The Ergodic Closing Lemma for $C^1$ Regular Maps
Kazumine Moriyasu · Tokyo Journal of Mathematics · 1992
In order to solve the $C^{1}$ Structural Stability Conjecture by Palis and Smale, Mane [1] established the ergodic closing lemma for diffeomorphisms.In 1987 he gave in [2] an answer to the conjecture by using original ideas.The ergodic closing lemma is a result that captures the asymptotic behaviour of orbits in the ergodic theory.This result is based on Pugh's closing lemma [3].Our aim is to prove the ergodic closing lemma for regular maps ofclosed manifolds.Let $M$ be a closed $C^{\infty}$ manifold and $C^{1}(M)$ be the set of $C^{1}$ maps of $M$ with the $C^{1}$ -topology.$f\in C^{1}(M)$ is called to be regular if for every $x\in M$ the derivativeand Per $(f)$ denotes the set of periodic points.Recently, in [4] L. Wen showed the $C^{1}$ closing lemma for regular maps as follows.THEOREM (L.Wen).Let $f:M\rightarrow M$ be a regular map and $p$ be a nonwanderingThe idea of the proof is in Proposition 1 below.Let $f\in R^{1}(M)$ and $q_{0}\in M$ be a non periodic point of $f$ .We define an infinite sequence $Q=\{Q_{n} ; n\geq 0\}$ of disjoint non-empty finite sets $Q_{n}=\{f^{-n}(q_{0})\}$ for $n=0,1,2,$ $\cdots$ .Then $f:Q^{\prime}-Q_{0}\rightarrow Q^{\prime}$ , where ([4]).Under the above notations, for $\epsilon_{0}>0$ there are a number $\rho_{0}>2$ and an integer $\mu_{0}\geq 1$ such that for any finite ordered set $P=\{p_{0}, p_{1}, \cdots,p_{t}\}$ in $T_{qo}M$, there is $y\in P\cap U(p_{t}, \rho_{0}|p_{0}-p_{t}|)$ such that for any branch $\Sigma=\{q_{0}, q_{1}, \cdots, q_{n}, \cdots\}$ of $(Q, f)$ , there is $w\in P\cap U(p_{t}, \rho_{0}|p_{0}-p_{t}|)$ , where $w$ is before $y$ in the order of $P$ , together with $\mu_{0}+1$ points $c_{O},$ $c_{1},$ $\cdots,$ $c_{\mu 0}$ in $U(p_{t}, \rho_{0}|p_{0}-p_{t}|)$ , not necessarily distinct, satisfying the following two conditions (a) and $(b).$'