Portal de Periódicos da CAPES

Homeomorphisms with the Pseudo Orbit Tracing Property of the Cantor Set

1983; Publication Committee for the Tokyo Journal of Mathematics; Volume: 6; Issue: 2 Linguagem: Inglês

10.3836/tjm/1270213870

0387-3870

Masahito Dateyama,

Stability and Controllability of Differential Equations

Let $X$ be a compact metric space with metric $d$ , and $f$ be a homeomorphism from $X$ onto itself.A sequence $\{x_{i}\}_{i=-\infty}^{\infty}$ is said to be a $\delta$ -pseudo- orbit of $f$ if $ d(fx_{i}, x_{+1})<\delta$ holds for all $i\in Z.$ (X, f ) is said to have the pseudo orbit tracing property (abbrev.P.0.T.P.) if for every $\epsilon>0$ there is $\delta>0$ such that, for every $\delta$ -pseudo-orbit $\{x_{i}\}_{i=-\infty}^{\infty}\subset X$ , there exists an $x\in X$ such that $ d(fx, x_{i})<\epsilon$ for all $i\in Z$ .Let $C\subset[0,1]$ be the Cantor set: i.e. $C$ is the set of the numbers $xe[0,1]$ with $ x=3^{-1}a_{1}+3^{-z}a_{z}+\cdots$ ( $a_{i}=0$ or 2 for $i\geqq 1$ ).We denote by $\ovalbox{\tt\small REJECT}(C)$ the set of all homeomorphisms on $C$ , and by $\ovalbox{\tt\small REJECT}(C)$ the set of all homeomorphisms with the P.0.T.P.. Define the metric $\overline{d}$ on $\ovalbox{\tt\small REJECT}(C)$ by $\overline{d}(f, g)=\max_{x}$ , $cd(fx, gx),$ $f,$ $g\in\ovalbox{\tt\small REJECT}(C)$ .Then $\mathscr{G}(C)$ is a Banach space.In this paper we prove:THEOREM.$\ovalbox{\tt\small REJECT}(C)$ is dense in $\ovalbox{\tt\small REJECT}(C)$ .For $r\geqq 1$ , we call the set $C\cap[3^{-r}i, 3^{-r}(i+1)](0\leqq i\leqq 3'-1)$ a Cantor subinterval with rank $r$ if $ C\cap(3^{-r}i, 3^{-r}(i+1))\neq\emptyset$ .We denote by $I(i, r)$ , the i-th Cantor subinterval with the rank $r$ from the left.Clearly $C=\bigcup_{i=1}^{2r}I(i, \gamma)$ and $I(i, r)=I(2i-1, r+1)\cup I(2i, r+1)$ .We call $g\in\ovalbox{\tt\small REJECT}(C)$ a generalized permutation if there exists $r\geqq 1$ such that the following i) and ii) hold:i) For every $1\leqq i\leqq 2^{f}$ , there exist $s=s(i)\geqq 1$ and $1\leqq j=j(i)\leqq 2^{8}$ such that $g(I(i, r))=I(j, s)$ , and ii) For every $1\leqq i\leqq 2^{r}$ , there exists $k=k(i)\in R$ such that $g(x)=$ $3^{r-\cdot(i)}x+k,$ $x\in I(i, r)$ .Denote by $\mathcal{G}$ the set of all generalized permutations.Then $\mathcal{G}$ is dense in $\ovalbox{\tt\small REJECT}(C)$ .In fact, take $f\in\ovalbox{\tt\small REJECT}(C)$ and $r\geqq 1$ .Choose $s\geqq 1$ such that $d(x, y)<3^{-}$ implies $d(fx, fy)<3^{-r}$ .Then for every $1\leqq i\leqq 2$ there exists