting of linear flows on flag bundles to the setting of linear actions of semigroups on flag bundles. We construct shadowing semigroups from a family of subsets of the semigroup by using the topological methodology as in [15]. We investigate linear actions of semigroups on vector bundles to show the Selgrade theorem (see Theorem 5.1).

Linearized system is one of the motivations of considering linear actions of semigroups on fiber bundles. A control system $\Sigma$ on a compact $n$-dimensional manifold $M$ lifts to a control system $T\Sigma$ on the tangent bundle $TM$. One has a linear control system on the vector bundle $TM\rightarrow M$ whose fibers are the tangent spaces $T_{p}M$ and the structural group is $\mathrm{Gl}( n,\mathbb R )$. From the Selgrade theorem for linear actions, we can construct a Whitney decomposition of $TM$ into invariant subbundles corresponding to the maximal chain transitive sets of the induced system on the projective bundle $PTM$, when the system is chain transitive in $M$.

2 Semigroup actions

We start collecting some concepts of semigroup actions and fiber bundles to be used afterward.

Assume that $M$ is a topological space and $S$ is a semigroup. An action (or a left action) of $S$ on $M$ is a mapping

$$\begin{aligned} \mu : \begin{array}[t]{l} S\times M \rightarrow M \\ (s,x) \mapsto \mu (s,x)=sx \end{array} \end{aligned}$$

satisfying $s( tx) =( st) x$ for all $x\in M$ and $s,t\in S$. In this case, we say that $S$ acts on $M$. We denote by $\mu _{s}:M\rightarrow M$ the map defined by $\mu _{s}( x) =\mu ( s,x)$. In this paper, we assume that $\mu _{s}$ is continuous for all $s\in S$. We often represent the action of $S$ on $M$ by $( S,M,\mu )$, or simply $( S,M)$.

For subsets $Y\subset X$ and $A\subset S$, we define

$$\begin{aligned} AY=\bigcup \limits _{s\in A}\mu _{s}\left( Y\right) \quad \text{ and}\quad A^{*}Y=\bigcup \limits _{s\in A}\mu _{s}^{-1}\left( Y\right). \end{aligned}$$

It is usual to say that $Y$ is forward invariant if $SY\subset Y$; it is backward invariant if $S^{*}Y\subset Y$; and it is invariant if it is forward and backward invariant.

Definition 2.1

Let $\mathcal F$ be a family of subsets of a semigroup $S$. The $\omega$ - limit set of $X\subset M$ for the family $\mathcal F$ by

$$\begin{aligned} \omega (X,\mathcal F )=\bigcap _{A\in \mathcal F }\mathrm{cls}(AX). \end{aligned}$$

The $\omega ^{*}$ - limit set of $X$ for $\mathcal F$ by

$$\begin{aligned} \omega ^{*}(X,\mathcal F )=\bigcap _{A\in \mathcal F }\mathrm{cls}\left( A^{*}X\right) \text{.} \end{aligned}$$

The $\omega$ -limit set and the $\omega ^{*}$-limit set of $X$ are called the limit sets of $X$.

Definition 2.2

Let $\mathcal F$ be a family of subsets of a semigroup $S$. An $\mathcal F$-attractor for an action of $S$ on the topological space $M$ is a set $\mathcal A$ which admits a neighborhood $V$ such that $\omega (V, \mathcal F )=\mathcal A$. An $\mathcal F$-repeller is a set $\mathcal R$ that has a neighborhood $U$ with $\omega ^{*}(U,\mathcal F )=\mathcal R$. The neighborhoods $V$ and $U$ are called respectively attractor neighborhood of $\mathcal A$ and repeller neighborhood of $\mathcal R$. We consider the empty set and $M$ as $\mathcal F$-attractors and $\mathcal F$-repellers.

For establishing a direction to asymptotic behavior of a semigroup action $( S,M)$, we choose a family $\mathcal F$ of subsets of $S$ that is a filter basis on the subsets of $S$ (that is, $\varnothing \notin \mathcal F$ and given $A,B\in \mathcal F$ there is $C\in \mathcal F$ with $C\subset A\cap B$) and satisfies the following hypotheses.

Definition 2.3

The family $\mathcal F$ is said to satisfy the translation hypothesis if:

• 1. for all $s\in S$ and $A\in \mathcal F$ there is $B\in \mathcal F$ such that $B\subset sA$.

• 2. for all $s\in S$ and $A\in \mathcal F$ there is $B\in \mathcal F$ such that $B\subset As$.

From now on, we assume that $M$ is compact. Let $\mathcal F$ be a family of subsets of $S$ that is a filter basis and satisfies translation hypothesis.

Let $\mathcal A$ be an $\mathcal F$-attractor in $M$. We define the set

$$\begin{aligned} \mathcal A ^{*}=M\backslash \left\{ x\in M:\omega \left( x,\mathcal F \right) \subset \mathcal A \right\}. \end{aligned}$$

We call $\mathcal A ^{*}$ the complementary repeller of $\mathcal A$. It is a compact invariant set and coincides with $\{ x\in M:\omega ( x,\mathcal F ) \cap \mathcal A =\emptyset \}$. Furthermore, $\mathcal A$ and $\mathcal A ^{*}$ are disjoint. We refer to [1, Section 3] for properties of complementary repellers.

Definition 2.4

Let $\emptyset =\mathcal A _{0}\subset \mathcal A _{1}\subset \cdots \subset \mathcal A _{n}=M$ be an increasing sequence of $\mathcal F$-attractors in $M$ and let $\mathcal M _{i}=\mathcal A _{i}\cap \mathcal A _{i-1}^{*}$, for $i=1,\ldots,n$. The ordered collection $\mathcal{M }=\{ \mathcal{C } _{1},\ldots, \mathcal{C }_{n} \}$ is called an $\mathcal F$ -Morse decomposition.

Now, we define control sets for local semigroups.

Definition 2.5

A local semigroup on $M$ is a family $\mathcal S$ of continuous maps $\phi :\mathrm{dom}\phi \rightarrow M$, with $\mathrm{dom}\phi \subset M$ an open set, such that, if $\phi,\psi \in \mathcal S$ and $\phi ^{-1}( \mathrm{dom}\psi ) \ne \emptyset$, then the composition $\psi \phi :\phi ^{-1}( \mathrm{dom}\psi ) \rightarrow M$ also belongs to $\mathcal S$.

We denote the forward and backward orbits of a local semigroup $\mathcal S$ in $M$ respectively by

$$\begin{aligned} \mathcal S x&= \left\{ \phi \left( x\right) :\phi \in \mathcal S,x\in \mathrm{dom}\phi \right\} \\&\text{ and} \\ \mathcal S ^{*}x&= \left\{ y:\exists \phi \in \mathcal S,\phi \left( y\right) =x\right\}. \end{aligned}$$

Definition 2.6

A nonempty set $D\subset M$ is a control set for the action of the local semigroup $\mathcal S$ if

• 1. $\mathrm{int}( D) \ne \emptyset$,

• 2. $D\subset \mathrm{cls}( \mathcal S x)$ for all $x\in D$, and

• 3. $D$ is maximal satisfying these properties.

The open set $D_{0}=\{ x\in D:x\in \mathrm{int}( \mathcal S x) \cap \mathrm{int}( \mathcal S ^{*}x) \}$ is called the transitivity set of $D$. When $D_{0}$ is nonempty, $D$ is called an effective control set for $\mathcal S$. In this case, $D_{0}$ is a dense subset in $D$. We refer to [6] for other properties of transitivity sets.

2.1 Fiber bundles

We now recall the main definitions and notations on principal bundles and their associated bundles. We refer to [11] and [12] for the theory of fiber bundles.

Let $G$ be a topological group acting on the right on a topological space $Q$. We denote by $( q,g) \in Q\times G\rightarrow qg\in Q$ the right action of $G$ on $Q$. A principal bundle is a quadruplet $(Q,\pi,B,G)$ where $\pi :Q\rightarrow B$ is an open and surjective map such that $\pi ( qg) =\pi ( q)$ for all $q\in Q$ and $g\in G$. The space $B$ is the base space, the space $Q$ is the total space, $G$ is the structure group. Assume that $G$ acts on the left on a topological space $F$. Then, $G$ acts on the right on $Q\times F$ by $( q,v) g=( qg,g^{-1}v)$. The quotient space $E=Q\times _{G}F$ is called bundle associated with the principal bundle $( Q,\pi,B,G)$. We denote an element of $E$ as $[ q,v]$. We also denote by $\pi$ the projection $\pi :E\rightarrow B$ defined by $\pi ( [ q,v ] ) =\pi ( q)$. Each fiber $\pi ^{-1}( x)$ of the associated bundle is homeomorphic with the topological space $F$, which is called the typical fiber of the associated bundle.

Let $\mathcal S$ be a semigroup acting on the left on the total space $Q$ of the principal bundle $( Q,\pi,B,G)$ satisfying $s( qg) =( sq) g$ for all $q\in Q$, $s\in \mathcal S$ and $g\in G$. The actions of $\mathcal S$ on $B$ and $E$ are defined by $s\pi ( q) =\pi ( sq)$ and $s[ q,v] =[ sq,v ]$, respectively.

Definition 2.7

A local endomorphism of $Q$ is a map $\phi :\mathrm{dom}\phi \rightarrow Q$ such that

• 1. $\mathrm{dom}\phi =\pi ^{-1}( U)$ where $U\subset B$ is an open set, and

• 2. $\phi ( qg) =\phi ( q) g$ for every $q\in \mathrm{dom}\phi$ and $g\in G$.

We denote by $\mathrm{End}_{l}( Q)$ the set of all local endomorphism of $Q$, which is clearly a local semigroup. A mapping $\phi \in \mathrm{End}_{l}( Q)$ maps fibers into fibers and hence induces a map from $\pi ( \mathrm{dom}\phi )$ into $B$. This induced map will be denoted by $\phi _{B}$. Also, if $E\rightarrow B$ is an associated bundle, we can define a map by $\phi _{E}( q\cdot v) =\phi ( q) \cdot v$. Its domain is the open set in $E$ above $\pi ( \mathrm{dom}\phi )$. We denote by $\mathrm{End}_{l}( E)$ the local semigroup in $E$ induced by $\mathrm{End}_{l}( Q)$.

Let $\mathcal S \subset \mathrm{End}_{l}( Q)$ be a local subsemigroup. It induces the semigroup $\mathcal S _{B}=\{ \phi _{B}\in C_{l}( B) :\phi \in \mathcal S \}$ on $B$ and the semigroup $\mathcal S _{E}=\{ \phi _{E}\in C_{l}( E) :\phi \in \mathcal S \}$ on the associated bundle $E$.

For $q\in Q$ we define the set

$$\begin{aligned} \mathcal S _{q}=\{g\in G:\exists s\in \mathcal S, sq=qg\}, \end{aligned}$$

which is a subsemigroup of $G$ if it is nonempty. Note that $\mathcal S _{q}$ is open in $G$ if the orbit $\mathcal S q$ is open in $Q$. For others properties of the semigroup $\mathcal S _{q}$, we refer to [6, pg 7].

In order the semigroup $\mathcal S _{q}$ has nonempty interior, we require the following accessibility property for $\mathcal S$.

Definition 2.8

Let $D$ be a control set of $\mathcal S _{B}$. We say that $\mathcal S$ is $*$ -accessible over $D$ if there exists $q\in \pi ^{-1}( D_{0})$ such that

$$\begin{aligned} \mathrm{int}\left( \mathcal S ^{*}q\right) \cap \pi ^{-1}\left( D\right) \ne \emptyset. \end{aligned}$$

If $\mathcal S$ is $*$-accessible over a control set $D\subset B$, then $\mathrm{int}( \mathcal S _{q}) \ne \emptyset$ in $G$ (see [15, Lemma 4.16]).

Let $( S,Q,\mu )$ be an action of a semigroup $S$ on the total space $Q$ of the principal bundle $(Q,\pi,B,G)$ such that $\mu _{s}\in \mathrm{End}_{l}( Q)$. The following result can be found in [15].

Theorem 2.1

Let $E=Q\times _{G}F$ be such that the fiber $F$ is compact and $G$ acts transitively on $F$. Assume that the local semigroup $\mathcal S$ is accessible over a control set $C\subset B$.

• 1. Let $D\subset E$ be a control set with $\pi ( D) \subset C$ and take $q\in Q$ with $D_{0}\cap E_{\pi ( q) }\ne \emptyset$. Then, there is a control set $A\subset F$ for the semigroup $\mathcal S _{q}$ such that

  $$\begin{aligned} D_{0}\cap E_{\pi \left( q\right) }=q\cdot A_{0}. \end{aligned}$$

• 2. Take $q\in Q$ above $C$ and let $A\subset F$ be a control set for $\mathcal S _{q}$. Suppose that there is $v\in A$ and $a\in i_{q}^{-1}( \mathrm{int}( \mathcal S ^{*}q) \cap qG)$ such that $av=v$. Then, there is a control set $D\subset E$ such that

  $$\begin{aligned} D_{0}\cap E_{\pi \left( q\right) }=q\cdot A_{0}. \end{aligned}$$

3 Chain recurrence

We now introduce the concept of chain recurrence for semigroup action. We refer to [1, Section 4] for details on chain recurrence.

Let $S$ be a semigroup acting on a topological space $M$. Let $\mathcal U$ and $\mathcal V$ be open coverings of $M$. We write $\mathcal V \leqslant \mathcal U$ if $\mathcal V$ is a refinement of $\mathcal U$. We write $\mathcal V \leqslant \frac{1}{2}\mathcal U$ if for every $V,V^{\prime }\in \mathcal V$, with $V\cap V^{\prime }\ne \emptyset$, there exist $U\in \mathcal U$ such that $V\cup V^{\prime }\subset U$. For an open covering $\mathcal U$ of $M$ and a compact subset $K\subset M$, we denote

$$\begin{aligned} \left[ \mathcal U,K\right] =\left\{ U\in \mathcal U :K\cap U\ne \emptyset \right\} \text{.} \end{aligned}$$

Let $Y\subset M$ be an open subset and suppose that $K\subset Y$ is a compact subset of $M$. We recall that an open covering $\mathcal U$ of $M$ is called $K$ - subordinated to $Y$ if for any $U\in [ \mathcal U,K]$ one has $U\subset Y$.

A family $\mathcal O$ of open coverings of $M$ is said admissible if it satisfies the following properties:

• 1. For each $\mathcal U \in \mathcal O$, there exists an open covering $\mathcal V \in \mathcal O$ such that $\mathcal V \leqslant \frac{1}{2} \mathcal U$;

• 2. Let $Y\subset M$ be an open set and $K$ a compact of $M$ contained in $Y$. Then, there exists an open covering $\mathcal U \in \mathcal O$ which is $K$-subordinated to $Y$;

• 3. For any $\mathcal U,\mathcal V \in \mathcal O$, there exists $\mathcal W \in \mathcal O$ which is a refinement of both $\mathcal U$ and $\mathcal V$.

It is known that the family of all open coverings is admissible if $M$ is paracompact (see [20, Section 20]).

Definition 3.1

For $x,y\in M$, an open cover $\mathcal U$ of $M$, and $A\subset S$, we define a $( \mathcal U,A)$-chain from $x$ to $y$ as a sequence $x_{0}=x,x_{1},\ldots,x_{n}=y$ in $M$, $a_{0},\ldots,a_{n-1}\in A$, and open sets $U_{0},\ldots,U_{n-1}\in \mathcal U$ such that $a_{i}x_{i},x_{i+1} \in U_{i}$, for all $i=0,\ldots,n-1$.

Definition 3.2

Let $\mathcal O$ be an admissible family of open coverings of $M$ and $\mathcal F$ a family of subsets of $S$. Given a nonempty subset $X\subset M$, we define the $\Omega$-chain limit set of $X$ for the family $\mathcal F$ as

$$\begin{aligned} \Omega \left( X,\mathcal F \right) ={\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\Omega \left( Y,\mathcal U,A\right), \end{aligned}$$

where $\Omega ( X,\mathcal U,A) =\{ y\in M:\text{ there} \text{ are} x\in X \text{ and} \text{ a} ( \mathcal U,A) \text{-chain} \text{ from} x \text{ to} y\}$, and we define the $\Omega ^{*}$ - chain limit set of $X$ as

$$\begin{aligned} \Omega ^{*}\left( X,\mathcal F \right) ={\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\Omega ^{*}\left( X,\mathcal U,A\right), \end{aligned}$$

where $\Omega ^{*}( X,\mathcal U,A) =\{ y\in M: \text{ there} \text{ are} x\in X \text{ and} \text{ a} ( \mathcal U,A) \text{-chain} \text{ from} y \text{ to} x\}$. A subset $Y\subset M$ is $\mathcal F$- chain transitive if $Y\subset \Omega ( y,\mathcal F )$ for all $y\in Y$. A point $x\in M$ is $\mathcal F$-chain recurrent if $x\in \Omega ( x,\mathcal F )$. We denote by $\mathfrak R$ the $\mathcal F$-chain recurrent set, that is, the set of all $\mathcal F$-chain recurrent points.

Remark 3.1

The maximal (with respect to set inclusion) $\mathcal F$-chain transitive sets are given by

$$\begin{aligned} \mathcal M _{x}=\Omega \left( x,\mathcal F \right) \cap \Omega ^{*}\left( x, \mathcal F \right) \end{aligned}$$

with $x\in \mathfrak R$.

The following result has been proved in [1, Proposition 4.6, Corollary 4.8].

Proposition 3.1

Let $\mathcal F$ be a family of subsets of $S$ which is a filter basis and satisfies the translation hypothesis. Then, the $\mathcal F$-chain recurrent set $\mathfrak R$ and the maximal $\mathcal F$-chain transitive sets are closed in $M$.

We also have an interesting relation between Morse decompositions and chain recurrence, which was proved in [2, Theorem 5.2].

Proposition 3.2

Let $\mathcal F$ be a family of subsets of $S$ which is a filter basis and satisfies the translation hypothesis. Assume that $M$ is compact. Then, there exists a finest $\mathcal F$-Morse decomposition if and only if $\mathfrak R$ is the union of a finite number of maximal $\mathcal F$-chain transitive sets.

3.1 Shadowing semigroups

A description of the maximal chain transitive sets is provided in terms of the action of shadowing semigroups. We give a characterization of maximal chain transitive sets as intersections of transitivity sets for the actions of shadowing semigroups.

Let $\mathcal S$ be a local semigroup on $M$. Given an open covering $\mathcal U$ of $M$, we define the $\mathcal S$-neighborhood of the identity map of $M$ relative to $\mathcal U$ as

$$\begin{aligned} \mathcal N _\mathcal{S,\mathcal U }=\left\{ \phi \in \mathcal S :\forall x\in \mathrm{dom}\phi \text{,} \exists U\in \mathcal U \text{ such} \text{ that} x,\phi \left( x\right) \in U\right\}. \end{aligned}$$

Let $( S,M,\mu )$ be the action of $S$ on $M$ as above. The shadowing semigroups are perturbations of $( S,M,\mu )$ in $\mathcal S$, as follows.

Definition 3.3

For all open covering $\mathcal U$ of $M$ and $A\subset S$, we define the set

$$\begin{aligned} \mathcal N _\mathcal{S,\mathcal U }A=\left\{ \phi \mu _{a}:\phi \in \mathcal N _\mathcal{S,\mathcal U },a\in A\right\}. \end{aligned}$$

The $( \mathcal U,A)$ - shadowing semigroup $\mathcal S _\mathcal{U,A}$ is the local semigroup generated by $\mathcal N _\mathcal{S,\mathcal U }A$.

The orbits of the shadowing semigroups describe trajectories with jumps in open sets of a covering.

Definition 3.4

Let $\mathcal O$ be an admissible family of open coverings of $M$. We say that $\mathcal S$ is $\mathcal O$ - locally transitive if given a covering $\mathcal U \in \mathcal O$ and $U\in \mathcal U$, for all $x,y\in U$ there is $\phi \in \mathcal N _\mathcal{S,\mathcal U }$ such that $\phi ( x) =y$.

For instance, given $\mathcal U \in \mathcal O$, $U\in \mathcal U$, and $x\in U$, define $\phi _{U,x}:U\rightarrow M$ by $\phi _{U,x}( y) =x$. The local semigroup $\mathcal S =\{ \phi _{U,x}:U\in \mathcal U, \mathcal U \in \mathcal O,x\in U\}$ is $\mathcal O$-locally transitive. In particular, the local semigroup $C_{l}( M)$ of all continuous maps defined on open subsets is $\mathcal O$-locally transitive.

The following result presents the link between the chains of $( S,M)$ and the action of the shadowing semigroups.

Proposition 3.3

Given $x\in M$, an open covering $\mathcal U$ of $M$, and $A\subset S$, we have $\mathcal S _\mathcal{U,A}x\subset \Omega ( x, \mathcal U,A)$ and $(\mathcal S _\mathcal{U,A})^{*}x\subset \Omega ^{*}( x,\mathcal U,A)$. Let $\mathcal O$ be an admissible family of open coverings of $M$. If $\mathcal S$ is $\mathcal O$-locally transitive, then $\mathcal S _\mathcal{U,A}x=\Omega ( x, \mathcal U,A)$ and $\mathcal S _\mathcal{U,A}^{*}x=\Omega ^{*}( x,\mathcal U,A)$, for all $\mathcal U \in \mathcal O$ and $A\subset S$.

Proof

Take $y\in \mathcal S _\mathcal{U,A}x$ and $\psi \in \mathcal S _\mathcal{ U,A}$ such that $\psi ( x) =y$. Write $\psi =\phi _{n}\sigma _{a_{n}},\ldots, \phi _{0}\sigma _{a_{0}}$, with $\phi _{i}\in N_\mathcal{S, \mathcal U }$ and $a_{i}\in A$, $i=0,\ldots,n$. Denote $x_{0}=x,x_{1}=\phi _{0}( x_{0}a_{0}),x_{2}=\phi _{1}( x_{1}a_{1}),\ldots,x_{n}=\phi _{n}( x_{n}a_{n}) =y$ in $M$. Then, for each $i$ there is $U_{i}\in \mathcal U$ such that $x_{i}a_{i},\phi _{i}( x_{i}a_{i}) \in U_{i}$, that is, $x_{i}a_{i},x_{i+1}\in U_{i}$. Hence, $y\in \Omega ( x,\mathcal U,A)$, whence $\mathcal S _\mathcal{U,A}x\subset \Omega ( x,\mathcal U,A)$. Now, if $z\in (\mathcal S _\mathcal{U,A})^{*}x$, then $x\in \mathcal S _\mathcal{U,A}z$. Hence, $x\in \Omega ( z,\mathcal U,A)$, which means $z\in \Omega ^{*}( x,\mathcal U,A)$. Thus, $(\mathcal S _{ \mathcal U,A})^{*}x\subset \Omega ^{*}( x,\mathcal U,A)$. For the second part of the proposition, assume that $\mathcal S$ is $\mathcal O$-locally transitive. Let $y\in \Omega ( x,\mathcal U,A)$. Take $x_{0}=x,\ldots,x_{n}=y\in M$, $a_{0},\ldots,a_{n-1}\in A$ and $U_{0},\ldots,U_{n-1}\in \mathcal U$ such that $x_{i}a_{i},x_{i+1}\in U_{i}$, $i=0,\ldots,n-1$. For each $i$ there is $\phi _{i}\in N_\mathcal{S,\mathcal U }$ such that $\phi _{i}( x_{i}a_{i}) =x_{i+1}$. Then, $y=\phi _{n-1}\sigma _{a_{n-1}},\ldots, \phi _{0}\sigma _{a_{0}}( x) \in \mathcal S _\mathcal{U,A}x$. Hence, $\Omega ( x,\mathcal U,A) \subset \mathcal S _\mathcal{U,A}x$. Now, if $z\in \Omega ^{*}( x, \mathcal U,A)$, then $x\in \Omega ( z,\mathcal U,A) \subset \mathcal S _\mathcal{U,A}z$. Hence, $z\in (\mathcal S _\mathcal{U,A})^{*}x$, whence $\Omega ^{*}( x,\mathcal U,A) \subset (\mathcal S _\mathcal{U,A})^{*}x$.$\square$

By Proposition 3.3, it follows the following characterization of the maximal $\mathcal F$-chain transitive sets as intersections of transitivity sets for the actions of shadowing semigroups.

Theorem 3.1

Let $\mathcal O$ be an admissible family of open coverings of $M$ and $\mathcal F$ a family of subsets of $S$. Assume that $\mathcal S$ is $\mathcal O$-locally transitive. Let $\mathcal M \subset M$ be a nonempty subset. Then, the following condition is necessary and sufficient for $\mathcal M$ to be a maximal $\mathcal F$-chain transitive set:

• For each shadowing semigroup $\mathcal S _\mathcal{U,A}$ there is an effective control set $D_\mathcal{U,A}$ of $\mathcal S _\mathcal{U,A}$ such that

  $$\begin{aligned} \mathcal M ={\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\left( D_{ \mathcal U,A}\right) _{0}={\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}D_\mathcal{U,A}={\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\mathrm{cls}\left( D_\mathcal{U,A}\right). \end{aligned}$$

Proof

Take $x\in \mathcal M.$ Then, $\mathcal S _\mathcal{U,A}x=\Omega ( x, \mathcal U,A)$ and $\mathcal S _\mathcal{U,A}^{*}x=\Omega ^{*}( x,\mathcal U,A)$, for all $\mathcal U \in \mathcal O$ and $A\subset \mathcal F$. Since the sets $\Omega ( x,\mathcal U,A)$ and $\Omega ^{*}( x,\mathcal U,A)$ are open in $M$, the orbits $\mathcal S _\mathcal{U,A}x$ and $\mathcal S _\mathcal{U,A}^{*}x$ are open in $M$. Hence,

$$\begin{aligned} \mathcal M&= \Omega \left( x,\mathcal F \right) \cap \Omega ^{*}\left( x,\mathcal F \right) \\&= \left\{ {\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\Omega \left( x,\mathcal U,A\right) \right\} \bigcap \left\{ {\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\Omega ^{*}\left( x,\mathcal U,A\right) \right\} \\&= {\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\mathcal S _{ \mathcal U,A}x\cap \mathcal S _\mathcal{U,A}^{*}x \\&= {\bigcap _\mathcal{U \in \mathcal O,A\in \mathcal F }}\left( D_\mathcal{ U,A}\right) _{0}\text{.} \end{aligned}$$

The other equalities follow since $( D_\mathcal{U,A}) _{0}$ is dense in $D_\mathcal{U,A}$.$\square$

4 Chain recurrence on fiber bundles

Our purpose now is to investigate the chain recurrence for semigroup actions on fiber bundles with metrizable fiber.

Let $\pi :Q\rightarrow B$ be a locally trivial bundle with structure group $G$ and paracompact base space. Assume that $G$ acts on the left on a metric space $( F,\mathrm{d})$ and let $\pi _{E}:E\rightarrow B$ be the bundle associated with $\pi$, where $E=Q\times _{G}F$. Consider a trivializing covering $\{ U_{i}\} _{i\in I}$ of $Q$. For each $i\in I$, there is a homeomorphism $\psi _{i}:\pi ^{-1}( U_{i}) \rightarrow U_{i}\times G$, $\psi _{i}=( \pi,\upsilon _{i})$, where $\upsilon _{i}:\pi ^{-1}( U_{i}) \rightarrow G$ is a continuous mapping such that $\upsilon _{i}( qg) =\upsilon _{i}( q) g$, for all $q\in \pi ^{-1}( U_{i})$ and $g\in G$. The family $\Psi =\{ ( U_{i},\psi _{i}) \} _{i\in I}$ is called a map of $Q$. For each $i\in I$, the application $\upsilon _{i}^{E}:\pi _{E}^{-1}( U_{i}) \rightarrow F$ given by $\upsilon _{i}^{E}( q\cdot u) =\upsilon _{i}( q) u$ is open. Hence, $\psi _{i}^{E}:\pi _{E}^{-1}( U_{i}) \rightarrow U_{i}\times F$, given by $\psi _{i}^{E}=( \pi _{E},\upsilon _{i}^{E})$, is a homeomorphism. Then, the family $\Psi ^{E}=\{ ( U_{i},\psi _{i}^{E}) \} _{i\in I}$ is a map of $E$.

Definition 4.1

Given $\varepsilon >0$ and an open covering $\mathcal U$ of $B$, a $\Psi$ -adapted covering of $E$ is defined as

$$\begin{aligned} \mathcal U _{\varepsilon }=\left\{ \left( \psi _{i}^{E}\right) ^{-1}\left( \left( U\cap U_{i}\right) \times B_{\varepsilon }\left( u\right) \right) :U\in \mathcal U,i\in I \text{ and} u\in F\right\}. \end{aligned}$$

We denote $\mathcal O _{\Psi }( E)$ the family of all $\Psi$-adapted coverings. Under general conditions, the family $\mathcal O _{\Psi }( E)$ is admissible. Indeed, $\mathcal O _{\Psi }( E)$ is admissible if the bundle $Q\rightarrow X$ can be reduced to a sub-bundle $P\rightarrow X$ whose structure group $K$ acts on the fiber by isometry. See [15] for details of $\mathcal O _{\Psi }( E)$. Moreover, the endomorphism semigroup $\mathrm{End}_{l}( E)$ satisfies the local transitivity property on $\mathcal O _{\Psi }( E)$ if some hypotheses are added. In fact, $\mathrm{End} _{l}( E)$ is $\mathcal O _{\Psi }( E)$-locally transitive if $( F,\mathrm{d})$ is convex and for every $u,v\in F$, there is $g\in G$ such that $v=gu$ and

$$\begin{aligned} \mathrm{d}\left( u,v\right) =\sup _{w\in F}\mathrm{d}\left( gw,w\right). \end{aligned}$$

We recall that $\left( F,\mathrm{d}\right)$ is convex if for any pair $u,v\in F$ there is $w\in F$ such that $\mathrm{d}\left( u,v\right) =\mathrm{d }\left( u,w\right) +\mathrm{d}\left( w,v\right)$. These conditions are satisfied if $G$ contains a compact metrizable subgroup $\left( K,\mathrm{d} _{K}\right)$ whose action is open and transitive on $F$. This fact occurs if there is a $K$-invariant metric in $F$ which is compatible with its topology. In this case, the fiber $F$ is identified with a quotient space of $K$ provided with the Hausdorff metric $\mathrm{d}_{H}$ obtained from $\mathrm{d}_{K}$.

From now on, the fiber bundle $Q\rightarrow B$ is reducible to a sub-bundle $P\rightarrow B$ with structure group $K$, where $K\subset G$ is a compact metrizable subgroup whose action is open and transitive on the fiber $F$ of the associated bundle $E\rightarrow B$. We provide $F$ with a Hausdorff metric $\mathrm{d}$, and we assume that $( F,\mathrm{d})$ is convex. In the special case of flag bundle, these conditions are satisfied.

Fix a map $\Psi =\{ ( U_{i},\psi _{i}) \} _{i\in I}$ of $Q$ and consider the family $\mathcal O =\mathcal O _{\Psi }( E)$. Then, $\mathcal O$ is admissible and the endomorphism semigroup $\mathrm{End}_{l}( E)$ is $\mathcal O$-locally transitive.

Let $( S,Q,\mu )$ be an endomorphism semigroup on $Q$ and $( S,E,\mu )$ the induced action of $S$ on $E$. Fix a family $\mathcal F$ of subsets of $S$ which is a filter basis on the subsets of $S$ and satisfies the translation hypothesis.

The following result has been proved in [3, Theorem 5.1].

Proposition 4.1

Let $E$ be a locally trivial bundle with projection $\pi :E\rightarrow B$ and typical fiber $F$, which is associated with a locally trivial bundle $Q$. Suppose that $E$ is compact and $B$ is $\mathcal F$-chain transitive. Then, each maximal $\mathcal F$-chain transitive set in $E$ intersects all fibers of $E$.

For each $\mathcal U \in \mathcal O$ and $A\in \mathcal F$, we denote by $\mathcal S _\mathcal{U,A}$ the local subsemigroup of $\mathrm{End} _{l}( Q)$ such that

$$\begin{aligned} \left( \mathcal S _\mathcal{U,A}\right) _{E}=\mathrm{End}_{l}\left( E\right) _\mathcal{U,A}. \end{aligned}$$

Then, each shadowing semigroup $\mathrm{End}_{l}( E) _\mathcal{U _{\varepsilon },A}$ is induced by the local semigroup

$$\begin{aligned} \mathcal S _\mathcal{U _{\varepsilon },A}=\left\{ \phi \in \mathrm{End} _{l}\left( Q\right) :\phi _{E}\in \mathrm{End}_{l}\left( E\right) _\mathcal{ U _{\varepsilon },A}\right\} \end{aligned}$$

which is generated from the set

$$\begin{aligned} N_\mathcal{U _{\varepsilon },A}=\left\{ \phi \circ \mu _{a}:\phi _{E}\in \mathcal N _{\mathrm{End}_{l}\left( E\right),\mathcal U _{\varepsilon }},a\in A\right\}. \end{aligned}$$

For $x\in B$ and $g\in G$, let $c_{x}:B\rightarrow B$ be the constant map $c_{x}( y) =x$ $( y\in B)$, and let $L_{g}:G\rightarrow G$ be the left translation by $g$. Consider the product map $c_{x}\times L_{g}( y,h) =( x,gh)$. For each $i\in I$ and $U\in \mathcal U$, we define the application

$$\begin{aligned} \phi _{U,x,g}^{i}=\psi _{i}^{-1}\circ \left( c_{x}\times L_{g}\right) \circ \psi _{i}\mid _{\pi ^{-1}\left( U\cap U_{i}\right) } \end{aligned}$$

with $x\in U\cap U_{i}$ and $g\in G$. Then, $\phi _{U,x,g}^{i}$ is a local endomorphism of $Q$.

The following result has been proved in [3, Proposition 3.2].

Proposition 4.2

Let $G/H$ be a compact homogeneous space of $G$ and $\mathcal V$ an open covering of $G/H$. Then, the $G$-neighborhood $\mathcal N _{G, \mathcal V }$ of the identity $e\in G$ relative to $\mathcal V$ is a neighborhood of $e$ in $G$.

Next, we show that the semigroups $\mathcal S _\mathcal{U _{\varepsilon },A}\$are accessible.

Proposition 4.3

For every $q\in Q$, $A\in \mathcal F$, and $\mathcal U _{\varepsilon }\in \mathcal O$, the orbits $\mathcal S _\mathcal{U _{\varepsilon },A}q$ and $\mathcal S _\mathcal{U _{\varepsilon },A}^{*}q$ have interior points.

Proof

It is enough to show that the generators $N_\mathcal{U _{\varepsilon },A}q$ and $N_\mathcal{U _{\varepsilon },A}^{*}q$ have interior points. Let $\mathcal B _{\varepsilon }$ be an open covering by $\varepsilon$-balls and take $a\in A$. For $U\in \mathcal U$ and $i\in I$, take $p\in \psi _{i}^{-1}( ( U\cap U_{i}) \times \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }}) v_{i}( aq) )$. Then, $\psi _{i}( p) =( \pi ( p),v_{i}( p) ) \in ( U\cap U_{i}) \times \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }}) v_{i}( aq)$. Hence, $p\in \pi ^{-1}( U\cap U_{i})$, and there is $g\in \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }})$ such that $gv_{i}( aq) =v_{i}( p)$. Thus, we have

$$\begin{aligned} \phi _{U,\pi \left( p\right),g}^{i}\left( aq\right) =\psi _{i}^{-1}\left( \pi \left( p\right),gv_{i}\left( aq\right) \right) =\psi _{i}^{-1}\left( \pi \left( p\right),v_{i}\left( p\right) \right) =p. \end{aligned}$$

It remains to show that $( \phi _{U,\pi ( p),g}^{i}) _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$. In fact, for $r\cdot u\in \pi ^{-1}( U\cap U_{i})$, there is an $\varepsilon$-ball $B_{\varepsilon }( w)$ such that $v_{i}( r) u,gv_{i}( r) u\in B_{\varepsilon }( w)$. Then,

$$\begin{aligned} \psi _{i}^{E}\left( r\cdot u\right) =\left( \pi \left( r\right),v_{i}\left( r\right) u\right) \in U\cap U_{i}\times B_{\varepsilon }\left( w\right) \end{aligned}$$

and

$$\begin{aligned} \psi _{i}^{E}\left( \left( \phi _{U,\pi \left( p\right),g}^{i}\right) _{E}\left( r\cdot u\right) \right)&= \psi _{i}^{E}\left( \phi _{U,\pi \left( p\right),g}^{i}\left( r\right) \cdot u\right) \\&= \psi _{i}^{E}\left( \psi _{i}^{-1}\left( \pi \left( p\right),gv_{i}\left( r\right) \right) \cdot u\right) \\&= \psi _{i}^{E}\left( \psi _{i}^{-1}\left( \pi \left( p\right),gv_{i}\left( r\right) \right) \cdot u\right) \\&= \left( \pi \left( p\right),gv_{i}\left( r\right) u\right) \in U\cap U_{i}\times B_{\varepsilon }\left( w\right). \end{aligned}$$

Hence, $r\cdot u,( \phi _{U,\pi ( p),g}^{i}) _{E}\ ( r\cdot u) \in ( \psi _{i}^{E}) ^{-1}( ( U\cap U_{i}) \times B_{\varepsilon }( w) )$. Thus, $( \phi _{U,\pi ( p),g}^{i}) _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$, and

$$\begin{aligned} \psi _{i}^{-1}\left( \left( U\cap U_{i}\right) \times \mathrm{int}\left( \mathcal N _{G,\mathcal B _{\varepsilon }}\right) v_{i}\left( aq\right) \right) \subset \mathrm{int}\left( N_\mathcal{U _{\varepsilon },A}q\right). \end{aligned}$$

Now, take $p\in \mu _{a}^{-1}( \psi _{i}^{-1}( ( U\cap U_{i}) \times \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }}) ^{-1}\upsilon _{i}( q) ) )$. Then, $\psi _{i}( ap) \in ( U\cap U_{i}) \times \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }}) ^{-1}\upsilon _{i}( q)$, which means $\pi ( ap) \in U\cap U_{i}$ and there is $g\in \mathrm{int}( \mathcal N _{G,\mathcal B _{\varepsilon }})$ such that $v_{i}( ap) =g^{-1}v_{i}( q)$. Hence,

$$\begin{aligned} \phi _{U,\pi \left( q\right),g}^{i}\left( ap\right) =\psi _{i}^{-1}\left( \pi \left( q\right),gv_{i}\left( ap\right) \right) =\psi _{i}^{-1}\left( \pi \left( q\right),v_{i}\left( q\right) \right) =q\text{.} \end{aligned}$$

As before we have $( \phi _{U,\pi ( q),g}^{i}) _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$. Therefore,

$$\begin{aligned} \mu _{a}^{-1}\left( \psi _{i}^{-1}\left( \left( U\cap U_{i}\right) \times \mathrm{int}\left( \mathcal N _{G,\mathcal B _{\varepsilon }}\right) ^{-1}\upsilon _{i}\left( q\right) \right) \right) \subset \mathrm{int}\left( N_\mathcal{U _{\varepsilon },A}^{*}q\right). \end{aligned}$$

$\square$

Now, we denote by $c_{l}( B)$ the semigroup of all constant maps whose domains are open sets of $B$ such that each one is contained in only one set of the family $\{ U_{i}:i\in I\}$. Take $\mathcal U \in \mathcal O ( B)$ and $U\in \mathcal U$. Given $x,y\in U$, take $\phi \in c_{l}( B)$ such that $x\in \mathrm{dom}( \phi )$, $\mathrm{dom}( \phi ) \subset U$ and $\phi ( z) =y$, for every $z\in \mathrm{dom}( \phi )$. Then, $\phi ( x) =y$. Moreover, given $z\in \mathrm{dom}( \phi )$, we have $z,\phi ( z) =y\in U$. It follows that $\phi \in \mathcal N _{c_{l}( B),\mathcal U }$. Thus, $c_{l}( B)$ is $\mathcal O ( B)$-locally transitive. In particular, the local semigroup $C_{l}( B)$ is $\mathcal O ( B)$-locally transitive.

Lemma 4.1

For every $A\in \mathcal F$, $\mathcal U \in \mathcal O \left( B\right)$ and $\varepsilon >0$, one has

$$\begin{aligned} c_{l}\left( B\right) _\mathcal{U,A}\subset \left( \mathcal S _\mathcal{U _{\varepsilon },A}\right) _{B}\subset C_{l}\left( B\right) _\mathcal{U,A}. \end{aligned}$$

Proof

It is enough to show that

$$\begin{aligned} \mathcal N _{c_{l}\left( B\right),\mathcal U }A\subset \left( N_\mathcal{U _{\varepsilon },A}\right) _{B}\subset \mathcal N _{C_{l}\left( B\right), \mathcal U }A. \end{aligned}$$

If $\eta \in \mathcal N _{c_{l}( B),\mathcal U }A$, then $\eta =\varphi \circ \mu _{s}$, with $\varphi \in \mathcal N _{c_{l}( B),\mathcal U }$ and $s\in A$. Then, there is $y\in B$ and $i\in I$ such that $\varphi ( x) =y$, for every $x\in \mathrm{dom}( \varphi )$, and $\mathrm{dom}( \varphi ) \subset U_{i}$ such that $\mathrm{dom}( \varphi ) \nsubseteq U_{j}$ if $j\ne i$. We have $\varphi =\phi _{X}$ for some $\phi \in \mathrm{End}_{l}( Q)$ such that $\phi _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$. Indeed, consider the set $\pi ^{-1}( \mathrm{dom}( \varphi ) ) =\psi _{i}^{-1}( \mathrm{dom}( \varphi ) \times G)$ and define the application $\phi :\pi ^{-1}( \mathrm{dom}( \varphi ) ) \rightarrow Q$ by

$$\begin{aligned} \phi \left( \psi _{i}^{-1}\left( x,g\right) \right) =\psi _{i}^{-1}\left( y,g\right) \end{aligned}$$

for $\psi _{i}^{-1}( x,g) \in \pi ^{-1}( \mathrm{dom}( \varphi ) )$. Since $\psi _{i}^{-1}( x,g) =\chi _{i}( x) g$, $\phi$ is an endomorphism. Moreover, given $q\in \pi ^{-1}( \mathrm{dom}( \varphi ) )$, we have

$$\begin{aligned} \phi _{B}\left( \pi \left( q\right) \right) =\pi \left( \phi \left( \psi _{i}^{-1}\left( \pi \left( q\right),\upsilon _{i}\left( q\right) \right) \right) \right) =\pi \left( \psi _{i}^{-1}\left( y,\upsilon _{i}\left( q\right) \right) \right) =y\text{.} \end{aligned}$$

Hence, $\phi _{B}=\varphi$ and $( \phi \circ \mu _{s}) _{B}=\eta$. Now, given $q\cdot u\in \mathrm{dom}( \phi _{E})$, we have $\pi ( q) \in \mathrm{dom}( \varphi )$, and there is $U\in \mathcal U$ such that $\pi ( q),y\in U$. Hence,

$$\begin{aligned} \psi _{i}^{E}\left( \phi _{E}\left( q\cdot u\right) \right) =\psi _{i}^{E}\left( \psi _{i}^{-1}\left( y,\upsilon _{i}\left( q\right) \right) \cdot u\right) =\left( y,\upsilon _{i}\left( q\right) u\right) \end{aligned}$$

and

$$\begin{aligned} \psi _{i}^{E}\left( q\cdot u\right) =\left( \pi \left( q\right),\upsilon _{i}\left( q\right) u\right). \end{aligned}$$

It follows that $\phi _{E}( q\cdot u),q\cdot u\in \psi _{i}^{-1}( U\cap U_{i}\times B_{\varepsilon }( \upsilon _{i}( q) u) ) \in \mathcal U _{\varepsilon }$. Thus, $\phi _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$ and $\eta \in ( N_\mathcal{U _{\varepsilon },A}) _{B}$. On the other hand, take $\eta \in ( N_\mathcal{U _{\varepsilon },A}) _{B}$. Then, $\eta =( \phi \circ \mu _{s}) _{B}$, where $\phi _{E}\in \mathcal N _{\mathrm{End}_{l}( E),\mathcal U _{\varepsilon }}$. To see that $\eta \in \mathcal N _{C_{l}( B),\mathcal U }A$, it is enough to show that $\phi _{B}\in \mathcal N _{C_{l}( B),\mathcal U }$. In fact, given $\pi ( q) \in \mathrm{dom}( \phi _{B})$ and $u\in F$, we have $q\cdot u\in \mathrm{dom}( \phi _{E})$. Hence, there is $\psi _{i}^{-1}( U\cap U_{i}\times B_{\varepsilon }( v) ) \in \mathcal U _{\varepsilon }$ such that $q\cdot u,\phi _{E}( q\cdot u) \in \psi _{i}^{-1}( U\cap U_{i}\times B_{\varepsilon }( v) )$. It follows that $\pi ( q),\phi _{B}( \pi ( q) ) \in U$, and we conclude the proof.$\square$

Since $B$ is paracompact, the family $\mathcal O ( B)$ is admissible and the local semigroups $c_{l}( B)$ and $C_{l}( B)$ are $\mathcal O ( B)$-locally transitive. Then, we can relate the chain transitivity of $( S,B)$ to the transitivity of the shadowing semigroups induced on $B$.

Proposition 4.4

The action $( S,B)$ is $\mathcal F$-chain transitive if, and only if, each semigroup $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ is transitive.

Proof

Assume that $( S,B)$ is chain transitive. By Theorem 3.1, it follows that all shadowing semigroup $c_{l}( B) _\mathcal{U,A}$ is transitive. By Lemma 4.1, it follows that $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ is transitive. Conversely, if all shadowing semigroup induced on the base space is transitive, by Lemma 4.1, it follows that all shadowing semigroup $C_{l}( B) _{ \mathcal U,A}$ is transitive. By Theorem 3.1 again, $( S,B)$ is chain transitive.$\square$

Thus, $B$ is a control set for each semigroup $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ if the action of $S$ on the base space $B$ is $\mathcal F$-chain transitive. By Proposition 4.3, it follows that each semigroup $\mathcal S _\mathcal{U _{\varepsilon },A}$ is accessible over $B$.

Finally, we present a relation between the chain transitivity in $E$ and the chain transitivity in $B$. It is done when an invariant probability measure in $F$ exists, in particular, if $G$ is compact or solvable.

Proposition 4.5

Assume that there is a probability measure in $F$ which is invariant by $G$. Then, $( S,E)$ is $\mathcal F$-chain transitive if, and only if, $( S,B)$ is $\mathcal F$-chain transitive.

Proof

If $( S,B)$ is $\mathcal F$-chain transitive, by Proposition 4.4, it follows that the semigroups $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ are transitive, for all $A\in \mathcal F$ and $\mathcal U _{\varepsilon }\in \mathcal O _{\Psi }( E)$. Then, $B$ is the only control set of $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$. Since the orbits of $\mathcal S _\mathcal{U _{\varepsilon },A}$ have interior points in $Q$, the semigroup $\mathcal S _\mathcal{U _{\varepsilon },A}$ is accessible on $B$. By Lemma 6.2 in [16], it follows that the semigroup $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{q}$ of $G$ is transitive in $F$ for all $q\in Q$. By Theorem 2.1, it follows that $E$ is the only control set of $\mathrm{End}_{l}( E) _\mathcal{U _{\varepsilon },A}$. Finally, by Theorem 3.1, $( S,E)$ is $\mathcal F$-chain transitive. Conversely, suppose that $( S,E)$ is $\mathcal F$-chain transitive. By Theorem 3.1 again, it follows that $E$ is the only control set of $\mathrm{End}_{l}( E) _\mathcal{U _{\varepsilon },A}$, for all $A\in \mathcal F$ and $\mathcal U _{\varepsilon }\in \mathcal O _{\Psi }( E)$. Then, $\pi ( E) =B$ is a control set of $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$, that is, $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ is transitive for all $A\in \mathcal F$ and $\mathcal U _{\varepsilon }\in \mathcal O _{\Psi }( E)$. From Proposition 4.4, $( S,B)$ is $\mathcal F$-chain transitive. $\square$

4.1 Flag bundles

Now, we describe the maximal chain transitive sets for semigroup actions on a flag bundle. The characterization of effective control sets in flag manifolds is fundamental to study endomorphisms of flag bundles. Under transitivity on the base space and accessibility of the local semigroup of endomorphism, we describe fully the effective control sets on the flag bundle.

Let $G$ be a reducible Lie group with connected semi-simple component, and let $\mathfrak g$ be the semi-simple component of the Lie algebra of $G$. Given a locally trivial bundle $\pi :Qarrow B$ with structure group $G$, we call flag bundle a bundle associated with $\pi$ whose typical fiber is a flag manifold of $\mathfrak g$. Take $\Theta \subset \Sigma$, where $\Sigma$ is the root simple system of $\mathfrak g$. Then, $\mathbb F _{\Theta }$ denote the flag manifold of type $\Theta$, and the flag bundle of type $\Theta$ associated with $\pi$ is the bundle $\pi _\mathbb{E _{\Theta }}:\mathbb E _{\Theta }\rightarrow B$, where $\mathbb E _{\Theta }=Q\times _{G}\mathbb F _{\Theta }$. We denote $\mathbb F$ the maximal flag manifold of $\mathfrak g$, and $\mathbb E =Q\times _{G}\mathbb F$. In this case, $\pi _\mathbb{E }:\mathbb E \rightarrow B$ is called maximal flag bundle.

Let $\mathcal S \subset \mathrm{End}_{l}( Q)$ be a local semigroup which is transitive in $B$, and its progressive and regressive orbits in $Q$ are open sets. Let $W$ be the Weyl group of $\mathfrak g$. For each $q\in Q$, the subsemigroup $\mathcal S _{q}\subset G$ is open. In [17], the effective control sets for the action of $\mathcal S _{q}$ on the maximal flag manifold were described by means of the Weyl group $W$. In this description we have a mapping

$$\begin{aligned} w\rightarrow \mathbb A ^{q}\left( w\right) \end{aligned}$$

which associates to $w\in W$ a control set $\mathbb A ^{q}( w)$ in such a way that the set of transitivity $\mathbb A ^{q}( w) _{0}$ is the set of the fixed points of type $w$ for the split-regular elements in $\mathrm{int}( \mathcal S _{q})$. For each $\Theta \subset \Sigma$, we denote $\mathbb A _{\Theta }^{q}( w)$ the effective control set of $\mathcal S _{q}$ in $\mathbb F _{\Theta }$ such that $\pi _{\Theta }( \mathbb A ^{q}( w) ) \subset \mathbb A _{\Theta }^{q}( w)$ and $\pi _{\Theta }( \mathbb A ^{q}( w) _{0}) =\mathbb A _{\Theta }^{q}( w) _{0}$, where $\pi _{\Theta }:\mathbb F \rightarrow \mathbb F _{\Theta }$ is the natural fibration. By Theorem 2.1, there is an effective control set $\mathbb D _{\Theta }^{q}( w)$ of $\mathcal S _\mathbb{E _{\Theta }}$ such that $\mathbb D _{\Theta }^{q}( w) _{0}\cap ( \mathbb E _{\Theta }) _{\pi ( q) }\ne \emptyset$ and

$$\begin{aligned} \mathbb D _{\Theta }^{q}\left( w\right) _{0}\cap \left( \mathbb E _{\Theta }\right) _{\pi \left( q\right) }=q\cdot \mathbb A _{\Theta }^{q}\left( w\right) _{0}, \end{aligned}$$

and these are all the effective control sets of $\mathcal S _\mathbb{E _{\Theta }}$ whose transitivity sets intersect the fiber $( \mathbb E _{\Theta }) _{\pi ( q) }$. Since $\mathcal S$ is transitive in $B$, the transitivity set of an effective control set $\mathbb D$ of $\mathcal S _\mathbb{E _{\Theta }}$ is projected onto the base space, that is, $\pi _\mathbb{E _{\Theta }}( \mathbb D _{0}) =B$. Hence, $\mathbb D _{0}$ intersect the fiber $( \mathbb E _{\Theta }) _{\pi ( q) }$. Furthermore, the control set $\mathbb D _{\Theta }^{q}( w)$ does not depend on $q\in Q$ (see [13, Theorem 5.8]). Therefore, $\{ \mathbb D _{\Theta }^{q}( w) :w\in W\}$ is the collection of all effective control sets of $\mathcal S _\mathbb{E _{\Theta }}$.

The bundle $Q\rightarrow B$ can be reduced to a sub-bundle $P\rightarrow B$ whose structure group is a metrizable compact subgroup of the semi-simple component of $G$, which is transitive on the flag manifolds of $\mathfrak g$. By fixing $\Theta \subset \Sigma$ and a map $\Psi =\{ ( U_{i},\psi _{i}) \} _{i\in I}$ of $Q$, we consider the family $\mathcal O _{\Theta }=\mathcal O _{\Psi }( \mathbb E _{\Theta })$ of all $\Psi$-adapted coverings. Then, $\mathcal O _{\Theta }$ is admissible and $\mathrm{End}_{l}( \mathbb E _{\Theta })$ is $\mathcal O _{\Theta }$-locally transitive. We assume that the semigroup $S$ acts on $Q$ by endomorphisms and consider a family $\mathcal F$ of subsets of $S$ which is a filter basis. We also assume that $B$ is $\mathcal F$-chain transitive. From Proposition 4.4, the semigroup $( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}$ is transitive for all $\mathcal U _{\varepsilon }\in \mathcal O _{\Theta }$, $\Theta \subset \Sigma$, and $A\in \mathcal F$. Since $\mathcal S _\mathcal{U _{\varepsilon },A}\subset \mathrm{End}_{l}( Q)$, the control sets of $( \mathcal S _\mathcal{U _{\varepsilon },A}) _\mathbb{E _{\Theta }}=\mathrm{End}_{l}( \mathbb E _{\Theta }) _\mathcal{U _{\varepsilon },A}$ in $\mathbb E _{\Theta }$ are $\mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }( w)$ with $w\in W$, where $\mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }( w) _{0}\cap ( \mathbb E _{\Theta }) _{x}\ne \emptyset$, for all $x\in B$. The following theorem presents all maximal $\mathcal F$-chain transitive sets in $\mathbb E _{\Theta }$.

Theorem 4.1

Let $\Theta$ be a set in the root simple system $\Sigma$. For each $w$ in the Weyl group $W$, set

$$\begin{aligned} E_{\Theta }\left( w\right) ={\bigcap _\mathcal{U _{\varepsilon }\in \mathcal O _{\Theta },A\in \mathcal F }}\mathrm{cls}\left( \mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }\left( w\right) _{0}\right) \end{aligned}$$

in the flag bundle $\mathbb E _{\Theta }$, where $\mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }( w)$ is the $w$-control set of the shadowing semigroup $\mathrm{End}_{l}( \mathbb E _{\Theta }) _{ \mathcal U _{\varepsilon },A}$. Then, $E_{\Theta }( w) \ne \emptyset$, and $\{ E_{\Theta }( w) :w\in W\}$ is the collection of all maximal $\mathcal F$-chain transitive sets in $\mathbb E _{\Theta }$.

Proof

Fix $x\in X$. Take the family $\mathcal C$ of closed sets of $( \mathbb E _{\Theta }) _{x}$ given by

$$\begin{aligned} \mathcal C =\left\{ \mathrm{cls}\left( \mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }\left( w\right) _{0}\right) \cap \left( \mathbb E _{\Theta }\right) _{x}:\mathcal U _{\varepsilon }\in \mathcal O _{\Theta }\ \text{ and} A\in \mathcal F \right\}. \end{aligned}$$

For $A_{1},\ldots,A_{n}\in \mathcal F$ and $\mathcal U _{\varepsilon _{1}}^{1},\ldots,\mathcal U _{\varepsilon _{n}}^{n}\in \mathcal O _{\Theta }$, take $A\in \mathcal F$ and the open covering $\mathcal U$ such that

$$\begin{aligned} A\subset A_{1}\cap \cdots \cap A_{n}\quad \text{ and} \quad \mathcal U =\mathcal U ^{1}\wedge \cdots \wedge \mathcal U ^{n}. \end{aligned}$$

If $\varepsilon =\min \{ \varepsilon _{1},\ldots,\varepsilon _{n}\}$, then $\mathcal U _{\varepsilon }\leqslant \mathcal U _{\varepsilon _{i}}^{i}$, for $i=1,\ldots,n$. It follows that

$$\begin{aligned} \mathrm{End}_{l}\left( \mathbb E _{\Theta }\right) _{\mathcal{U } _{\varepsilon },A}\subset \mathrm{End}_{l}\left( \mathbb E _{\Theta }\right) _{\mathcal{U }_{\varepsilon _{i}}^{i},A_{i}} \end{aligned}$$

for each $i$. Hence, $\mathbb D _{\mathcal{U }_{\varepsilon },A}^{\Theta }\left( w\right) _{0}\subset \mathbb D _{\mathcal{U }_{\varepsilon _{i}}^{i},A_{i}}^{\Theta }\left( w\right) _{0}$, and

$$\begin{aligned} \emptyset \ne \mathbb D _\mathcal{U _{\varepsilon },A}^{\Theta }\left( w\right) _{0}\cap \left( \mathbb E _{\Theta }\right) _{x}\subset \mathbb D _{ \mathcal U _{\varepsilon _{i}}^{i},A_{i}}^{\Theta }\left( w\right) _{0}\cap \left( \mathbb E _{\Theta }\right) _{x} \end{aligned}$$

for each $i$, which implies

$$\begin{aligned} {\bigcap _{i=1}^{n}}\mathrm{cls}\left( \mathbb D _{\mathcal{U }_{\varepsilon _{i}}^{i},A_{i}}^{\Theta }\left( w\right) _{0}\right) \cap \left( \mathbb E _{\Theta }\right) _{x}\ne \emptyset. \end{aligned}$$

Thus, $\mathcal C$ has the finite intersection property. Since $( \mathbb E _{\Theta }) _{x}$ is compact, it follows that $E_{\Theta }( w) \ne \emptyset$. The proof follows by Theorem 3.1. $\square$

5 Projective Bundles and Selgrade theorem

In this section, we present the main result of the paper. We consider a linear action of a semigroup on a vector bundle $\pi :\mathcal V \rightarrow B$ over a metric space $B$. From the results of semigroup actions on flag bundles in the last section above, we construct a Whitney decomposition of $\mathcal V$ into invariant subbundles corresponding to the maximal chain transitive sets of the induced action on the projective bundle. This is the Selgrade theorem.

Let $( B,d)$ be a compact metric space. An $n$-dimensional vector bundle $\pi :\mathcal V \rightarrow B$ is a fiber bundle whose typical fiber is $\mathbb R ^{n}$. It can be put in our principal bundle context by taking the bundle of frames $B\mathcal V \rightarrow B$ of $\mathcal V$. The elements of $B\mathcal V$ are the invertible linear maps $q:\mathbb R ^{n}\rightarrow \mathcal V _{b}$, where $\mathcal V _{b}$ denotes the fiber $\pi ^{-1}( b)$. The structural group of $B \mathcal V$ is $G=\mathrm{Gl}( n,\mathbb R )$ which acts on the right on $B\mathcal V$ by $qg=q\circ g$, $q\in B\mathcal V$, $g\in \mathrm{Gl}( n,\mathbb R )$. The vector bundle $\mathcal V \rightarrow B$ is recovered from $B\mathcal V$ as the associated bundle $B\mathcal V \times _{\mathrm{Gl}( n,\mathbb R ) }\mathbb R ^{n}$ obtained by the standard linear action of $\mathrm{Gl}( n,\mathbb R )$ in $\mathbb R ^{n}$. We fix a map $\Psi =\{ ( U_{i},\psi _{i}) \} _{i\in I}$ of $B\mathcal V$, where $\psi _{i}=( \pi,\upsilon _{i})$. For each $i\in I$, we have the open map $\upsilon _{i}^{ \mathcal V }:\pi ^{-1}( U_{i}) \rightarrow \mathbb R ^{n}$ given by $\upsilon _{i}^\mathcal{V }( q\cdot u) =\upsilon _{i}( q) u$. Then, we obtain the map $\Psi ^\mathcal{V }=\{ ( U_{i},\psi _{i}) \} _{i\in I}$, where $\psi _{i}=( \pi,\upsilon _{i}^\mathcal{V })$.

The vector space structure on the fiber $\mathcal V _{\pi ( q) }$ is recovered from $\mathbb R ^{n}$ as

$$\begin{aligned} \lambda \left( q\cdot u\right) +q\cdot v=q\cdot \left( \lambda u+v\right) \end{aligned}$$

for $\lambda \in \mathbb R$ and $u,v\in \mathbb R ^{n}$. Also, it is given a continuous and positive definite bilinear form

$$\begin{aligned} \left\langle q\cdot u,q\cdot v\right\rangle =\sum _{i}\mathrm{d}\left( \pi \left( q\right),B\setminus U_{i}\right) \left\langle q\cdot u,q\cdot v\right\rangle _{i} \end{aligned}$$

where $\langle q\cdot u,q\cdot v\rangle _{i}=\langle v_{i}( q) u,v_{i}( q) v\rangle$ and $\langle q\cdot u,q\cdot v\rangle _{i}=0$ for $\pi ( q) \notin U_{i}$.

The projective bundle $P\pi :P\mathcal V \rightarrow B$ of $\mathcal V$ is the flag bundle $P\mathcal V =B\mathcal V \times _{\mathrm{Gl }( n,\mathbb R ) }\mathbb P ^{n-1}$ obtained by the standard action of $\mathrm{Gl}( n,\mathbb R )$ on the projective space $\mathbb P ^{n-1}$.

The zero section of $\mathcal V$ is the set $Z=\{ q\cdot 0:q\in B \mathcal V \}$. The fibration $P:\mathcal V \setminus Z\rightarrow P \mathcal V$ is given by

$$\begin{aligned} P\left( q\cdot u\right) =q\cdot \mathbb P u \end{aligned}$$

where $\mathbb P :\mathbb R ^{n}-0\rightarrow \mathbb P ^{n-1}$ denotes the canonical projection map. Then, the following diagram commutes:

$$\begin{aligned} \begin{array}{lllll} \mathcal V \setminus Z&\,&\overset{P}{\longrightarrow }&\,&P\mathcal V \\&\searrow \pi&\,&P\pi \swarrow&\\&\,&\quad B&\,&\end{array}. \end{aligned}$$

For $\mathcal L \subset \mathcal V$, we define $P\mathcal L =\{ q\cdot \mathbb P u:q\cdot u\in \mathcal L \setminus Z\}$. In particular, $P \mathcal L =\emptyset$ if $\mathcal L \subset Z$. Thus, for a subset $X\subset P\mathcal V$, one has

$$\begin{aligned} P^{-1}\left( X\right) =\left\{ q\cdot u\in \mathcal V :u\ne 0 \text{ implies} q\cdot \mathbb P u\in X\right\}. \end{aligned}$$

It is well-known that $\mathcal V$ is a metrizable topological space (see [7, Lemma B.1.12]). We keep that metric $\mathrm{d}$ on $\mathcal V$ that is compatible with the original topology. Then, $P\mathcal V$ is a compact metric space under the metric

$$\begin{aligned} \mathrm{d}\left( q\cdot \mathbb P u,p\cdot \mathbb P v\right) =\min \left\{ \mathrm{d}\left( \frac{q\cdot u}{{\Vert } q\cdot u{\Vert } },\frac{ q\cdot v}{{\Vert } q\cdot v{\Vert } }\right),\mathrm{d}\left( \frac{ q\cdot u}{{\Vert } q\cdot u{\Vert } },-\frac{q\cdot v}{{\Vert } q\cdot v{\Vert } }\right) \right\} \end{aligned}$$

for $q\cdot u,q\cdot v\in \mathcal V \setminus Z$.

The following result is useful.

Proposition 5.1

There is a constant $\delta >0$ such that

$$\begin{aligned} \delta \mathrm{d}\left( q\cdot \mathbb P u,q\cdot \mathbb P v\right) \le 1- \frac{\left\langle q\cdot u,q\cdot v\right\rangle ^{2}}{{\Vert } q\cdot u{\Vert } ^{2}{\Vert } q\cdot v{\Vert } ^{2}}\le \delta ^{-1} \mathrm{d}\left( q\cdot \mathbb P u,q\cdot \mathbb P v\right) \end{aligned}$$

for all $q\cdot u,q\cdot v\in \mathcal V \setminus Z$.

Proof

See [7] Lemma B.1.17.$\square$

We now define subbundle and Whitney sum of subbundles.

Definition 5.1

A subbundle of $\mathcal V$ is a closed subset $\mathcal W \subset \mathcal V$ that intersects each fiber $\mathcal V _{b}$ in a linear subspace $\mathcal W _{b}$ such that $\mathrm{dim}\mathcal W _{b}=\mathrm{dim }\mathcal W _{b^{\prime }}$ for all $b,b^{\prime }\in B$. Let $\mathcal X, \mathcal Y \subset \mathcal V$ be subbundles with $\mathcal X \cap \mathcal Y \subset Z$. The Whitney sum of these subbundles is the vector bundle $\mathcal W =\mathcal X \oplus \mathcal Y \subset \mathcal V$ with fibers

$$\begin{aligned} \mathcal W _{b}=\mathcal X _{b}\oplus \mathcal Y _{b}\mathcal = \left\{ x+y:x\in \mathcal X,y\in \mathcal Y \right\}. \end{aligned}$$

We now consider linear actions on vector bundles and reproduce the Selgrade theorem.

Definition 5.2

A linear action $\mu :S\times \mathcal V \rightarrow \mathcal V$ of a semigroup $S$ on a vector bundle $\pi :\mathcal V \rightarrow B$ is an action $\mu$ on $\mathcal V$ such that, for all $\lambda \in \mathbb R$, $q\cdot u,q\cdot v\in \mathcal V$ and $s\in S$, one has

$$\begin{aligned} \mu _{s}\left( \lambda \left( q\cdot u+q\cdot v\right) \right) =\lambda \mu _{s}\left( q\cdot u\right) +\lambda \mu _{s}\left( q\cdot v\right). \end{aligned}$$

Let $\mu :S\times \mathcal V \rightarrow \mathcal V$ be a linear action on a vector bundle $\pi :\mathcal V \rightarrow B$ such that each $\mu _{s}$ is a homeomorphism of $\mathcal V$. We denote by $S^{-1}$ the semigroup of the inverse maps $\mu _{s}^{-1}$. Let $\mathcal F$ be a family of subsets of $S$ which is a filter basis and satisfies the translation hypothesis. We observe that the family $\mathcal F$ provided with the inclusion order is a direct set. We write $A\succcurlyeq B$ if $A\subset B$.

Next, we discuss some properties of the action of $S$ on $P\mathcal V$. We show that an $\mathcal F$-attractor in $P\mathcal V$ generates a subbundle in $\mathcal V$ if the base space $B$ is $\mathcal F$-chain transitive.

Proposition 5.2

Let $\mathcal A$ be an $\mathcal F$-attractor in $P\mathcal V$. Then, $P^{-1}( \mathcal A )$ and $P^{-1}( \mathcal A ^{*})$ intersect each fiber in linear spaces.

Proof

Consider a fiber $\mathcal V _{\pi ( q) }$. If $\mathcal A \cap P( \mathcal V _{\pi ( q) }\setminus Z) =\emptyset$, then $P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( q) }=\{ q\cdot 0\}$ is the trivial linear space. If $\mathcal A \cap P( \mathcal V _{\pi ( q) }\setminus Z)$ consists of a single point $q\cdot \mathbb P u$, then $P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( q) }$ is the one-dimensional linear space generated by $q\cdot u$. Now, suppose that $q\cdot \mathbb P u,q\cdot \mathbb P v\in \mathcal A \cap P( \mathcal V _{\pi ( q) }\setminus Z)$ and $q\cdot \mathbb P u\ne q\cdot \mathbb P v$. We have $q\cdot tu,q\cdot tv\in P^{-1}( \mathcal A )$ for all $t\in \mathbb R$. It remains to show that $q\cdot ( tu+v) \in P^{-1}( \mathcal A )$ for all $t\in \mathbb R$. Since $\mathcal A$ is invariant, the orbits $S^{-1}( q\cdot \mathbb P u)$ and $S^{-1}( q\cdot \mathbb P v)$ are contained in $\mathcal A$. Thus, for all $s\in S$ and $t\in \mathbb R$, we have $\mu _{s}^{-1}q\cdot tu,\mu _{s}^{-1}q\cdot tv\in P^{-1}( \mathcal A )$. Set

$$\begin{aligned} t^{*}=\sup \left\{ t\ge 0:q\cdot \mathbb P \left( t^{\prime }u+v\right) \in \mathcal A \quad \text{ for} \text{ all} \quad t^{\prime }\in \left[ 0,t\right] \right\} \end{aligned}$$

and

$$\begin{aligned} t_{-}^{*}=\inf \left\{ t\le 0:q\cdot \mathbb P \left( t^{\prime }u+v\right) \in \mathcal A \quad \text{ for} \text{ all} \quad t^{\prime }\in \left[ t,0\right] \right\}. \end{aligned}$$

The proposition will be demonstrated if we show that $t^{*}$ and $t_{-}^{*}$ are infinite. In fact, if $t^{*}$ and $t_{-}^{*}$ are infinite, then $q\cdot ( tu+v) \in P^{-1}( \mathcal A )$ for all $t\in \mathbb R$ and we conclude that $P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( q) }$ is a linear space. Analogously, we show that $P^{-1}( \mathcal A ^{*}) \cap \mathcal V _{\pi ( q) }$ is a linear space, since $S$ acts by homeomorphisms. We show that $t^{*}$ is infinite, and the proof for $t_{-}^{*}$ is similar.

Suppose otherwise that $t^{*}$ is finite. Since $\mathcal A$ is closed, it follows that $q\cdot \mathbb P ( t^{*}u+v) \in \mathcal A$, and we can choose $t_{0}>0$ such that $q\cdot \mathbb P ( t_{0}u+t^{*}u+v) \notin \mathcal A$. Hence, $\mu _{s}^{-1}q\cdot \mathbb P ( t_{0}u+t^{*}u+v) \notin \mathcal A$ for all $s\in S,$ since $\mathcal A$ is invariant. Thus, $\omega ^{*}( q\cdot \mathbb P ( t_{0}u+t^{*}u+v),\mathcal F ) \subset \mathcal A ^{*}$. Let $U$ be a compact neighborhood of $\mathcal A$ such that $q\cdot \mathbb P ( t_{0}u+t^{*}u+v) \notin U$ and choose $\varepsilon >0$ such that the $\varepsilon$-neighborhood of $\mathcal A$ is contained in $U$. It follows that, for some $A_{0}\in \mathcal F$ and each $A\succcurlyeq A_{0}$, there is $s_{A}\in A$ such that $\mu _{s_{A}}^{-1}q\cdot \mathbb P ( t_{0}u+t^{*}u+v) \notin U$ and therefore

$$\begin{aligned} \mathrm{d}\left( \mu _{s_{A}}^{-1}q\cdot \mathbb P \left( t_{0}u+t^{*}u+v\right),q\cdot \mathbb P \left( t^{*}u+v\right) \right) >\varepsilon. \end{aligned}$$

(1)

For each $A\succcurlyeq A_{0}$ denote by $s_{A}( t)$ the mapping

$$\begin{aligned} s_{A}\left( t\right) =\mu _{s_{A}}^{-1}q\cdot \mathbb P \left( tu+t^{*}u+v\right) \end{aligned}$$

for $t\in \mathbb R$, and define

$$\begin{aligned} t_{A}^{+}=\sup \left\{ t>0:s_{A}\left( \left[ 0,t\right] \right) \subset U\right\}. \end{aligned}$$

Since $U$ is compact, we have $s_{A}( t_{A}^{+}) \in U$. It follows that there is a convergent subnet $( t_{A_{\lambda }}^{+}) _{\lambda \in \Lambda }$ such that $\lim _{\lambda }t_{A_{\lambda }}^{+}\ne 0$. In fact, suppose otherwise that a such subnet does not exist, that is, any convergent subnet of $( t_{A}^{+})$ converges to $0$. This fact implies the net $\left( \frac{\Vert \mu _{s_{A}}^{-1}q\cdot ( t^{*}u+v) \Vert }{\Vert \mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \Vert }\right) _{A\succcurlyeq A_{0}}$ converges to $0$. In fact, by Proposition 5.1 there is $\delta >0$ such that, for all $p\cdot w,p\cdot w^{\prime }\in \mathcal V \setminus Z$

$$\begin{aligned} \frac{\left\langle p\cdot w,p\cdot w^{\prime }\right\rangle ^{2}}{{\Vert } p\cdot w{\Vert } ^{2}{\Vert } p\cdot w^{\prime }{\Vert } ^{2}}\ge 1-\delta \varepsilon \quad \text{ implies} \quad \mathrm{d}\left( p\cdot \mathbb P w,p\cdot \mathbb P w^{\prime }\right) \le \varepsilon. \end{aligned}$$

If otherwise $\left( \frac{\Vert \mu _{s_{A}}^{-1}q\cdot ( t^{*}u+v) \Vert }{\Vert \mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \Vert }\right)$ does not converge to $0$, there is a constant $M$ such that for a subnet

$$\begin{aligned} \frac{{\Vert } \mu _{s_{A^{i}}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) {\Vert } }{{\Vert } \mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) {\Vert } }\le M \quad \text{ for} \text{ all}\quad i\in \mathcal I \text{.} \end{aligned}$$

For $c_{0}>0$ sufficiently small it follows that, for all $i\in \mathcal I$ and $c\in [ 0,c_{0}]$

$$\begin{aligned}&\frac{\left\langle \mu _{s_{A^{i}}}^{-1}q\cdot \left( ct_{0}u+c\left( t^{*}u+v\right) \right) +\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right),\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\rangle ^{2}}{\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( ct_{0}u+c\left( t^{*}u+v\right) \right) +\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}} \\&= \frac{P\left( c,A^{i}\right) }{Q\left( c,A^{i}\right) }\ge 1-\delta \varepsilon \end{aligned}$$

where $P( c,A^{i})$ is the number

$$\begin{aligned}&c^{2}\left\langle \mu _{s_{A^{i}}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right),s_{A_{i}}^{-1}q\cdot \left( t^{*}u+v\right) \right\rangle ^{2} \\&\quad +2c\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\left\langle \mu _{s_{A^{i}}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right),\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\rangle \\&\quad +\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{4} \end{aligned}$$

and $Q( c,A^{i})$ is the number

$$\begin{aligned}&c^{2}\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2}\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\\&\quad +2c\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\left\langle \mu _{s_{A^{i}}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right),\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\rangle \\&\quad +\left\Vert\mu _{s_{A^{i}}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{4}. \end{aligned}$$

Hence, $\mathrm{d}\ ( \mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P \ ( ct_{0}u+c\ ( t^{*}u+v) +t^{*}u+v),\mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P ( t^{*}u+v) ) \le \varepsilon$ for all $i\in \mathcal I$ and $c\in [ 0,c_{0}]$, and therefore

$$\begin{aligned} \mathrm{d}\left( \mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P \left( \frac{ct_{0}}{c+1}u+\left( t^{*}u+v\right) \right),\mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P \left( t^{*}u+v\right) \right) \le \varepsilon. \end{aligned}$$

But since $\lim _{i}t_{A^{i}}^{+}=0$, there is $i\in \mathcal I$ and $t< \frac{c_{0}t_{0}}{c_{0}+1}$ such that $s_{A^{i}}( t) \notin U$. Hence

$$\begin{aligned} \mathrm{d}\left( \mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P \left( tu+t^{*}u+v\right),\mu _{s_{A^{i}}}^{-1}q\cdot \mathbb P \left( t^{*}u+v\right) \right) >\varepsilon \end{aligned}$$

and we have a contradiction. Thus, $\frac{\Vert \mu _{s_{A}}^{-1}q\cdot ( t^{*}u+v) \Vert }{\Vert \mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \Vert }\rightarrow 0$.

Nevertheless,

$$\begin{aligned}&\frac{\left\langle \mu _{s_{A}}^{-1}q\cdot t_{0}u,\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\rangle ^{2}}{\left\Vert\mu _{s_{A}}^{-1}q\cdot t_{0}u\right\Vert^{2}\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2}} \\&= \frac{\left\langle \mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right)-\mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right),\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\rangle ^{2}}{ \left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right)-\mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2}}= \frac{P\left( A\right) }{Q\left( A\right) } \end{aligned}$$

where $P( A)$ is the number

$$\begin{aligned}&\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{4}\\&\quad -\,2\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2}\left\langle \mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right),\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\rangle \\&\quad +\left\langle \mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right),\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\rangle ^{2} \end{aligned}$$

and $Q( A)$ is the number

$$\begin{aligned}&\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{4}\\&\quad -\,2\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2}\left\langle \mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right),\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\rangle \\&\quad +\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t^{*}u+v\right) \right\Vert^{2}\left\Vert\mu _{s_{A}}^{-1}q\cdot \left( t_{0}u+t^{*}u+v\right) \right\Vert^{2} \end{aligned}$$

which implies $\frac{\langle \mu _{s_{A}}^{-1}q\cdot t_{0}u,\mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \rangle ^{2}}{ \Vert \mu _{s_{A}}^{-1}q\cdot t_{0}u\Vert ^{2}\Vert \mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \Vert ^{2}} \rightarrow 1$ as $\frac{\Vert \mu _{s_{A}}^{-1}q\cdot ( t^{*}u+v) \Vert }{\Vert \mu _{s_{A}}^{-1}q\cdot ( t_{0}u+t^{*}u+v) \Vert }\rightarrow 0$. By Proposition 5.1 again, it follows that

$$\begin{aligned} \lim _{A}\mathrm{d}\left( \mu _{s_{A}}^{-1}q\cdot \mathbb P u,\mu _{s_{A}}^{-1}q\cdot \mathbb P \left( t_{0}u+t^{*}u+v\right) \right) =0. \end{aligned}$$

It implies

$$\begin{aligned} \mathrm{d}\left( \mu _{s_{A}}^{-1}q\cdot \mathbb P \left( t_{0}u+t^{*}u+v\right),q\cdot \mathbb P \left( t^{*}u+v\right) \right) <\varepsilon \end{aligned}$$

for some $A\succcurlyeq A_{0}$, which contradicts 1. Therefore, there is a subnet $( t_{A_{\lambda }}^{+}) _{\lambda \in \Lambda }$ such that $\lim _{\lambda }t_{A_{\lambda }}^{+}=\tau >0$.

Now, for $\eta >0$ with $\eta \le \tau$, there is $\lambda _{0}$ such that $t_{A_{\lambda }}^{+}>\tau -\eta$ for all $\lambda \geqslant \lambda _{0}$. Hence, $s_{A_{\lambda }}( \tau -\eta ) \in U$ for all $\lambda \geqslant \lambda _{0}$, and therefore $\omega ^{*}( q\cdot \mathbb P ( ( \tau -\eta ) u+t^{*}u+v) ) \nsubseteq \mathcal A ^{*}$, which implies $q\cdot \mathbb P ( ( \tau -\eta ) u+t^{*}u+v) \in \mathcal A$. Thus, $q\cdot \mathbb P ( tu+t^{*}u+v) \in \mathcal A$ for all $t\in [ 0,\tau ]$. This fact contradicts the assumption that $t^{*}$ is supreme with $q\cdot \mathbb P ( tu+v) \in \mathcal A$ for all $t\in [ 0,t^{*}]$. Therefore, $t^{*}$ is infinite.$\square$

It should be remembered that we take the admissible family $\mathcal O _{\Psi }( P\mathcal V )$ of the $\Psi$-adapted coverings in $P \mathcal V$, and denote by $\mathcal S _\mathcal{U _{\varepsilon },A}$ the local semigroup that induces the shadowing semigroup $End_{l}( P \mathcal V ) _\mathcal{U _{\varepsilon },A}$. The Proposition 5.2 implies the following one.

Proposition 5.3

Let $\mathcal A$ be an $\mathcal F$-attractor in $P\mathcal V$ and assume that $B$ is $\mathcal F$-chain transitive. Then, $P^{-1}( \mathcal A )$ and $P^{-1}( \mathcal A ^{*})$ are subbundles of $\mathcal V$.

Proof

Since $\mathcal A$ is compact, the set $P^{-1}( \mathcal A )$ is closed in $\mathcal V$. By Proposition 5.2, it follows that $P^{-1}( \mathcal A )$ intersects each fiber in a linear space. It remains to show that $\mathrm{dim}( P^{-1}( \mathcal A ) \cap \mathcal V _{b}) =\mathrm{dim}( P^{-1}( \mathcal A ) \cap \mathcal V _{b^{\prime }})$ for all $b,b^{\prime }\in B$. Consider $\pi ( q),\pi ( p) \in B$, $\mathcal U _{\varepsilon }\in \mathcal O _{\Psi }( P\mathcal V )$ and $A\in \mathcal F$. By Proposition 4.4, we have $\pi ( p) \in ( \mathcal S _\mathcal{U _{\varepsilon },A}) _{B}\pi ( q)$, which means there are $\phi \in \mathcal S _\mathcal{U _{\varepsilon },A}$ and $g\in \mathrm{Gl}( n,\mathbb R )$ such that $\phi ( q) =pg$. Suppose that $\mathrm{dim}( P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( q) }) =k$. Take a basis $\{ q\cdot u_{1},\ldots,q\cdot u_{k}\}$ of $P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( q) }$. For any $q\cdot u\in P^{-1}( \mathcal A ) \setminus Z$, we have

$$\begin{aligned} p\cdot g\mathbb P u=\phi \left( q\right) \cdot \mathbb P u\in \Omega \left( q\cdot \mathbb P u,\mathcal U _{\varepsilon },A\right) \cap p\cdot \mathbb P ^{n-1}\subset \Omega \left( \mathcal A,\mathcal U _{\varepsilon },A\right) \cap p\cdot \mathbb P ^{n-1}. \end{aligned}$$

Hence, $\{ p\cdot gu_{1},\ldots,p\cdot gu_{k}\}$ is a basis for a subspace in $P^{-1}( \Omega ( \mathcal A,\mathcal U _{\varepsilon },A) ) \cap \mathcal V _{\pi ( p) }$. Thus, for each open covering $\mathcal U _{\varepsilon }$ of $P\mathcal V$ and each $A\in \mathcal F$, there is a $k$-dimensional subspace $L_\mathcal{U _{\varepsilon },A}$ in $P^{-1}( \Omega ( \mathcal A,\mathcal U _{\varepsilon },A) ) \cap \mathcal V _{\pi ( p) }$. Let $\mathcal L ( \mathcal U _{\varepsilon },A)$ be the set of $k$-dimensional subspaces in $P^{-1}( \mathrm{cls}( \Omega ( \mathcal A,\mathcal U _{\varepsilon },A) ) ) \cap \mathcal V _{\pi ( p) }$. We can consider the set $\mathcal L ( \mathcal U _{\varepsilon },A)$ as a subset in the Grassmann manifold $G_{n,k}$ of $k$-dimensional subspaces in $\mathcal V _{\pi ( p) }$. For $\mathcal U _{\varepsilon _{1}}^{1},\ldots,\mathcal U _{\varepsilon _{n}}^{n}\in \mathcal O _{\Psi }( P\mathcal V )$ and $A_{1},\ldots, A_{n}\in \mathcal F$, take $A\subset A_{1}\cap \cdots \cap A_{n}$, $\mathcal U =\mathcal U ^{1}\wedge \cdots \wedge \mathcal U ^{n}$, and $\varepsilon =\min \{ \varepsilon _{1},\ldots,\varepsilon _{n}\}$. We have

$$\begin{aligned} \mathcal L \left( \mathcal U _{\varepsilon },A\right) \subset \mathcal L \left( \mathcal U _{\varepsilon _{1}}^{1},A_{1}\right) \cap \cdots \cap \mathcal L \left( \mathcal U _{\varepsilon _{n}}^{n},A_{n}\right). \end{aligned}$$

Since $G_{n,k}$ is compact, it follows that

$$\begin{aligned} \bigcap _\mathcal{U _{\varepsilon }\in \mathcal O _{\Psi }\left( P\mathcal V \right),A\in \mathcal F }\mathrm{cls}\left( \mathcal L \left( \mathcal U _{\varepsilon },A\right) \right) \ne \emptyset. \end{aligned}$$

This implies there are $k$ linearly independent vectors $p\cdot v_{1},\ldots, p\cdot v_{k}$ in $\underset{\mathcal{U }_{\varepsilon },A}{\bigcap } P^{-1}( \mathrm{cls}( \Omega ( \mathcal A,\mathcal U _{\varepsilon },A) ) ) \cap \mathcal V _{\pi ( p) }$. Hence, $p\cdot \mathbb P v_{1},\ldots, p\cdot \mathbb P v_{k}\in \Omega ( \mathcal A,\mathcal F )$. Since $\Omega ( \mathcal A,\mathcal F )$ is the intersection of the $\mathcal F$-attractors containing $\omega ( \mathcal A,\mathcal F )$, we have $p\cdot \mathbb P v_{1},\ldots,p\cdot \mathbb P v_{k}\in \mathcal A$. Hence, $P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( p) }$ is a linear space and contains $k$ linearly independent vectors, and therefore $\mathrm{dim}( P^{-1}( \mathcal A ) \cap \mathcal V _{\pi ( p) }) \ge k$. Since $q$ and $p$ are arbitrary and $B$ is $\mathcal F$-chain transitive, it follows that $P^{-1}( \mathcal A )$ is a subbundle of $\mathcal V$. Analogously, since $\mathcal F$-repellers are $\mathcal F$-attractor for the action of $S^{-1}$, we can show that $P^{-1}( \mathcal A ^{*})$ is a subbundle.$\square$

Now we are able to show that $\mathcal F$-Morse decompositions in $P \mathcal V$ form Whitney decompositions in $\mathcal V$.

Proposition 5.4

Assume that $B$ is $\mathcal F$-chain transitive. Let $\{ \mathcal C _{1},\ldots,\mathcal C _{l}\}$ be an $\mathcal F$-Morse decomposition in $P\mathcal V$. Then, $1\le l\le n=\mathrm{dim}\mathcal V _{b}$, $b\in B$, every $\mathcal F$-Morse set $\mathcal C _{i}$ defines a subbundle $P^{-1}( \mathcal C _{i})$ of $\mathcal V$, and the following decomposition into a Whitney sum holds:

$$\begin{aligned} \mathcal V =P^{-1}\left( \mathcal C _{1}\right) \oplus \cdots \oplus P^{-1}\left( \mathcal C _{l}\right). \end{aligned}$$

Proof

Take the sequence of $\mathcal F$-attractors in $P\mathcal V$

$$\begin{aligned} \emptyset =\mathcal A _{0}\subset \mathcal A _{1}\subset \cdots \subset \mathcal A _{l}=P\mathcal V \end{aligned}$$

such that $\mathcal C _{i}=\mathcal A _{i}\cap \mathcal A _{i-1}^{*}$, $i=1,\ldots, l$. From Proposition 5.3, it follows that $P^{-1}( \mathcal M _{i}) =P^{-1}( \mathcal A _{i}) \cap P^{-1}( \mathcal A _{i-1}^{*})$ is a subbundle of $\mathcal V$. Next, we show that the $\mathcal F$-Morse decomposition defines a Whitney decomposition of $\mathcal V$. This is immediate for $l=1$. Then, we assume that the assertion is true for all vector bundles and all $\mathcal F$-attractor sequences of length $l-1$, and prove it for $l$. Since $\mathcal C _{l}=\mathcal A _{l-1}^{*}$, we have $\{ \mathcal C _{1},\ldots,\mathcal C _{l-1}\} \subset \mathcal A _{l-1}$. Hence, $\{ \mathcal C _{1},\ldots,\mathcal C _{l-1}\}$ is an $\mathcal F$-Morse decomposition in $\mathcal A _{l-1}$. By induction assumption, we have the Whitney decomposition

$$\begin{aligned} P^{-1}\left( \mathcal A _{l-1}\right) =P^{-1}\left( \mathcal C _{1}\right) \oplus \cdots \oplus P^{-1}\left( \mathcal C _{l-1}\right). \end{aligned}$$

It remains to show that $P^{-1}( \mathcal A _{l-1})$ and $P^{-1}( \mathcal A _{l-1}^{*})$ form a Whitney decomposition of $\mathcal V$. Choose $\pi ( q) \in B$ and consider a subspace $L$ of $\mathcal V _{\pi ( q) }$ such that $\mathcal V _{\pi ( q) }=( P^{-1}( \mathcal A _{l-1}) \cap \mathcal V _{\pi ( q) }) \oplus L$. Then, we can take an attractor neighborhood $U$ of $\mathcal A _{l-1}$ such that $P( L) \cap U=\emptyset$. Hence, $\Omega ^{*}( P( L) ) \subset \mathcal A _{l-1}^{*}$. Suppose that $\mathrm{dim}( P^{-1}( \mathcal A _{l-1}) \cap \mathcal V _{\pi ( q) }) =k$, and therefore $\mathrm{dim}( L) =n-k$. Take a basis $\{ q\cdot u_{1},\ldots, q\cdot u_{n-k}\}$ of $L$. For a $\Psi$-adapted covering $\mathcal U _{\varepsilon }$ of $P\mathcal V$ and $A\in \mathcal F$, there are $\phi \in \mathcal S _\mathcal{U _{\varepsilon },A}$ and $g\in \mathrm{Gl}( n,\mathbb R )$ such that $\phi ( q) =qg$. Then, for any $q\cdot u\in L\setminus Z$, we have $q\cdot \mathbb P u=\phi ( q) \cdot g^{-1}\mathbb P u$, and therefore $q\cdot g^{-1} \mathbb P u\in \Omega ^{*}( P( L),\mathcal U _{\varepsilon },A) \cap q\cdot \mathbb P ^{n-1}$. This implies $\{ q\cdot g^{-1}u_{1},\ldots,q\cdot g^{-1}u_{n-k}\}$ is a basis for a subspace in $P^{-1}( \Omega ^{*}( P( L),\mathcal U _{\varepsilon },A) ) \cap \mathcal V _{\pi ( q) }$. Hence, $P^{-1}( \Omega ^{*}( P( L),\mathcal F ) ) \cap \mathcal V _{\pi ( q) }$ contains a subspace of dimension $n-k$, which implies that $\mathrm{dim}( P^{-1}( \mathcal A _{l-1}^{*}) \cap \mathcal V _{\pi ( q) }) \ge n-k$. Since $\mathcal A _{l-1}$ and $\mathcal A _{l-1}^{*}$ are disjoint, it follows that

$$\begin{aligned} \mathcal V _{\pi \left( q\right) }=P^{-1}\left( \mathcal A _{l-1}\right) \cap \mathcal V _{\pi \left( q\right) }\oplus P^{-1}\left( \mathcal A _{l-1}^{*}\right) \cap \mathcal V _{\pi \left( q\right) }. \end{aligned}$$

Therefore, $P^{-1}( \mathcal A _{l-1})$ and $P^{-1}( \mathcal A _{l-1}^{*})$ form a Whitney decomposition of $\mathcal V$.$\square$

Finally, we are able to demonstrate the Selgrade theorem for linear action in $\mathcal V$.

Theorem 5.1

(Selgrade) Let $( S,\mathcal V )$ be a linear action of $S$ on $\mathcal V$ and $\mathcal F$ a family of subsets of $S$ which is a filter basis and satisfies the translation hypothesis. Assume that the base space $B$ is $\mathcal F$-chain transitive. Then, there is a finest $\mathcal F$-Morse decomposition $\{ \mathcal C _{1},\ldots,\mathcal C _{l}\}$ in the projective bundle $P\mathcal V$, and $1\le l\le n=\mathrm{dim}\mathcal V _{b}$, $b\in B$. Every $\mathcal F$-Morse set $\mathcal C _{i}$ defines an invariant subbundle $P^{-1}( \mathcal C _{i})$ of $\mathcal V$ and the following decomposition into a Whitney sum holds:

$$\begin{aligned} \mathcal V =P^{-1}\left( \mathcal C _{1}\right) \oplus \cdots \oplus P^{-1}\left( \mathcal C _{l}\right). \end{aligned}$$

Proof

Since $P\mathcal V$ is a flag bundle, we apply Theorem 4.1 and Proposition 3.2 to conclude there is a finest $\mathcal F$-Morse decomposition $\{ \mathcal C _{1},\ldots,\mathcal C _{l}\}$ in $P \mathcal V$, where each $\mathcal F$-Morse set $\mathcal C _{i}$ is a maximal $\mathcal F$-chain transitive set. Since each $\mathcal C _{i}$ is compact and invariant, $P^{-1}( \mathcal M _{i})$ is closed and invariant in $\mathcal V$. The proof follows from Proposition 5.4. $\square$

Let us see an application of Theorem 5.1 to control systems.

Example 5.1

It is well-known that a control system $\Sigma$ on a compact $n$-dimensional manifold $M$ lifts to a control system $T\Sigma$ on the tangent bundle $TM$. From the Selgrade theorem, we can construct a Whitney decomposition of $TM$ into invariant subbundles corresponding to the maximal chain transitive sets of the induced system on the projective bundle $PTM$, when the system is chain transitive in $M$. We consider a control system

$$\begin{aligned} \dot{x}\left( t\right) =X_{0}\left( x\left( t\right) \right) +\underset{i=1}{ \overset{m}{\sum }}u_{i}\left( t\right) X_{i}\left( x\left( t\right) \right),\quad t\in \mathbb R, \end{aligned}$$

on a $d$-dimensional compact manifold $M$, with control range $U\subset \mathbb R ^{m}$ and set of admissible control functions $\mathcal U _{ \mathrm{cp}}=\{u=(u_{1},\ldots,u_{m}):\mathbb R \rightarrow U$, piecewise constant$\}$. Then, linearization along the trajectories yields a bilinear control system on the tangent bundle $TM:=\underset{p\in M}{\bigcup }T_{p}M$. The linearized system is described by

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}Tx\left( t\right) =TX_{0}\left( T x\left( t\right) \right) +\underset{i=1}{\overset{m}{\sum }}u_{i}\left( t\right) TX_{i}\left( Tx\left( t\right) \right),\quad t\in \mathbb R, \end{aligned}$$

where $u\in \mathcal U$ and, for a vector field $X_{i}$ on $M$, its linearization is denoted by $TX_{i}=( X_{i},DX_{i})$. For $v\in TM$ and $x=( x_{1},\ldots,x_{m}) \in \mathbb R ^{m}$, we define $TX( v,x) =TX_{0}( v) \,+\, \sum _{i=1}^{m} x_{i}TX_{i}( v)$. The linearized system is determined by the corresponding set of vector fields $F=\{ TX_{u}:\;u\in U\}$. The system semigroup for the linearized system is defined as

$$\begin{aligned} \mathcal S =\left\{ \exp t_{n}Y_{n}\exp t_{n-1}Y_{n-1},\ldots, \exp t_{0}Y_{0}:Y_{j}\in F,\overset{n}{\underset{j=0}{\sum }}t_{j}\ge 0,\quad t_{j}\ge 0,\quad n\in \mathbb N \right\}. \end{aligned}$$

For $T>0$, we define the set

$$\begin{aligned} U(T) =\left\{ \exp t_{n}Y_{n}\exp t_{n-1}Y_{n-1},\ldots, \exp t_{0}Y_{0}:Y_{j}\in F,\overset{n}{\underset{j=0}{\sum }}t_{j}\ge T,\quad n\in \mathbb N \right\}. \end{aligned}$$

It is immediate that the family $\mathcal F =\{ \mathrm{U}( T) :T\ge 0\}$ of subsets of $\mathcal S$ is a filter basis on them. Now we consider the vector bundle $\pi :TM\rightarrow M$ given by $\pi ( v) =p$ for $v\in T_{p}M$. Since the linearized control system on $TM$ induces naturally the control system on $M$, the $\mathcal F$-chain recurrence on $M$ covers the chain recurrence of the control system on $M$. Therefore, $M$ is $\mathcal F$-chain transitive if the control system on $M$ is chain transitive. By assuming that the control system on $M$ is chain transitive and the family $\mathcal F$ satisfies the translation hypothesis, there is a finest $\mathcal F$-Morse decomposition $\{ \mathcal C _{1},\ldots,\mathcal C _{l}\}$ in the projective bundle $PTM$, with $l\le d$. Every $\mathcal F$-Morse set $\mathcal C _{i}$ defines an invariant subbundle $P^{-1}( \mathcal C _{i})$ of $TM$ and the following decomposition into a Whitney sum holds:

$$\begin{aligned} TM=P^{-1}\left( \mathcal C _{1}\right) \oplus \cdots \oplus P^{-1}\left( \mathcal C _{l}\right). \end{aligned}$$