数理解析研究所講究録
1118 巻 1999 年 115-118

115

INVERSE LIMIT STABILITY FOR SEMIFLOWS

早稲田大理工池田宏 (HIROSHI IKEDA)

ABSTRACT. On semiflows on Banach manifolds, Quandt stated that some special. However, the concepts and proofs used
semiflows were inverse limit
to show inverse limit stability of semiflows are rough and ambiguous. We consider
semiflows on finite dimensional compact manifolds or on finite dimensional compact
non-singular branched manifolds. In this paper we announce that more special semiflows are inverse limit stable. That is, Anosov semiflows are inverse limit stable.
$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}[15,16]$

1. INTRODUCTION
For diffeomorphisms (resp. flows) of closed manifolds, the theory of structural
stability has been established. For a geometric study of evolution equations, Quandt$[15, 16]$ outlined a corresponding theory for endomorphisms (resp. semiflows) of
Banach manifolds (or Banach space). To extend the stability theory from diffeomorphisms (resp. flows) to endomorphisms (resp. semifiows) structural stability
was generalized to inverse limit. Inverse limit stability does notguarantee preservation of the topological dynamics. However inverse limit stability gives thecorrespondence between all global solutions for any two
semiflows near original one. On semiflows on Banach manifolds,
statedspecialthat some
semiflows were inverse limit stable. However, the concept and theproof used to show inverse limit stability of semiflows are rough and ambiguous. In
order to give rigorous proof we consider an appropriate setting in this paper. So
we consider semiflows on finite dimensional smooth manifolds without boundaryor on finite dimensional compact smooth non-singular branched manifolds withoutboundary. We call finite dimensional compact (resp. noncompact) smooth manifold
without boundary closed (resp. open) manifold. Non-singular branched manifolds
were introduced by R. F.
. By a non-singular branched n-dimensional
of
class
is
a
meant metrizable space $K$ together with:manifold(i) a collectionof closed subsets of $K$ ;a finite collection(ii) for eachof closed subsets of(iii) for each , a map
a closed -disk of class:;insubject to the following axioms:
and $\bigcup_{i}IntU_{i}=K$ ;(b)
is a homeomorphism onto;(c) there is a cocycle of diffeomorphismsof classsuch thatdomainiswhen defined. The
of.
To show the existence of Anosov semiflows which are not flows, we construct afinite dimensional compact smooth non-singular branched manifold without boundary. In this paper we announce:$\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{b}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}[10,13]$$\mathrm{o}\mathrm{n}\mathrm{e}- \mathrm{t}_{0}$$\mathrm{Q}\mathrm{u}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{t}[16]$$\mathrm{W}\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{i}\mathrm{a}\mathrm{m}\mathrm{S}[20]$$C^{k}$$\{U_{i}\}$$\{D_{ij}\}$$U_{i}$$i$$\pi_{i}$$U_{i}arrow D_{i}^{n},$$U_{i;}$$C^{k}$$n$$D_{i}^{n}$$R^{n}$$( \mathrm{a})\bigcup_{j}D_{ij}=U_{i}$$\pi_{i}|D_{ij}$$D_{i}^{n}$$\{\alpha_{i’i}\}$$\alpha_{i’i}$$C^{k}$$\pi_{i’}=\alpha_{i’i^{\circ\pi_{i}}}$$\pi_{i}(U_{i}\cap U_{i^{\prime)}}$Typeset by$A_{\mathcal{M}^{[}}\mathrm{a}\mathrm{I}\mathrm{p}\zeta$

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Theorem. Let $M$ be either amanifold or a finite dimensional compact
smooth non-ningular branched manifold without boundary. Then Anosov semiflows.
ininverse limit
on $M$ are$clo\mathit{8}ed$$C^{1}$$\mathit{8}table$$\mathrm{S}\mathrm{e}\mathrm{m}_{s}^{1}(M)$

2. PRELIMINARIES
Let $M$ be a finite dimensional smooth connected boundaryless (non-singularbranched) manifold with a Riemannian metric. Let be a metric on $M$ induced
by its Riemannian metric. A
semiflow $F(t)$ on $M,$ $r\geq 1$ , is a one parameter
maps $\{F(t) : Marrow M|t\geq 0\}$ such thatfamily of(1) $F(s+t)=F(s)F(t)$ for all $s,t\geq 0$ ,is the identity map of $M$ ,(2)(3) $F(\cdot)(X):[0, \infty)arrow M$ is continuous for every $x\in M$ .
$d$$C^{r}$$C^{r}$$F(\mathrm{O})$We shall sometimes use the notation $F(t, x)=F(t)(x)$ for $(t, x)\in[0, +\infty)\cross M$ .
semiflows on $M,$ $r\geq 0$ , with the metric ,Sem $(M)$ denotes the class of allwhere $d_{r}(F(t), G(t))= \sup\{d(F(t)(x), c(t)(X)),$ $|DiF(t)(X)-D^{i}c(t)(x)||$.
$t\in[0,1],$ $x\in M$ and $i\in\{1, \cdots , r\}\}$
for $F(t),$$C^{r}$$d_{r}$$G(t)\in \mathrm{s}_{\mathrm{e}\mathrm{m}^{r}}(M)$such that
A global solution of a semiflow $F(t)$ is a function$F(t)(v(\tau))=v(\tau+t)$ for all $t\geq 0$ and. (3) of the above definition implies
be the spacethat a global solution is a continuous function of to $M$ . Let$M$
the set of all global solutions ofof continuous functions from to . Let$F(t)$ . For each
for
on $S(F)$ by
we define a shifl
be a projection defined by
. Let
$v\in
C(\mathrm{R},M)$
for. Let $A(F)= \bigcap_{t\geq 0}F(t)[M]$ for any semiflow $F(t)$ . Note that
. We sometimes call $A(F)$ an attractor of $F(t)$ .
for every
A subset of $M$ is -invariant for $F(t)$ if $F(t)J\subset J$ for all $t\geq 0$ . Thenor denotes the set of all global solutions of $F(t)$ contained in . We will say thata subset of $M$ is
for $F(t)$ if $F(t)J=J$ for all $t\geq 0$ . We introduce a$M)$ ) by(more generally on
metric$v:\mathrm{R}arrow M$$\tau\in \mathrm{R}$$C(\mathrm{R}, M)$$\mathrm{R}$$\mathrm{R}$$\overline{S}(F)\underline{\mathrm{b}}\mathrm{e}$$\tau\in \mathrm{R},$$v\in\overline{S}(F)$$\overline{p}_{s}(v)=v(s)$$\overline{p}_{s}:C(\mathrm{R}, M)arrow M$$A(F)=\overline{p}_{s}(\overline{S}(F))$$J$$[\overline{F}(t)v](\mathcal{T})=v(\tau+t)$$\overline{F}(t)$$t\in \mathrm{R}$$s\in \mathrm{R}$$\overline{J}(F)$$\omega$$J$$\overline{J}$$J$$inva\dot{\mathcal{H}}ant$$\overline{d}\mathrm{o}\mathrm{n}\overline{S}(F)$$C(\mathrm{R},$$\overline{d}(v, w)=\sup_{t\in \mathrm{R}}e^{-|t|d}(v(t),w(t))$

for$v,w\in C(\mathrm{R}, M)$.

is endowed with a topology induced by the metric We shall say thaton $M$ is strong if $F(t)$ satisfies the following:aR.
is differentiable for(4) every global solution

Then$\overline{d.}$$\overline{S}(F)$$C^{r_{\mathrm{S}\mathrm{e}}}\mathrm{m}\mathrm{i}\mathrm{f}\mathrm{l}\mathrm{o}\mathrm{w}F(t)$$v(t)\in\overline{S}(F)$$t\in$

Let{$r\geq0$ , with the metric$M,$
semiflows on
with the metric . It is obvious that$\mathrm{S}\mathrm{e}\mathrm{m}_{\mathrm{s}}^{r}(M)=$is strong} be the class of all. Let
be the space of all$C^{r}$$F(t)\in \mathrm{S}\mathrm{e}\mathrm{m}^{r}(M)|F(t)$$d_{r}$$\mathcal{F}^{r}(M)$strongflows$C^{r}$.
if there exists a point
A global solution of $F(t)$ is called an equilibrium, and asuch that $v(t)=p$ for all
solution if there exists a constant$T>0$ such that $v(t+T)=v(t)$ for all
andfor $0<t<T$ . The
time $T>0$ is called the period of the solution. Thus we distinguish equilibrium
be the set of all points on periodic
solutions from periodic solutions. Let
is an equilibrium solution.} be the set of all
solutions of $F(t)$ . Let{.
fixed points for $F(t)$ . For a global solution of $F(t)$ define$d_{r}$$\mathcal{F}^{\Gamma}(M)\subset \mathrm{S}\mathrm{e}\mathrm{m}_{s}^{r}(M)\subset \mathrm{S}\mathrm{e}\mathrm{m}^{r}(M)$$\mathit{8}\mathit{0}\iota_{u}tion$

$v$

$p$

$t\in \mathrm{R}$$pe7\dot{\mathrm{v}}odic$$t\in \mathrm{R}$$v(t)\neq v(\mathrm{O})$$\mathrm{p}_{\mathrm{e}\mathrm{r}}(F)$$\mathrm{E}(F)=$$v(\mathrm{O})|v$$v$$\mathcal{O}(v)=\{v(t)|t\in \mathrm{R}\}$

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Let $F(t),$. We say that) is inverse limit semiconjugate to$F(t)$ if there exists a continuous surjection $H:S(F)arrow\overline{S}(G)$ which takes the global
solutions of $F(t)$ onto the global solutions of $G(t)$ and preserves the orientation intime; i.e. for each, there exists a nondecreasing automorphism
such that$c(t)\in \mathrm{S}\mathrm{e}\mathrm{m}^{r}(M)$$G\underline{(t}$$v\in\overline{S}(F)$$\beta_{v}:\mathrm{R}arrow \mathrm{R}$for all$[H\circ\overline{F}(t)](v)=[\overline{G}(\beta v(t))\circ H](v)$$t\in \mathrm{R}$.

Furthermore, if $H$ is injective then $F(t)$ and $G(t)$ are inverse limit conjugate. A
semiflow $F(t)$ islimit stable if there exists a neighborhood of $F(t)$ in$F(t)$ and $G(t)$ are inverse limit conjugate.
Sem $(M)$ such that for each
Similarly $F(t)$ isinverse limit stable inif there exists a neighborhoodof $F(t)$ in$F(t)$ and $G(t)$ are inverse limit
such that for eachconjugate. Equilibrium solutions and periodic solutions are preserved by inverse
limit conjugacy.
$C^{r}$$C^{r}inver\mathit{8}e$$\mathcal{U}$$G(t)\in \mathcal{U},$$C^{r}$$\mathcal{U}$$\mathrm{S}\mathrm{e}\mathrm{m}_{s}^{r}(M)$$G(t)\in \mathcal{U},$$\mathrm{S}\mathrm{e}\mathrm{m}_{s}^{r}(M)$Definition. Let $F(t)$ be asemiflow on $M,$ $r\geq 1$ . We willthat an -invariant
set is a hyperbolic set for $F(t)$ if every global solution $v\in J(F)$ is differentiable
for and there exist a Riemannian norm
on $M$ and constants $\mu>0$ suchthat for every
there exists a continuous splitting
into adirect sum, where$C^{r}$$\omega$$\mathrm{s}\mathrm{a}\underline{\mathrm{y}}$

$J$

$t$

$|\cdot|$$c,$$v\in\overline{J}(F)$$\mathrm{o}\mathrm{f}\cup^{\infty}t=-\infty^{\tau M}v(t)$$E^{s}\oplus E^{o}\oplus E^{u}$(1)$E^{i}= \bigcup_{t=-\infty}^{\infty}E_{v}^{i}(t)$for$i=s,$$\mathit{0},u$for$(T_{v}(t)F(\mathcal{T}))[E_{v}i](t)=E_{v}i)(t+\tau$(2)$|(T_{v(t)}F(\mathcal{T}))(w\mathfrak{j}|\leq ce^{-\mu\tau}|w|$$|(T_{v(t})F(\mathcal{T}))(w)|\geq c^{-1}e^{\mu\tau}|w|$

,

for

for

for each$E_{v(t)}^{O}= \mathrm{R}\cdot\frac{d}{dt}v(t)$$i=s,$$\mathit{0},$$w\in E_{v(t)}^{s},$$w\in E_{v(t)}^{u},$$u$,and every

and every$t\in \mathrm{R}$$t\in \mathrm{R}$$\tau\geq 0$

and every$t\in \mathrm{R}$,$\tau\geq 0$$\tau\geq 0$,.

If A is a hyperbolic set for $F(t)$ such that, then) $=j$ for all
we call theindex of A for $F(t)$ . For convenience we will say thatis hyperbolic for $F(t)$ if is hyperbolic for $F(t)$ .
For
we can define a tangent spaceofat . Let$\Gamma(v^{*}TM)$ be the space of continuous sections of the pullback
bundle of $TM$ by .
$\Gamma(v^{*}TM)$ is endowed with a norm
, whereis a Riemannian
norm on $M$ . Then we can consider that $T_{v}C(\mathrm{R}, M)=C(\mathrm{R},v^{*}TM)=\Gamma(v^{*}TM)$
is a Banach space. Thus we can take an open neighborhood $U^{*}(v)=\Gamma_{\epsilon}(v^{*\tau}M)=$$\{\xi\in\Gamma(v^{*}TM)|||\xi||<\epsilon\}\subset C(\mathrm{R},v^{*}\tau M)$ of zerosection of $v^{*}TM$ such that $U^{*}(v)$can be identified with a neighborhood $U(v)$ of in
by a homeomorphism$G:U^{*}(v)arrow U(v)$ defined by $G(\xi)(t)=\exp_{v(t)}\xi(t)$ for all. Let
be the space offunctions
to $M,$ $r\geq 1$ . If it is further assumed thatevery
, then we can argue the similarity to the above and obtainisthat $T_{v}C^{r}(\mathrm{R}, M)=C^{r}(\mathrm{R},v^{*}TM)=\Gamma^{r}(v^{*}TM)$.
$v\in\overline{\Lambda}(F)$$\dim E_{v(}^{s}0$$j$$\mathit{8}table$$v\in\overline{\Lambda}(F)$$\Lambda$$v\in\overline{S}(F)$$T_{v}C(\mathrm{R}, M)$$C(\mathrm{R}, M)$

$v$

$v$

$|| \xi||=\sup_{t\in \mathrm{R}}|\xi(t)|$$v$$|\cdot|$$C(\mathrm{R}, M)$$t\in \mathrm{R}$$C^{r}$$v\in\overline{S}(F)$$C^{r}(\mathrm{R}, M)$$\mathrm{h}\mathrm{o}\mathrm{m}\mathrm{R}$$C^{r}$

Definition. We say that asemiflow $F(t)$ is an Anosov
the following:(i) $A(F)$ is a compact hyperbolic set for $F(t)$ with stable index(ii) $A(F)$ contains no equilibrium solutions.
$C^{1}$$\mathit{8}emiflow$

if $F(t)$ satisfies;$j=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}>0$

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Intensionally we don’t refer to manifolds in the definition. Because we consider
general case in consideration of extension to infinite dimensional case. We shalldescribe the motivation of the above definition. First of all remember the definitionof Anosov flows. If the entire manifold $M$ possesses a hyperbolic structure thenthe flow is called an
flow. In the case of semiflows, it is appropriate that
hyperbolic structure (i.e. condition ). In the caseapossesses
only the attractorof Anosov flows on a compact manifold $M$ , hyperbolicity of $M$ implies that $M$ hasconstant stable index. In finite dimensional case of semiflows in this paper, we easily
have that $A(F)$ is equal to $M$ and has constant stable index. However, in infinite
dimensional case or special finite dimensional case (i.e. open manifolds) we cannot
guarantee that $A(F)=M$ . So constant stable index of $A(F)$ is not guaranteedby only hyperbolicity of $A(F)$ . If $A(F)\neq\subset M$ then there exists the possibilityof decomposition of $A(F)$ into disjoint hyperbolic sets which have different stableindex. Therefore, from intuition of original Anosov flows or diffeomorphisms it isnatural to require constant stable index of $A(F)$ and no equilibrium solutions.
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$Pe\dot{n}odi_{C}$$sem\ovalbox{\tt\small REJECT} ws$$\Omega$$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\text{\’{e}}$$\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\text{\’{e}}$$\gamma \mathrm{p}$$\Omega$$sem\ovalbox{\tt\small REJECT} wS$