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DISCRETE VERSION OF LIAO'S CLOSING LEMMA AND THE $C^1$ STABILITY CONJECTURE : HAS THE $C^1$ STABILITY CONJECTURE BEEN SOLVED? (Complex Systems and Theory of Dynamical Systems)

Author(s) Ikeda, Hiroshi

Citation 数理解析研究所講究録 (2002), 1244: 17-23

Issue Date 2002-01

URL http://hdl.handle.net/2433/41667

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

DISCRETE VERSION OF LIAO’S CLOSING LEMMA AND

THE $C^{1}$ STABILITY CONJECTURE :HAS THE $C^{1}$ STABILITY CONJECTURE BEEN SOLVED ?

池田 宏 (HIROSHI IKEDA) 早大 理エ

ABSTRACT. R. Maii6 published aproof of the $C^{1}$ stability conjecture for diffeomor- phisms[5]. In the proof R. Mane used the discrete version of Liao’s Closing Lemma without proof. However, the author cannot be convinced of this version of Liao’s Closing Lemma. We consider length of $\gamma$-strings. We prove the discrete version of Liao’s Closing Lemma in consideration of length of $\gamma$-strings. In this paper we claim need of reconstruction of aproof of the $C^{1}$ stability and $\Omega$ stability conjecture for diffeomorphisms and flows.

1. INTRODUCTION

R. Maii6 published aproof of the $C^{1}$ stability conjecture for diffeomorphisms[5]. In [5] R. Mane used the discrete version of Liao’s Closing Lemma without proof. Liao’s Closing Lemma is akind of Shadowing Lemma to show existence of aperiodic orbit near agiven periodic pseud0-0rbit. Marie cited this lemma from [3]. However, in [3] the original flow version of the Closing Lemma is only applied to aproof of atheorem. The original version of the Closing Lemma is stated in [2] in Chinese. Moreover, aproof of Lemma 3.6 in [2] is incorrect. Thus, there exists acounter example. But the original flow version maybe holds by minor corrections or at least in similar setting to Mane’s diffeomorphism version. The author however cannot be convinced of Mane’s discrete version of Liao’s Closing Lemma, Lemma II.2[5]. Mane’s version has no bounds for length of $\gamma$-strings(that is, length of parts of agiven pseud0-0rbit). Mane’s discrete version is very powerful because there exist no bounds for length of $\gamma$-strings. However we need bound for length of $\gamma$-strings to guarantee shadowing property. We consider length of $\gamma$-strings to guarantee shadowing property. We prove the discrete version of Liao’s Closing Lemma in consideration of length of $\gamma$-strings. In the framework of the argument of $\mathrm{M}\mathrm{a}\tilde{\mathrm{n}}\acute{\mathrm{e}}[5]$ , we need not only the existence of aperiodic orbit but also the periodic orbit to shadow agiven periodic pseud0-0rbit. If Lemma II.2[5] does not hold, then Theorem 1.4 and Theorem II.1 in [5] collapse. If one would like to declare that the $C^{1}$ stability conjecture has been solved, one should show us clear and rigorous proof of Lemma II.2[5]. In this paper we claim need of reconstruction of aproof of the $C^{1}$ stability and $\Omega$-stability conjecture for diffeomorphisms $[5,6]$ and flows[l].

1991 Mathematics Subject Classification. Primary $58\mathrm{F}10$;Secondary $58\mathrm{F}15$ .

Typeset by $\mathrm{A}\lambda 4\theta \mathrm{I}\mathrm{k}\mathrm{X}$

数理解析研究所講究録 1244巻 2002年 17-23

17

In section 2we give definitions and precise statements of results. After we inves- tigate several information obtained from uniform 7-strings, we prove the discrete version of Liao’s Closing Lemma in consideration of length of 7-strings. Also we prove Lemma II(Pliss’s Lemma).

2. DISCRETE VERSION 0F LIAO’S cL0S1NG LEMMA

Let $M$ be aclosed manifold with dimension $m\geq 2$ and let Diff (Af), $r\geq 1$ , be the space of $C^{f}$ diffeomorphisms of $M$ endowed with the $C^{r}$ topology. Given acompact $f$-invariant subset Aof $f\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}^{r}(\mathrm{A}\mathrm{f})$ we say that asplitting $TM|\Lambda=E$ ce $F$ is a dominated splitting if it is acontinuous, $Df$-invariant and there exist aRiemannian norm $||\cdot||$ on $TM$ , and $C>0$ , $0

bound of length of $\mathrm{j}$-strings respectively. Moreover, Liao’s original flow version[2] has lower bound for length of $\ovalbox{\tt\small REJECT}$ -string.

From now on, we shall call above $\alpha$ connecting range, above $\beta$ shadowing range, and above $\hat{\gamma}$ contracting rate. The essence of our problem is not the number of $\gamma$-strings but the length of $\gamma$-strings consisting of aperiodic pseud0-0rbit. More precisely, the main problem is whether asufficiently long uniform $\gamma$-string can be decomposed into appropriate (uniform) $\gamma’$-strings with $\gamma\hat{\gamma}$ . Hence there exists asequence $0=m_{0}

Let $\hat{F}_{z,x}$ be the map with the diagonal block matrix

$(\begin{array}{ll}A_{z,x} OO D_{z,x}\end{array})$ .

In this setting we obtain two preliminary lemmas. Lemma 1. For all $\eta>0$ we find a constant $0

(Because $E$ is contracting and ($x_{j}$ , $f^{n_{j}}(xj)$ ) is $\hat{\gamma}$-string for $j=1$ , $\cdots$ , $k.$ ) Hence for some $0

In order to apply Proposition 7.7 [8], we must have

(e) $\hat{\lambda}+c^{2}(K\eta+\eta’)

Continuing in this fashion, we obtain asequence of numbers $\{\nu_{j}|1\leq j\leq s\}$ satisfying

(I) $f(\nu_{j})\geq f(\nu)$ for $\nu_{j}\leq\nu\leq n$ ; (II) $0\leq f(\nu j-1)-f(\nu_{j})