\documentstyle[fullpage,amsfonts,12pt]{article} %\def\N{{{I\negthinspace\!N}}} \newcommand{\N}{{\Bbb N}} \def\SBIMSMark#1#2#3{ % \font\SBF=cmss10 at 10 true pt \font\SBI=cmssi10 at 10 true pt \setbox0=\hbox{\SBF Stony Brook IMS Preprint \##1} \setbox2=\hbox to \wd0{\hfil \SBI #2} \setbox4=\hbox to \wd0{\hfil \SBI #3} \setbox6=\hbox to \wd0{\hss \vbox{\hsize=\wd0 \parskip=0pt \baselineskip=10 true pt \copy0 \break% \copy2 \break% \copy4 \break}} \dimen0=\ht6 \advance\dimen0 by \vsize \advance\dimen0 by 8 true pt \advance\dimen0 by -\pagetotal \dimen2=\hsize \advance\dimen2 by .25 true in % % Check for publication info % % \newread\jref % \openin2=publishd.tex % \ifeof2\setbox0=\hbox to 0pt{} % \else \setbox0=\hbox to 3.1 true in{ \vbox to \ht6{\hsize=3 true in \parskip=0pt \noindent {\it Fund. Math.}~{\bf 155} (1998), 237--249. \vfill}} % \fi % \closein2 \ht0=0pt \dp0=0pt % % \ht6=0pt \dp6=0pt \setbox8=\vbox to \dimen0{\vfill \hbox to \dimen2{\copy0 \hss \copy6}} \ht8=0pt \dp8=0pt \wd8=0pt \copy8 \message{*** Stony Brook IMS Preprint #1, #2 ***} } \begin{document} \title{Period Doubling, Entropy, and Renormalization} \author{Jun Hu$^{1}$ and Charles Tresser$^{2}$} \addtocounter{footnote}{1}\footnotetext {Department of Mathematics, Graduate Center of CUNY, 33W 42nd Str., New York, NY 10036} \addtocounter{footnote}{1}\footnotetext{IBM, Po Box 218, Yorktown Heights, NY 10598} \date{\relax} \maketitle \SBIMSMark{1995/13}{October 1995}{} \thispagestyle{empty} \begin{abstract} We show that in any family of stunted sawtooth maps, the set of maps whose set of periods is the set of all powers of $2$ has no interior point, {\em i.e.}, the combinatorial description of the boundary of chaos coincides with the topological description. We also show that, under mild assumptions, smooth multimodal maps whose set of periods is the set of all powers of $2$ are infinitely renormalizable. \end{abstract} \null % \newpage \noindent {\bf 1 Introduction} \noindent The present work is motivated by the following folklore conjecture (see also [OT]): \begin{description}\item[Conjecture A.] A real polynomial map of $f$ with set of periods (of its periodic orbits) \[P(f)=\{1, 2, 4, \cdots, 2^{i}, \cdots\}=\{2^{n}: n\in \N\} \] can be approximated by polynomials maps with positive entropy and by polynomials maps with finitely many periodic orbits. \end{description} \noindent This conjecture is now established for quadratic polynomials (as a consequence of [Su] or [La]) and work is in progress toward generalization for higher degree polynomials [Hu]. The interest in such a conjecture comes from Theorems A and B below (see section 2.1) and the fact that {\em topological entropy} (conceived as an invariant of topological conjugacy [AKM]) is also one way to measure the complexity of the dynamics of a map (see section 2.1): one is trying to describe how maps with simple dynamics can be deformed to maps with complicated dynamics, or, as one says, chaotic maps. Tradition, as well as the availability in this framework of a greater set of techniques, have put some emphasis on the particular case of polynomial maps, as in Conjecture A. However, the problem of the transition to chaos is more generally interesting in the category of smooth maps, in particular smooth endomorphisms of the interval, for which we recall the following: \begin{description} \item[Conjecture B.] All endomorphisms of the interval $f \in C^{k}(I),\, k\geq 1$, with $P(f)=\{2^{n}: n\in \N\}$ are on the boundary of chaos and on the boundary of the interior of the set of zero entropy. \end{description} \noindent We first show that any {\em stunted sawtooth map} (see section 2.2), whose set of periods is the set of all powers of $2$, can be approximated by stunted sawtooth maps with positive entropy and by stunted sawtooth maps with only finitely many periodic orbits (see section 2.3). This result solves the symbolic dynamic version of the above conjectures in the sense that stunted sawtooth maps carry all possible {\em kneading data} (see section 2.2) of multimodal maps. \noindent We also make a second step toward Conjecture A by proving that maps with $P(f)$ as above, which satisfy some smoothness conditions (and in particular polynomial maps), are {\em infinitely renormalizable} (see section 2.4). \noindent Sections 3 and 4 contain proofs of the results formulated in section 2. \noindent {\em Acknowledgements.} Both authors would like to thank Dennis Sullivan for his constant interest and encouragements. The first author is grateful to the IMS at Stony Brook for support and hospitality during his visit during the Spring of 1995. The second author also benefited from regular visits to this institute. \vspace{.3in} \noindent {\bf 2 Preliminary definitions and results} \bigskip \noindent {\bf 2.1 Topological entropy of one dimensional maps} \noindent A point $x$ is a $periodic$ $point$ of period $n$ of a map $f$ if $f^{i}(x)\neq x, 00$, $n\in Z_{+}$, we say a subset $S\subset X$ is {\em $(n,\epsilon )$-separated} if \[x,y\in S, x\neq y\Rightarrow \exists m:0\leq m\epsilon .\] Set $H(f,n,\epsilon )$ to be the maximal cardinality of $(n,\epsilon )$-separated sets, then \[h(f)=\lim _{\epsilon \rightarrow 0}\limsup _{n\rightarrow \infty } \frac{1}{n}\log H(f,n,\epsilon ).\] \noindent For maps on an interval, the following result gives a necessary and sufficient condition for the positivity of topological entropy. \begin{description} \item[Theorem A.] ([BF], [Ml]) A continuous map of an interval to itself it has positive topological entropy if and only if it has a periodic point whose period is not a power of $2$. \end{description} \noindent {\bf Remark:} The ``if" part is from [BF], the ``only if" part from [Ml]. \bigskip \noindent From Theorem A and [BlH], one gets the following. \begin{description} \item[Theorem B.] In the space $C^{k}(I), k\geq 1$, of $C^{k}$ endomorphisms of an interval $I$, if a map $f$ is on the boundary of positive topological entropy then the set $P(f)$ of its periods is $\{2^{n}:n\in \N\}$. The same is true for $f$ on the boundary of the interior of the set of maps with zero topological entropy. \end{description} \noindent {\bf Remark:} Conjectures A and B are about the converse of Theorem B. \bigskip \noindent We next give another necessary and sufficient condition for the positivity of topological entropy, which will be an important tool for us. This requires some more terminology. \noindent So let $f\in C^{0}(I)$ and $p$ be a fixed point of $f$. A point $x$ of $I$ belongs to the {\em unstable manifold} $W^{u}(p,f)$ of $p$ if, for every neighborhood of $V$ of $p$, $x\in f^{n}(V)$ for some positive integer $n$. It is easy to check that $W^{u}(p,f)$ is connected and invariant under $f$. A point $x\in I$ is a $homoclinic$ $point$ of $f$ if there is a periodic point $p$ of $f$ of period $n$ such that $x\neq p$, $x\in W^{u}(p,f^{n})$ and $f^{mn}(x)=p$ for some $m\in \N$ [Bl]. \noindent \begin{description} \item[Theorem C]([Bl]) A map $f\in C^{0}(I)$ has positive topological entropy if and only if it has a homoclinic point. \end{description} \noindent {\bf 2.2 Multimodal and stunted sawtooth maps} \noindent Consider the continuous map $f:I\to I$ where $I=[c_0,c_{d+1}]$. For $d\geq 0$, assume there are points $c_i$, $0< i < d+1$ with $c_j< c_{j+1}$ for $0\leq j\leq {d}$ such that $f$ is monotone on each {\em lap} $[c_j,c_{j+1}]$, and not monotone on any segment of the form $[c_j,c_{j+2}]$. Such a map is then called {\em d-modal} or {\em multimodal} with {\em modality} $d$ (one says {\em amodal} if $d=0$, {\em unimodal} if $d=1$, and then {\em bimodal}, and so on). The maximal interval $[a_i,b_i]$ containing $c_i$ on which $f$ is constant is called a {\em turning interval} and, more precilsely, a {\em plateau} if $a_i0$ and $1\leq i,j\leq d$. The $i$-ordered collection of signs with $j$ fixed is the $j^{\rm th}$ {\em kneading sequence} of $f$, and the $j$-ordered collection of kneading sequences is the {\em kneading invariant} of $f$ (for more on kneading theory, we refer to [MiT] and [BORT]). One might wish to first understand at the symbolic level some questions one formulates for polynomials or smooth maps. It is in fact more practical to consider continuous families of $d$-modal maps rich enough to exhibit all possible kneading data for $d$-modal maps, yet significantly easier to study than smooth maps. Such families exist, and we next recall the construction of one of them. \noindent By the {\em sawtooth map} of shape $s_{1}s_{2}\cdots s_{d+1}, s_{i}\in \{+,-\}, s_{i+1}=-s_{i}$, $d\geq1$, we mean the unique map $S_{d}:I\rightarrow I$ which is piecewise linear with slope $s_{1}(d+1), s_{2}(d+1), \cdots s_{d+1}(d+1)$ ($d+1$ alternate values). This is a $d$-modal map with topological entropy $\log (d+1)$, the largest possible value for $d$-modal maps. \noindent Given any {\em critical value vector} $w=(w_{1}, w_{2}, \cdots , w_{d})$ satisfying $$(w_{j}-w_{j+1})\cdot s_{(j+1)}<0,\,w_{j}\in I,\, j=1, 2, \cdots ,d\,,$$ we obtain the {\em stunted sawtooth map} $S_w$ from $S_{d}$ by cutting off the tops and bottoms of the graph at heights $w_{j}, j=1,2,\cdots ,d$. The $d$-parameter family of stunted sawtooth maps $S_w$ is complete in the following sense [MiT, DGMT]. \begin{description}\item[Theorem D.] To any $d$-modal map $f$ there is a canonical $d$-modal stunted sawtooth map $S_w$ which has exactly the same kneading data as $f$. \end{description} \noindent {\bf 2.3 The first main result} \begin{description}\item[Theorem 1.] Suppose $S_{w}$ is a stunted sawtooth map with \[P(S_{w})=\{2^{i}:i\in \N\}.\] Then for any $\epsilon >0$, there exist $w^{'}$ and $w^{''}$ such that \[|w^{'}-w|<\epsilon ,\;\;\;\;\;|w^{''}-w|<\epsilon ,\] $h(S_{w^{'}})>0$ and $S_{w^{''}}$ has only finitely many periods. \end{description} \begin{description}\item[Corollary 1.] In the parameter space, the set $\{w: P(S_{w})=\{2^{n}: n\in \N\}\}$ has no interior point, {\em i.e.}, the combinatorial and the topological descriptions of the boundary of chaos coincide. \end{description} \noindent {\bf Remark:} Let $f_i$, $i\in \{1,2,3\}$ be three $d$-modal maps of same shape with kneading invariants $$K_i=\{K_{i,1},K_{i,2},\dots ,K_{i,d}\}\,.$$ Assume that $P(f_2)=\{2^{n}: n\in \N\}$ and that, with the usual order on kneading sequences (see, {\em e.g.}, [BORT]) $$K_{1,j}< K_{2,j} < K_{3,j}\,\,{\rm if}\,\,s_j=+\,,$$ $$K_{1,j}> K_{2,j} > K_{3,j}\,\,{\rm if}\,\,s_j=-\,,$$ for $1\leq j\leq d$. It then also follows from Theorem 1 that $f_1$ has only finitely many periodic orbits and that $f_3$ has positive topological entropy. \bigskip \noindent {\bf 2.4 Renormalization} \noindent Let $I$ be an interval. A map $f:I\rightarrow I$ is called $renormalizable$ if there exists a proper subinterval $J$ of $I$ and an integer $p$ such that (1) $f^{i}(J), i=0,1,\cdots ,p-1$, have no pairwise interior intersection, (2) $f^{p}(J)\subset J$. \noindent Then $f^{p}|_{J}:J\rightarrow J$ is called a $renormalization$ of $f$. A map $f: I\rightarrow I$ is {\em infinitely renormalizable} if there exist an infinite sequence $\{I_{n}\}_{n=1}^{\infty }$ of nested intervals and an infinite sequence $\{u(n)\}_{n=1}^{\infty }$ of integers such that $f^{u(n)}|_{I_{n}}: I_{n}\rightarrow I_{n}$ are renormalizations of $f$ and the length of $I_{n}$ tends to zero as $n\rightarrow \infty $. \noindent Another purpose of this paper is to prove that maps $f$ with $P(f)=\{2^{i}:i\in \N\}$, which satisfy some smooth conditions, are infinitely renormalizable. To this end, we shall prove the following abstract result for which we first recall some definitions. \noindent Let $f\in C^{0}(I)$. A periodic point $x$ of period $n$ is {\em attracting } (resp. {\em one-sided attracting }) if there exists a neighborhood (resp. a one-sided neighborhood) $U$ of $x$ such that for any $y\in U$, $f^{nl}(x)\rightarrow x$ as $l\rightarrow \infty $. In both case we say the orbit of $x$ is a $periodic$ $attractor$. An open interval $J\subset I$ is called a {\em wandering interval} of $f$ if 1) $f^{n}(J)\cap f^{m}(J) =\emptyset$ for any $n\neq m, n,m\in \N$ and 2) $f^{n}(J)$ does not converge to a periodic orbit. \begin{description}\item[Theorem 2] Assume the multimodal map $f: I\rightarrow I$ with $P(f)=\{2^{n}: n\in \N\}$ has no wandering intervals, no plateaus, and no more than finitely many periodic attractors. Then $f$ is infinitely renormalizable. \end{description} \noindent Let $f: I\rightarrow I$ be a map which is not constant on any open set. We say that $f$ belongs to $\Gamma (2)$ if a) $f$ is $C^{2}$ away from the turning points; b) For every $x_{0} \in T_{f} $, there exists $\alpha >1 $, a neighborhood $U(x_{0}) \; of \; x_{0} $ and a $C^{2}$-diffeomorphism $\phi :U(x_{0}) \rightarrow (-1,1) $ such that $\phi (x_{0})=0$ and \[f(x)=f(x_{0})\pm |\phi (x)|^{\alpha } ,\;\; \forall x \in U(x_{0}) .\] \begin{description}\item[Theorem E.]$([Ma], [MMS])$ (1) If $f\in \Gamma (2)$ then $f$ has no wandering intervals. (2) Any $f\in \Gamma (2)$ has at most finitely many periodic attractors. \end {description} \noindent Combining this result with Theorem 2 yields: \begin{description}\item[Theorem 3.] Any $d$-modal map $f\in \Gamma (2)$, with $P(f)=\{2^{n}: n\in Z_{+}\}$ is infinitely renormalizable. \end{description} \noindent In particular, we have also: \begin{description}\item[Theorem 4] Any real polynomial map $f$ with $P(f)=\{2^{n}:n\in \N\}$ is infinitely renormalizable. \end{description} \bigskip \noindent {\bf Remark:} Using results from [HS], the smoothness condition in (1) of Theorem E can be relaxed, which allows a proof of Theorem 3 with relaxed smoothness condition for unimodal maps. \bigskip \noindent Using more language from kneading theory, one could formulate a conjecture corresponding to Theorems 2 to 4 for general renormalizations (not just those at the boundary of chaos). Such a generalization completely escape the methods of the present paper. \bigskip \noindent {\bf 3 Proof of Theorem 1} \begin{description}\item[Lemma 1.] Let $f: I\rightarrow I$ be a multimodal map with $P(f)=\{2^{i}:i\in Z_{+}\}$, and let $\Omega (f)$ be the set of accumulation points of the periodic points of $f$. Then no point in $\Omega (f)$ is periodic, so that $\Omega (f)$ is not a finite set. \end{description} {\bf Proof:} Let $p\in \Omega (f)$ be a periodic point of period $2^{n}$. Denote $g=f^{2^{n}}$. Then $p$ is a fixed point of $g$. Look at the map $g$ near $p$. Because $f$ hence $g$ has isolated turning intervals, there are only three types of local behaviors for $g$. \noindent 1. $g$ is monotone in a small neighborhood of $p$ (if $g$ is monotone reversing then $g^{2}$ is monotone preserving in a small neighborhood of $p$). \noindent 2. $p$ is in the interior or at the end of one of the plateaus of $g$. \noindent 3. $p$ is a turning point of $g$. \noindent Look at three iterates of $g$ or $g^{2}$ near $p$, one can easily see that neither $g$ nor $g^{2}$ has any periodic points with higher period in a small neighborhood of $p$. This contradicts that $p$ be an accumulation point of period-doubling periodic points of $f$. $\Box $ \bigskip \noindent {\bf Remark:} Lemma 1 is false for continuous maps: examples with $\Omega (f)$ reduced to a point are easily provided. \bigskip \noindent Let $\sum=\{0,1\}^{\N}$ and let $\sigma$ stand for the {\em adding machine}, {\em i.e.}, the map $\sigma :\sum \rightarrow \sum $ defined by $\sigma (x_i)_0^\infty = (y_i)_0^\infty $, where $y_i=1-x_i$ if $x_j=1$ for all $j0$ stand for the distance $d(\Omega (S_{w}), \Sigma(S_{w}))$. Let $K\subset \Omega (S_{w})$ be as in Theorem F. Clearly $d(K, \Sigma(S_{w}))>\epsilon$. Suppose that $x\in K$ is recurrent. There exists $l=2^{k}, k\geq 1$, (by Theorem F) such that $|f^{l}(x)-x|<\frac{1}{8}\epsilon $. Again by Theorem F, one can assume that $f^{l}(x)>x$. Let $V$ be the largest neighborhood of $x$ on which $f^{l}$ is monotone. The slope of $f^{l}$ on $V$ is $d^{l}$ and clearly $V\supset (x-\epsilon /d^{l}, x+\epsilon /d^{l})$. \noindent Assume first that $f^{l}$ is orientation preserving on $U$. Since $f^{l}(x)>x$ and $f^{l}(x-\frac{7}{8}\epsilon /d^{l})=f^{l}(x)-\frac{7}{8}\epsilon 0$, we push the concave plateaus up a little bit and the convex plateaus down a little bit, if one of their end points belongs to $\Lambda $, to get another stunted sawtooth map $S_{w^{'}}$ with $|w-w^{'}|<\epsilon $. If a periodic orbit of $S_{w}$ has no point in the interior of any plateau of $S_{w}$ then it is also a periodic orbit of $S_{w^{'}}$. Thus $S_{w}\subset S_{w^{'}}$so that the arguments in the proof of Lemma 2 show that $S_{w^{'}}$ has a homoclinic point. Thus $S_{w^{'}}$ has positive topological entropy. \noindent We next push all concave plateaus down a little bit and all convex plateaus up a little bit to get another stunted sawtooth map $S_{w^{''}}$ with $|w-w^{''}|<\epsilon $. Suppose that $P(S_{w^{''}})=\{2^{n}: n\in \N\}$. Then by the proof of Lemma 2, $S_{w}$ has positive topological entropy, a contradiction. $\Box $ \vspace{.2in} \goodbreak \noindent {\bf 4 Proof of Theorem 2} \begin{description}\item[Lemma 3.] Suppose $f\in C^{0}(I)$ has no attracting cycles and $P(f)=\{2^{n}: n\in \N\}$. Let $K$ be as in Theorem F, and $K=K_{0}\cup K_{1}$ with $f(K_{i})=K_{1-i}, i=0, 1$. Assume that $\sup \{x: x\in K_{0}\}<\inf \{x: x\in K_{1}\}$. Let $[K_{i}]$ denote the smallest closed interval containing $K_{i}, i=0, 1$. Then for $i=0, 1$ there exists a periodic point $q$ of $f$ with period $n=1$ or $2$ in the gap between $[K_{0}]$ and $[K_{1}]$ such that $U(f^{2}, q)$ contains $[K_{i}]$. \end{description} \noindent {\bf Proof:} By Theorem F, there exists a fixed point $p$ of $f$ in the gap between $[K_{0}]$ and $[K_{1}]$. For $i=1$, let $s=\inf \{x: x\in K_{1}\}$ and consider the map $f^{2}$ on the interval $[p, s]$. $p$ is fixed by $f^{2}$. Let $q$ be the largest fixed point of $f^{2}$ in $[p, s]$. Because of Lemma 1, $q\neq s$ and then $f^{2}(y)>y$ for any $y\in (q, s)$. Clearly $f^{2}(K_{1})\subset K_{1}$ and $f^{2}(s)>s$. Therefore $s$, hence $[K_{1}]$by connectivity, is in $U(f^{2}, q)$.The case $i=0$ is treated similarly. $\Box $. \bigskip \noindent From Theorem F and Lemma 3, one has \begin{description}\item[Corollary 2.] Suppose $f\in C^{0}(I)$ has no attracting cycles and $P(f)=\{2^{n}: n\in \N\}$. Let $K$ be as in Theorem F, with $K=\bigcup _{i=1}^{2^{n}}K_{i}^{(n)}$, $n\in \N$. Then for any $n\in \N$ there exists a periodic point $p$ of period $m$, where $m=2^{n}$ or $2^{n+1}$, whose orbit is contained in the set $\bigcup _{i=1}^{2^{n}}[K_{i}^{(n)}]\setminus \bigcup _{i=1}^{2^{n+1}}[K_{i}^{(n+1)}]$ and such that $U(f^{m}, p)$ contains some $K_{i}^{(n+1)}$, where $1\leq i\leq 2^{n+1}$. \end{description} \begin{description}\item[Lemma 4] Assume a continuous map $f: I\rightarrow I$ with $P(f)=\{2^{n}: n\in Z_{+}\}$ has no wandering interval, no plateau, only finitely many turning points and finitely many periodic attractors. Let $K$ be as in Theorem F. Then the semi-conjugacy $h$ in Theorem F is actually a conjugacy. \end{description} {\bf Proof:} Suppose that $h$ is not a conjugacy. Then by Theorem F there exists a point $s\in \sum $ such that $h^{-1}(s)=\{x, y\}$, $x\neq y$. We claim that $h^{-1}(\sigma ^{n}(s))$ eventually contains a single point when $n$ is large enough. Let $I_{n}$ denote the supporting interval of $ h^{-1}(\sigma ^{n}(s))$, $n\geq 0$. Since there are only finitely many turning points and $I_{n}$, $n\geq 0$, are pairwise disjoint, there exists $m>0$ such that $I_{n}$ contains no turning points for any $n\geq m$. If the claim is false, then $I_{m}$ is a wandering interval, a contradiction. Therefore one can assume that $f(x)=f(y)$, where $x, y\in K$ and $x\neq y$. We separate our considerations into three cases. Case 1. Suppose that $f$ is monotone preserving in some neighborhoods of $x$ and $y$ (the proof is similar when $f$ is monotone reversing near $x$ and $y$). By Corollary 2, there exists a periodic point $p$ of $f$ of period $n$, $n=2^{k}$, $k>0$, such that the unstable manifold $U(f^{n}, p)$ of $f^{n}$ at $p$ contains the interval $[x, y]$, $p$ is not in $[x, y]$ and $p$ is close enough to $x$ or $y$. By continuity of $f$, when $p$ is close enough to $x$ (or $q$), $f(p)$ is very close to $f(x)=f(y)$. By the intermediate value theorem, there are at least two points $a, b\in (x, y)$ such that $f(a)=f(b)=f(p)$. Then $f^{n}(a)=f^{n}(p)=p$. Hence $a$ is a homoclinic point, a contradiction. Case 2. Suppose that $f$ is monotone preserving in a neighborhood of $x$ and monotone reserving in a neighborhood of $y$ (the proof is similar when the situation is reversed). Denote $x_{t}=f^{t}(x)=f^{t}(y)$, $t\in \N$. Let $\{c_{i}\}_{i\in \alpha }$ denote the set of turning points of $f$, where $\alpha $ is a finite set. Clearly there exists $t_{0}>0$ such that $(x_{t}, c_{i})\bigcap K\neq \emptyset $ for any $t\geq t_{0}$, denoted by $K_{(t, i)}$. In the interval $(x, y)$ we select $v$ and $w$ near $x$ and $y$ respectively with $f(v)=f(w)$. Because of the nonexistence of wandering intervals, the itineraries of $f(p)=f(q)$ and $f(v)=f(w)$ under $f$ are eventually different. Let $v_{t}=f^{t}(v)$, $t\in \N$. Then there exists $t>t_{0}$ such that there is at least one $c_{i}\in (x_{t}, v_{t})$. Clearly $K_{(t, i)}\subset (x_{t}, v_{t})$ and $f^{t}((x, v))=f^{t}((w, y))\supset (x_{t}, v_{t})$. From Theorem F and Corollary 2, there exists a periodic point $p$ of period $n=2^{k}$, $k>0$, where $p$ is not in $[x, y]$, such that $p$ is very near $x$ (or $y$), $f^{j}(p)$ is in the supporting interval $[K_{(t, i)}]$ and the unstable manifold $U(f^{n}, p)\supset [x, y]$. Clearly there exists a point $u\in (x, v)$ (or $u\in (w, y)$) such that $f^{t}(u)=f^{j}(p)$. Hence $u\in U(f^{n}, p)$, $u\neq p$ and $f^{nt}(u)=f^{n}(f^{j}(p))=p$. So $u$ is a homoclinic point, a contradiction. Case 3. If at least one of $x$ and $y$ is a turning point, the proof is just a slight modification of that of Case 1 or Case 2. $\Box $ \bigskip \noindent {\bf Proof of Theorem 2:} Let $K$ be as in Theorem F. By Lemma 4, $f: K\rightarrow K$ is conjugate to $\sigma : \sum \rightarrow \sum $ by $h$. Separate the turning points of $f$ into two parts. Let $S_{1}$ (resp. $S_{2}$) denote the turning points of $f$ contained (resp. not contained) in $K$. Denote $K=\cup _{i=1}^{2^{n}}K_{i}^{(n)}, n\in \N$. There exists $n_{0}\in \N$ such that for any $n>n_{0}$, $S_{2}\cap \cup _{i}^{2^{n}}[K_{i}^{(n)}]=\emptyset $. It is sufficient to prove for any $x\in S_{2}$ there exists $n_{0}$ such that for any $n>n_{0}$, $x$ is not in $\cup _{i}^{2^{n}}[K_{i}^{(n)}]$. Suppose the contrary. Then there exists an infinite sequence of nested intervals $[K_{i(n)}^{(n)}]$ containing $x$. Hence $x\in \cap _{n\in \N}[K_{i(n)}^{(n)}]$. Since the end points of $\cap _{n\in \N}[K_{i(n)}^{(n)}]$ are in the same fiber of $h$, which has to be a point, $x$ is in $K$, a contradiction. Now let $n>n_{0}$. Then $f$ maps each $[K_{i}^{(n)}]$ onto another $[K_{j}^{(n)}]$. Therefore $f$ is infinitely renormalizable. $\Box $ \begin{thebibliography}{Dillo 83} \bibitem[BORT]{BORT} H. Bass, M.V. Otero-Espinar, D. Rockmore, and C. Tresser, {\em Cyclic Renormalization and Automorphisms Groups of Rooted Trees}, Springer Verlag's Lecture Notes in Mathematics (to appear). \bibitem[Bl]{Bl}L. Block, Homoclinic points of mappings of the interval, {\em Bull. Amer. Math. Soc.}, Vol. 72, No. 3 (1978) 576-580. \bibitem[BlH]{BlH}L. Block and D. Hart, The bifurcation of periodic orbits one-dimensional maps, {\em Ergod. Th. \& Dynam. Sys.} 2 (1982) 125-129. \bibitem[Bo]{Bo}R. Bowen, Entropy for group endomorphisms and homogeneous spaces, {\em Trans. Amer. Math. Soc.} 153 (1971) 401-414. \bibitem[BF]{BF}R. Bowen and J. Franks, The periodic points of the disc and the interval, {\em Topology} 15 (1976) 337-342. \bibitem[DGMT]{DGMT}S. P. Dawson, R. Galeeva, J. Milnor, and C. Tresser, A monotonicity conjecture for real cubic maps, in {\em Real and Complex Dynamical Systems}, B. Branner and P. Hjorth Eds. (Kluwer, Dordrecht/Boston/London, 1995). \bibitem[Hu]{Hu}J. Hu, Renormalization, rigidity, and universality in bifurcation theory, Ph.D. Dissertation, Department of Math., City Univ. of New York, 1995. \bibitem[HS]{HS}J. Hu and D. Sullivan, Topological conjugacy of circle diffeomorphisms, to appear in {\em Ergod. Th. \& Dynam. Sys.}. \bibitem[La]{La}O. E. Lanford III, A computer-assisted proof of the Feigenbaum conjecture, {\em Bull. Amer. Math. Soc.} N.S. {\bf 6}:127 (1984). \bibitem[Ma]{Ma} M. Martens: Interval dynamics. Thesis Delft (1990). \bibitem[MMS]{MMS}M. Martens, W. de Melo and S. van Strien, Julia-Fatou-Sullivan theory for real one-dimensional dynamics, {\em Acta Math.} 168 (1992) 273-318. \bibitem[MiT]{MiT}J. Milnor and W. Thurston, On iterated maps of the interval, in {\em Lecture Notes in Mathematics}, No. 1342, (Springer-Verlag, New York, 1988). \bibitem[M1]{M1}M. Misurewicz, Horseshoes for mappings of the interval, {\em Bull. Acad. Polon. Sci. Ser. Sci. Math.} 27 (1979) 167-169. \bibitem[M2]{M2}---, Invariant measures for continuous transformations of [0,1] with zero topological entropy, in {\em Lecture Notes in Mathematics}, No. 729, (Springer-Verlag, New York, 1979). \bibitem[OT]{OT}M. V. Otero-Espinar and C. Tresser, Global complexity and essential simplicity: A conjectural picture of the boundary of chaos for smooth endomorphisms of the interval, {\em Physica D} {\bf 39} (1989) 163-168. \bibitem[Su]{Su}D. Sullivan, Bounds, quadratic differentials and renormalization conjectures, in {\em Mathematics into Twenty-first Century}, Vol. 2, (Amer. Math. Soc., Providence, RI, 1992). \end{thebibliography} \end{document}