instructions: first separate the AmS TeX part (ending at "\enddocument") from the (encapsulated) postscript part (the rest). The postscript file should be called fig2.eps. Then compile the AmS TeX file using TeX. If psfig version 1.9 is correctly installed, this should produce a complete version of the article. If the figure is not inserted correctly, print fig2.eps separately. BODY \magnification=\magstep1 \input amstex \input psfig \documentstyle{amsppt} \vsize=22 truecm \hsize=16 truecm \TagsOnRight \NoRunningHeads %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\today {\ifcase\month\or January \or February \or March \or April \or May \or June \or July \or August \or September \or October \or November \or December \fi \number\day~\number\year} \def \real{{\Bbb R}} \def \complex{{\Bbb C}} \def \integer{{\Bbb Z}} \def \rational{{\Bbb Q}} \redefine \natural{{\Bbb N}} \def \varr{{\text{var}\,}} \def \sgn{{\text{sgn}\,}} \def \modd{{\text{mod}\,\,}} \def \Fix{{\text{Fix}\,}} \def \Fixs{{\text{Fix}\star}} \def \Per{{\text{Per}\,}} \def \Pers{{\text{Per}\star}} % \def\AA{{\Cal A}} \def\BB{{\Cal B}} \def\CC{{\Cal C}} \def\DD{{\Cal D}} \def\EE{{\Cal E}} \def\FF{{\Cal F}} \def\HH{{\Cal H}} \def\II{{\Cal I}} \def\JJ{{\Cal J}} \def\KK{{\Cal K}} \def\LL{{\Cal L}} \def\MM{{\Cal M}} \def\NN{{\Cal N}} \def\OO{{\Cal O}} \def\PP{{\Cal P}} \def\QQ{{\Cal Q}} \def\RR{{\Cal R}} \def\SS{{\Cal S}} \def\TT{{\Cal T}} \def\UU{{\Cal U}} \def\VV{{\Cal V}} \def\XX{{\Cal X}} \def\ZZ{{\Cal Z}} %!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! %=============postscript======= %figure number psfile caption % (will be centered) \def\figure #1 #2 #3\cr { \bigskip \centerline{\psfig {figure=#2,height=7cm,width=16cm} } \smallskip \vbox{\eightpoint\noindent {\bf Figure: #1} #3} \bigskip } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \topmatter \title\nofrills An extension of the theorem of Milnor and Thurston \\ on the zeta functions of interval maps \endtitle \author V. Baladi and D. Ruelle \endauthor \address CNRS, UMR 128, UMPA, ENS Lyon, 46, all\'ee d'Italie, F-69364 Lyon Cedex 07, France \endaddress \email baladi\@umpa.ens-lyon.fr\endemail \address Institut des Hautes \'Etudes Scientifiques, F-91440 Bures-sur-Yvette, France \endaddress \date{March 1993} \enddate \dedicatory Dedicated to Huzihiro Araki. \enddedicatory \subjclass 58F20 58F03 \endsubjclass \abstract {We consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor-Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.} \endabstract \endtopmatter %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \document \head Introduction \endhead Let $f$ be a continuous map from a compact interval, say $[0,1]$, to itself. ``Counting'' the periodic orbits of $f$ is a basic dynamical problem. A natural assumption is that $f$ is piecewise strictly monotone with a finite number $N$ of monotonicity intervals $[a_{i-1},a_i]$ where $0 =a_0 < a_1 < a_2 < \ldots < a_{N-1} < a_N=1$. Suppose that each iterate $f^{\circ n}$ can have at most one fixed point in each monotonicity interval (this happens for instance when $f$ is piecewise expanding); then it is natural to define the {\it zeta function} $$ \zeta(t) = \exp \sum_{n \ge 1} {t^n \over n} \# \{ x \, : \, f^{\circ n} x = x \} $$ which simply counts periodic points {\it \`a la} Artin-Mazur. For more general situations we shall use (see below) a ``reduced'' zeta function similar to the one introduced by Milnor and Thurston [1988] (this article was widely circulated as a preprint in the late `70s). The main result of Milnor and Thurston relates the zeta function to the determinant of a finite {\it kneading matrix} (an invariant of the map defined by Milnor and Thurston in the same paper). Using this relation, they are able to show that $\zeta(t)$ is meromorphic in the unit disc and that the first pole occurs at $e^{-s}$ where $s$ is the topological entropy of $f$. More recently, {\it weighted} zeta functions for piecewise monotone interval maps have attracted interest. Here, a weight $g: [0,1]\to \complex$ is given, which is supposed either constant on each $(a_{i-1},a_i)$, or of bounded variation on $[0,1]$ (see e.g. Hofbauer-Keller [1984], Baladi-Keller [1990]; for more general $g$ see Keller-Nowicki [1992], Ruelle [1992]). Assuming that the periodic points are isolated, one sets $$ \zeta_g(t)= \exp \sum_{n \ge 1} {t^n\over n} \sum_{x : f^{\circ n} x = x} \prod_{k=0}^{n-1} g(f^{\circ k} x) \, . $$ The analytic properties of $\zeta_g(t)$ are studied with the help of a {\it transfer operator} acting on a suitable Banach space. (Transfer operators have their origin in statistical mechanics; their study in infinite dimension was initiated by Araki, Ruelle, and Mayer, see for instance Mayer [1980].) Suppose for simplicity that $g$ is real-valued and positive. Under suitable assumptions, one shows that $\zeta_g(t)$ is meromorphic in a disc and that its first pole is the exponential of minus the topological pressure of $\log g$. In some cases the dynamical meaning of the other poles is understood. In particular, when $f$ is topologically mixing, piecewise $\CC^2$ and expanding, and when $g = 1/\log|f'|$, the second pole corresponds to a rate of mixing (in this case the first pole occurs at $t=1$, and the corresponding eigenfunction of the transfer operator gives rise to an absolutely continuous invariant measure for $f$). The introduction of weights is also useful when studying the unweighted zeta functions of semi-flows obtained by suspending interval maps under non necessarily constant return times. In this article, we introduce weighted kneading matrices and determinants and extend Milnor and Thurston's main result to the case of weighted zeta functions of the type of $\zeta_g(t)$. (See Section 1 for precise definitions and results.) The weight $g$ is supposed to be constant on each interval $(a_{i-1},a_i)$ (the case of a weight of bounded variation, where a countable matrix is needed, will be treated in a further work). We also generalise the setting in that we only require the map $f$ to be continuous on the intervals $(a_{i-1},a_i)$. (The piecewise continuous setting is useful for example in the study of the Lorenz attractor. See e.g. Williams [1979] and Rand [1978] for a definition of kneading series in this case.) We do not need each $a_i$ to be a turning point, i.e., $f$ can be monotone in a neighbourhood of some of the $a_i$. Finally, our proof, while based on the homotopy argument used by Milnor and Thurston, is somewhat simpler due to the fact that $f$ is allowed to be discontinuous at the $a_i$ (and we can use more elementary transversality arguments). ``Fredholm'' matrices analogous to the kneading matrix have been introduced by Mori [1990,1991]. He considers weights which are either locally constant or of bounded variation and uses a renewal equation to relate the determinant of the Fredholm matrix to the corresponding transfer operator and zeta function. However, the relationship with the Milnor-Thurston theory is not made explicit, and the method of proof is completely different from ours. \bigskip The first author is very grateful to J. Milnor for his interest expressed in useful conversations and an enlightening letter. She also thanks the IHES for an invitation during which work on this article was initiated and H.H. Rugh for useful comments. \head 1. Definitions and statement of results \endhead Let $a_0 < a_1 < \ldots < a_N$, and $f_i : [a_{i-1}, a_i] \to (a_0,a_N)$ be strictly monotone and continuous maps for $i=1, \ldots, N$. We write $f = (f_1, \ldots, f_N)$. By abuse of language we shall also denote by $f$ the ``multivalued map'' $[a_0,a_N] \to [a_0,a_N]$ whose graph is the union of the graphs of the $f_i$. We let $\epsilon_i = \pm 1$ depending on whether $f_i$ is increasing or decreasing. We also let $z_i \in \complex$ and write $Z=(z_1, \ldots, z_N)$. We define functions $\epsilon, z$ on $[a_0, a_N]$ such that they have the constant values $\epsilon_i, z_i$ on $(a_{i-1},a_i)$ and $0$ on $\{ a_0, a_1, \ldots, a_N\}$. We define the {\it address} of $x \in [a_0, a_N]$ to be the vector $$ \vec \alpha (x) = (\sgn (x-a_1), \ldots, \sgn (x-a_{N-1})) \in \integer^{N-1}\, . $$ The {\it invariant coordinate} of $x$ is the $(N-1)$-tuple of formal power series $$ \vec \theta (x,Z)=\vec \theta_f(x,Z)= \sum_{n=0}^\infty \bigl ( \prod_{k=0}^{n-1} (\epsilon z) (f^{\circ k} x) \bigr ) \vec \alpha(f^{\circ n} x) \in \integer [[z_1, \ldots, z_N]]^{N-1} \, . $$ Note that $\vec \theta$ is single valued because if $f^{\circ k} x \in \{a_0, \ldots, a_N\}$ for some $k < n$ then $(\epsilon z)(f^{\circ k} x)=0$. Writing $\phi(a\pm) = \lim \phi(x)$ when $x \downarrow a$ or $x \uparrow a$, we let $$ \eqalign { \vec K_i (Z) &= {1\over 2} \biggl [ \vec \theta(a_i +,Z) - \vec \theta(a_i -, Z) \biggr ] \cr &= (K_{i,1} (Z), \ldots, K_{i,N-1} (Z)) \, ,\cr } $$ for $i=1, \ldots, N-1$. The $(N-1)\times (N-1)$ matrix $\bigl [ K_{ij} (Z) \bigr ]$ is the {\it kneading matrix}, and the determinant $$ \Delta (Z) =\Delta_f(Z)= \det \bigl [ K_{ij} (Z) \bigr ] \in \rational[[z_1, \ldots, z_n]] $$ is the {\it kneading determinant}. Since $K_{ij}(Z)= \delta_{ij} + \text{higher order}$, we have $\Delta (Z) = 1 + \text{higher order}$. Note that if we set $z_1 = \ldots = z_N= t$, and if we assume that $\epsilon_i=- \epsilon_{i-1}$ for each $1\le i\le N$, then we recover a simple modification of the kneading determinant of Milnor and Thurston [1988] (see also Preston [1989] for another presentation). We denote by $\Fix f^{\circ m}$ the set of fixed points of $f^{\circ m}$ which have an orbit disjoint from $\{ a_0, \ldots, a_N\}$. We assume that for each $m$ the set $\Fix f^{\circ m}$ is finite. For $x \in \Fix f^{\circ m}$, if the graph of $f^{\circ m}$ does not cross the diagonal at $x$ we set $$ L (x,f^{\circ m})=0\, . $$ If the graph of $f^{\circ m}$ crosses the diagonal, we may define $$ L(x,f^{\circ m}) = \lim_{y\to x} { \sgn(f^{\circ m} y-y) \over \sgn(x-y)} \, , $$ ($L(x,f^{\circ m})$ is a Lefschetz index for $x$) and $$ \nu(x,f^{\circ m}) = -L(x,f^{\circ m}) \cdot \prod_{k=0}^{m-1} \epsilon(f^{\circ k} x) \, . $$ (If $f^{\circ m}$ is decreasing at $x \in \Fix f^{\circ m}$ then the assumption that $\# \Fix f^{\circ m} < \infty$ for all $m$ implies that $x$ is either an attracting or repelling fixed point for $f^{\circ m}$. In all cases where the graph of $f^{\circ m}$ crosses the diagonal at $x$, the periodic point $x$ is hence either attracting or repelling, and $\nu (x,f^{\circ m}) = -1$ if and only if $f^{\circ m}$ is increasing and attracting at $x$.) We extend now the set $\Fix f^{\circ m}$ to a set $\Fixs f^{\circ m}$ obtained by adding some symbols $x *$ where $x \in (a_0, a_N)$ and $*$ is $+$ or $-$: $$ \eqalign { \Fixs f^{\circ m} &= \Fix f^{\circ m} \bigcup \{ x* \, : \, f^{\circ m}(x*)= x \, ,\cr &\qquad\qquad f^{\circ k} (x*) = a_i \text{ for some $k,i$ and } \prod_{s=0}^{m-1} \epsilon(f^{\circ s}(x *)) = +1 \} \, . } $$ If $x* \in \Fixs f^{\circ m} \setminus \Fix f^{\circ m}$, we let $$ L(x*,f^{\circ m}) = \cases 0 &\text{if $x*$ is (one-sided) repelling} \cr 1&\text{if $x*$ is (one-sided) attracting} \cr \endcases $$ and $\nu(x*,f^{\circ m}) = -L(x*,f^{\circ m})$. We define now the ``reduced'' zeta function $$ \zeta(Z) = \zeta_f(Z) = \exp \sum_{m=1}^\infty {1 \over m} \sum_{\xi \in \Fixs f^{\circ m}} \nu(\xi,f^{\circ m}) \cdot \prod_{k=0}^{m-1} z(f^{\circ k} \xi) \, . $$ If $f^{\circ m} (a_i\pm)\ne a_i$ for all $i$ and all $m \ge 1$, we may replace $\Fixs f^{\circ m}$ by $\Fix f^{\circ m}$ in this formula. Denote by $\Per$ the set of periodic orbits of $f$ which do not contain any of the $a_i$, and by $\Pers$ the extended set of periodic orbits formed of elements of $\cup_m \Fixs f^{\circ m}$. If $\gamma \in \Pers$ is of period $p(\gamma)$ (i.e., $p(\gamma) = \min \{ m \ge 1 \, : \, \gamma \subset \Fixs f^{\circ m} \}$) and contains the element $\xi_\gamma$ we let $$ Z (\gamma) = \prod_{k=0}^{p(\gamma)-1} z(f^{\circ k} \xi_\gamma) \, . $$ With this notation, we may write $$ \zeta(Z)=\prod_{\gamma \in \Pers} F(\gamma) $$ where $$ F(\gamma) = \exp \sum_{m =1}^\infty {1 \over m} \nu(\xi_\gamma, f^{\circ m\cdot p(\gamma)}) Z(\gamma)^m \, . $$ Let $\gamma \in \Per$, with $p(\gamma)=p$, and $x_\gamma \in \gamma$, then \roster \item"(0)" $F(\gamma) = 1$ if the graph of $f^{\circ p}$ does not cross the diagonal at $x_\gamma$; \item $F(\gamma) = {1\over 1-Z(\gamma)}$ if $f^{\circ p}$ is increasing at $x_\gamma$ and $\gamma$ repelling; \item $F(\gamma) = 1-Z(\gamma)$ if $f^{\circ p}$ is increasing at $x_\gamma$ and $\gamma$ attracting; \item $F(\gamma) = 1+Z(\gamma)$ if $f^{\circ p}$ is decreasing at $x_\gamma$ and $\gamma$ attracting; \item $F(\gamma) = {1\over 1-Z(\gamma)}$ if $f^{\circ p}$ is decreasing at $x_\gamma$ and $\gamma$ repelling. \endroster Let $\gamma \in \Pers \setminus \Per$, then \roster \item"(5)" $F(\gamma) = 1$ if $\gamma$ is one-sided repelling; \item "(6)" $F(\gamma) = 1-Z(\gamma)$ if $\gamma$ is one-sided attracting. \endroster It follows that $\zeta (Z) \in \integer [[z_1, \ldots, z_n]]$. Notice that the cases \therosteritem{1} and \therosteritem{4} where $\gamma$ is repelling give the same expression for $F(\gamma)$. If all periodic points are repelling, and if $f^{\circ m} a_i* \ne a_i$, for all $m \ge 1 $ and $1 \le i \le N-1$, then we recover the usual (weighted) one-dimensional zeta function (see e.g. Hofbauer-Keller [1984]). We can now state our main results. \proclaim{1.1. Theorem} With the above notation and assumptions we have $$ \zeta(Z) \Delta(Z) =1 -{1\over 2} (\epsilon_1 z_1+ \epsilon_N z_N) \, . $$ \endproclaim \proclaim{1.2. Corollary} If all periodic points are repelling we have $$ \zeta(Z) = \prod_{\gamma \in \Per} {1 \over 1-Z(\gamma)} ={1 -{1\over 2} (\epsilon_1 z_1+ \epsilon_N z_N) \over \Delta(Z)} \, . $$ \endproclaim For each fixed choice of $a_0, \ldots, a_N$ and $\epsilon_1, \ldots, \epsilon_N$ we shall prove Theorem~1.1 by a ``homotopy argument'' very similar to the one used by Milnor and Thurston [1988, \S 11]. This means that we check the theorem for some particular choice of $f$ and we then compare the changes in $1 / \zeta$ and $\Delta$ for a suitable one-parameter family of $f$'s, showing that they are both multiplied by the same factor at each bifurcation (see the crucial Lemma~2.4). \head 2. Proofs \endhead Let $N\ge 1$, $a_0, \ldots, a_N \in \real$ (with $a_0 < a_1< \ldots < a_N$) and $\epsilon_1, \ldots, \epsilon_N \in \{-1,+1\}$ be fixed. As outlined in Section~1, we start by checking Theorem~1.1 on a simple example: \proclaim{2.1. Lemma} Suppose that $f_i^0 [a_{i-1},a_i] \subset (a_0, a_1)$ for $i=1, \ldots, N$ and that $f_1^0$ is differentiable with $|(f_1^0)'| < 1$. Then $$ \zeta_{f^0}(Z) \Delta_{f^0}(Z) = 1 -{1\over 2 } (\epsilon_1 z_1 + \epsilon_N z_N ) \, . $$ (In particular, the $f_i^0$ may be taken to be polynomials with derivatives vanishing at $a_{i-1}, a_i$.) \endproclaim \smallskip Our assumptions imply that the set of periodic points of $f^0$ consists of a single fixed point $x \in (a_0, a_1)$, and $x$ is of type \therosteritem{2} if $\epsilon_1=+1$, \therosteritem{3} if $\epsilon_1=-1$. Therefore, we have in either case $$ \zeta_{f^0} (Z) = 1-\epsilon_1 z_1 \, . $$ We also have $$ \eqalign { \vec \theta(a_i +, Z) &= (\ldots,+1, +1, -1,\ldots) +\epsilon_{i+1} z_{i+1} \cdot \sum_{n=1}^\infty (\epsilon_1 z_1 )^{n-1} \cdot (-1, \ldots, -1) \cr \vec \theta(a_i -, Z) &= (\ldots,+1, -1,-1,\ldots) +\epsilon_i z_i \cdot \sum_{n=1}^\infty (\epsilon_1 z_1 )^{n-1} \cdot (-1, \ldots, -1) \, , \cr } $$ and $$ \eqalign { \vec K_i(Z) &= (0, \ldots,1, \ldots, 0) \cr &+{1\over 2}( \epsilon_{i+1} z_{i+1} -\epsilon_i z_i) \cdot {1 \over 1-\epsilon_1 z_1 } \cdot (-1, \ldots, -1) \, . } $$ Writing $\alpha_i = \frac {1} {2} (\epsilon_{i+1} z_{i+1} -\epsilon_i z_i ) \cdot \frac{1} {1-\epsilon_1 z_1 }$ gives $K_{ij} (Z) = \delta_{ij} - \alpha_i$, hence $\Delta_{f^0}(Z)=\det (1-T)$, where the matrix $T_{ij}=\alpha_i$ is of rank one. Hence $$ \eqalign { \Delta_{f^0} (Z)&=\sum_{n=0}^{N-1} (-1)^n \text{Tr}\, \wedge^n T = 1-\text{Tr} \, T \cr & = 1-\sum_i \alpha_i\cr &=1+ {1\over 2} (\epsilon_1 z_1 - \epsilon_N z_N) \cdot {1 \over 1-\epsilon_1 z_1} \cr &= [1 - {1\over 2} (\epsilon_1 z_z + \epsilon_N z_N ) ] \cdot {1 \over 1-\epsilon_1 z_1} \, , \cr } $$ and finally $$ \zeta_{f^0}(Z) \Delta _{f^0}(Z) = 1 -{1\over 2 } (\epsilon_1 z_1 + \epsilon_N z_N ) \, .\qed $$ \smallskip \bigskip It is convenient to use the $C^1$ topology. For $r= 0$ or $1$, let $P^r=P^r(N, (a_i),(\epsilon_i))$ denote the set of $N$-tuples $f= (f_1, \ldots, f_N )$ where each $f_i:[a_{i-1},a_i]\to (a_0, a_N)$ is $\CC^r$ and strictly monotone increasing or monotone decreasing according to whether $\epsilon_i$ equals $+1$ or $-1$. If $r=1$ we further impose that $f_i'(a_{i-1})=f_i'(a_i) =0$ (vanishing of the derivative at interval endpoints). The $d_r$ distance between $f$ and $g$ in $P^r$ is given by the sum of the $\CC^r$ distances between the $f_i$'s and $g_i$'s. As already mentioned, we view $f$ and its iterates $f^{\circ m}$ as multivalued maps $[a_0, a_N] \to (a_0, a_N)$. For the definition of $\zeta_f$, we have required that the sets $\Fix f^{\circ m}$ be finite, but this condition will be lifted after Lemma 2.2. Let $P^r_M$ consist of those $f \in P^r$ such that $f^{\circ m}(a_i *) \ne a_i$ whenever $1 \le m \le M $, $1 \le i \le N-1$ and $*=\pm$. \proclaim{2.2. Lemma} $P^r_M$ is an open subset of $P^r$. If $\II_{M+1}$ is the ideal of elements of order $\ge M+1$ in $\rational[[z_1, \ldots, z_N]]$, the map $$ f \mapsto \zeta_f \quad (\modd \II_{M+1}) $$ defined on the set $\{ f \in P^0_M \, : \, \# \Fix f^{\circ m} < \infty \, , \, \forall m \ge 1 \}$ extends to a locally constant map $$ P^0_M \to \integer [[z_1, \ldots, z_N]]/(\II_{M+1} \cap \integer [[z_1, \ldots, z_N]])\, . $$ \endproclaim \smallskip $P^r_M$ is defined by a finite number of open conditions, hence is open in $P^r$. For general $f \in P^0$, we may define $L(f_{\ell_m} \circ \cdots \circ f_{\ell_1})$ to be: \roster \item"" $-1$ if the left end of the graph of $f_{\ell_m} \circ \cdots \circ f_{\ell_1}$ is $<$ the diagonal and the right end $>$ the diagonal, \item "" $+1$ if $f_{\ell_m} \circ \cdots \circ f_{\ell_1}$ is increasing and the left end of the graph is $\ge$ the diagonal and the right end is $\le$ the diagonal, \item "" $+1$ if $f_{\ell_m} \circ \cdots \circ f_{\ell_1}$ is decreasing and the left end of the graph is $>$ the diagonal and the right end is $<$ the diagonal, \item "" $0$ in all other cases (in particular when the domain of $f_{\ell_m} \circ \cdots \circ f_{\ell_1}$ is empty or reduced to a point). \endroster Note that if $f \in P^0_M$ then neither the left nor the right end of the graph of $f_{\ell_m} \circ \cdots \circ f_{\ell_1}$ can intersect the diagonal for $1 \le m \le M$. When $\Fix f^{\circ m}$ is finite we have thus $$ \eqalign { \sum_{\xi \in \Fixs f^{\circ m}} \nu(\xi,f^{\circ m}) &= \sum_{\ell_1, \ldots, \ell_m} \, \sum_{\xi \in \Fixs f_{\ell_m} \circ \cdots \circ f_{\ell_1}} \nu(\xi,f_{\ell_m} \circ \cdots \circ f_{\ell_1})\cr &= -\sum_{\ell_1, \ldots, \ell_m} (\epsilon_{\ell_1} \cdots \epsilon_{\ell_m}) L(f_{\ell_m} \circ \cdots \circ f_{\ell_1}) \, .\cr } $$ Note that the $L(f_{\ell_m} \circ \cdots \circ f_{\ell_1})$ are locally constant on $P^0_M$, and that $$ \{ f \in P^0_M \, : \, \Fix f^{\circ m} \text{ is finite for all } m \ge 1 \} $$ is dense in $P^0_M$ [approximate by $N$-tuples of non-affine polynomials]. Note also that the map $\xi \mapsto \prod_{k=0}^{m-1} z(f^{\circ k} \xi)$ is constant on $\Fixs f_{\ell_m} \circ \cdots\circ f_{\ell_1}$. Therefore the lemma results from the definition of $\zeta$. \qed \smallskip \smallskip The formula given above yields a natural definition of $\zeta_f$ ($\modd \II_{M+1}$) for general $f \in P^0$, and we shall use this definition to prove Theorem 1.1. (Note that our definition of $\Delta$ did not use the condition that the sets $\Fix f^{\circ m}$ be finite.) \smallskip \proclaim{2.3. Lemma} Let $M \ge 1$ and $\tilde f$ be a sufficiently small perturbation of $f$ in $P^0$ such that the ``shrinking condition'' $$ \text{range } \tilde f_{\ell_m} \circ \cdots \circ \tilde f_{\ell_1} \subset \text{range } f_{\ell_m} \circ \cdots \circ f_{\ell_1} $$ holds when $m \le M$. Then $$ \eqalign { \zeta_{\tilde f}(Z) &= \zeta_f (Z) \quad (\modd \II_{M+1})\cr \Delta_{\tilde f} (Z) &= \Delta_f(Z) \quad (\modd \II_{M+1}) \, .\cr } $$ Furthermore, if we have ``strict shrinking'' $$ \text{range } \tilde f_{\ell_m} \circ \cdots \circ \tilde f_{\ell_1} \subset \text{ interior range } f_{\ell_m} \circ \cdots \circ f_{\ell_1} $$ for $m \le M$, then $\tilde f^{\circ m} (a_i \pm) \notin \{ a_1, \ldots, a_{N-1} \}$ for all $1 \le i\le N$ and $m \le M$, in particular $\tilde f \in P^0_M$. \endproclaim \smallskip Our definition of $L(f_{\ell_m} \circ \cdots \circ f_{\ell_1})$ shows that this quantity does not change if the range of $f_{\ell_m} \circ \cdots\circ f_{\ell_1}$ is shrunk by a small amount. Hence, $\zeta_{\tilde f} (Z)= \zeta_f(Z)$ ($\modd \II_{M+1}$). When $\tilde f$ tends to $f$, then $$ \tilde f^{\circ m} (a_i \pm) \to f^{\circ m} (a_i \pm) $$ and the shrinking condition implies that the limit is reached on the same side as the limit $$ f^{\circ m}(x)\to f^{\circ m}(a_i \pm) $$ when $\pm(x-a_i) \downarrow 0$. Therefore the $\vec \theta(a_i \pm, Z)$ coincide ($\modd \II_{M+1}$) for $\tilde f$ and $f$ and the same holds for the $\vec K_i$ so that $\Delta_{\tilde f}(Z) = \Delta_f(Z)$ ($\modd \II_{M+1}$). The last statement of the lemma follows from the fact that the numbers $$ | \tilde f ^{\circ m} (a_i \pm) - f^{\circ m} (a_i\pm)| $$ are in an arbitrarily small interval $(0,\delta)$. \qed \smallskip \smallskip \demo{Proof of Theorem 1.1} Given $f$ we may take $\tilde f$ to be arbitrarily close to $f$ in $P^0$, such that $\tilde f_i$ and $f_i$ differ only in $[a_{i-1}, a_i+\delta]$, $[a_i-\delta, a_i]$ and $\tilde f_i [a_{i-1}, a_i] \subset f_i(a_{i-1},a_i)$. This will imply the strict shrinking condition of Lemma 2.3 for some fixed $M \ge 1$ if $\delta=\delta(M)$ is sufficiently small. Let now $f^1 \in P^1_M$ be an $N$-tuple of polynomials sufficiently close to $\tilde f \in P^0_M$; then the invariant coordinates of the $a_i*$ are the same for $f^1$ and $\tilde f$ up to terms of order $> M$. Therefore $$ \eqalign { \zeta_{f} &= \zeta_{f^1} \quad (\modd \II_{M+1}) \cr \Delta_{f} &= \Delta_{f^1} \quad (\modd \II_{M+1}) \cr } $$ and the proof of the theorem reduces to showing that $$ \zeta_{f^1} \, \Delta_{f^1} = 1-{1\over 2} (\epsilon_1 z_1 + \epsilon_N z_N) \quad (\modd \II_{M+1}) \, . $$ Using $f^0$ as defined in Lemma 2.1, let $f^{\lambda} = (1-\lambda) f^0 + \lambda f^1 \in P^1$. By definition, $f^\lambda=(f^{\lambda}_1, \ldots, f^{\lambda}_N)$ is an $N$-tuple of polynomials, none of which is affine [$f^\lambda_i$ is non constant with derivative vanishing at $a_{i-1}$ and $a_i$]. Therefore, $\Fix (f^\lambda)^{\circ m}$ is finite for all $m$. Note that the maps $(x,\lambda) \to f^\lambda_i(x)$, being also polynomials, are naturally defined on $\real^2$. Every composition $f^\lambda_{\ell_m} \circ \cdots \circ f^\lambda_{\ell_1}$ is hence allowed and defined on $\real$. Since $f^\lambda_{\ell_m} \circ \cdots \circ f^\lambda_{\ell_1} (a_i) -a_j$ is a polynomial in $\lambda$, of degree $D$ (depending on $\ell_1, \ldots, \ell_m, a_i$), which we may take $\ne 0$ at $\lambda=0,1$, there is a finite set $\Lambda$ of values of $\lambda$ outside of which $$ (f^\lambda)^{\circ m} (a_i *) \ne a_j \quad \text{if } 1 \le m \le M\, , 1 \le i \le N-1, 1\le j \le N-1 \, . $$ Therefore $\zeta_{f^\lambda} \Delta_{f^\lambda}$ remains constant ($\modd \II_{M+1}$) for $\lambda$ in each interval of $[0,1]\setminus \Lambda$. (In fact, both $\zeta_{f^\lambda}$ and $\Delta_{f^\lambda}$ remain constant individually; the zeta function because of Lemma 2.2 and the determinant because each $\vec \theta_{f^\lambda}(a_i \pm,Z)$ is constant.) There remains to show that $1/ \zeta_{f^\lambda}$ and $\Delta_{f^\lambda}$ are multiplied by the same factor ($\modd \II_{M+1}$) when $\lambda$ crosses a point of $\Lambda$. The changes of sign of the $(f^\lambda)^{\circ m} (a_i *) - a_j$ when $\lambda$ crosses an element of $\Lambda$ may be complicated. We shall make them simpler by modifying the family $(f^\lambda)$ near each $\lambda \in \Lambda$ to obtain a family $(g^\lambda)$ with nonlinear (but $\CC^\infty$) dependence on $\lambda$. If the maps $(x,\lambda) \mapsto f^\lambda_i(x)$ are approximated in the $\CC^\infty$ topology by maps $(x,\lambda) \mapsto g^\lambda_i(x)$, then $\lambda \mapsto f^\lambda_{\ell_m} \circ \ldots \circ f^\lambda_{\ell_1} (a_i)-a_j$, and $\lambda \mapsto g^\lambda_{\ell_m} \circ \cdots \circ g^\lambda_{\ell_1} (a_i) -a_j$ are $\CC^\infty$ close and therefore $g^\lambda_{\ell_m} \circ \cdots \circ g^\lambda_{\ell_1} (a_i) -a_j$ can have at most $D$ zeros and $(g^\lambda)^m(a_i *) -a_j$ at most $D'$ zeros for some fixed $D'$ uniform in the choice of $(g^\lambda)$. The uniformity of this bound (when $m \le M$) will be used in a moment. For a given $\lambda_0 \in \Lambda$ we construct an oriented graph $\Gamma$ as follows. Let $(f^{\lambda_0})^{\circ m} (a_i*) = a_j$ with $1 \le m \le M$, and suppose that $(f^{\lambda_0})^{\circ k} (a_i*) \notin \{ a_1, \ldots, a_{N-1}\}$ for $k=1, \ldots, m-1$; we define a sign $\pm$ by $$ *\prod_{k=0}^{m-1} \epsilon(f^{\circ k}(a_i*)) = \pm1 $$ and place an arrow $(a_i *) \Rightarrow (a_j \pm)$. Note that there may be {\it simple loops} $(a_i *) \Rightarrow (a_i *)$. An arrow starting from $(a_{i-1}+)$ (respectively $(a_i-)$) may be removed from the graph corresponding to $\lambda_0$ by a $\CC^\infty$ small change of $f_i^{\lambda}$ near $(a_{i-1}, \lambda_0)$ (respectively $(a_i, \lambda_0)$) while the other arrows are left unchanged. Repeating this operation, we can arrange that the graph corresponding to $\lambda_0$ consists of a single arrow (which may be a simple loop). We have thus replaced the family $(f^\lambda)$ by a new family $(\tilde f ^\lambda)$, the set $\Lambda$ by a new finite set $\tilde \Lambda$, and the new graph $\tilde \Gamma$ corresponding to $\lambda_0$ has only one arrow. By a $\CC^\infty$ small change of $\tilde f^\lambda$ near $\lambda=\lambda_0$ we may assume that $\Fix (\tilde f^{\lambda_0})^{\circ m}$ is finite for $1 \le m \le M$, and that the derivative of $(\tilde f^{\lambda_0})^{\circ m}$ at $x \in \Fix (\tilde f^{\lambda_0})^{\circ m}$ is not equal to $1$ (i.e., the periodic points of period $\le M$ for $\tilde f^{\lambda_0}$ are not neutral). Note that the families $(f^\lambda)$, $(\tilde f ^\lambda)$ coincide outside of a small neighbourhood of $\lambda_0$; to obtain $\tilde \Lambda$ from $\Lambda$ we have replaced $\lambda_0$ by a finite set $\{ \lambda_0, \lambda_0', \ldots \}$. We now start again the above process with a new element $\tilde \lambda_0$ of $\tilde \Lambda$ (being careful to leave $\tilde f^{\lambda_0}$ unchanged). Since the cardinality of the sets $\Lambda$, $\tilde \Lambda$, \dots, is uniformly bounded, after a finite number of steps the family $(f^\lambda)$ is replaced by $(g^\lambda)$ with the following properties: \roster \item"(a)" $g^\lambda \in P^1$, the map $(x,\lambda) \mapsto g^\lambda(x)$ is $\CC^\infty$, $g^0=f^0$, $g^1=f^1$; \item"(b)" for $\lambda$ outside of a finite set $G$, $(g^\lambda)^{\circ m} (a_i*) \ne a_j$, if $1 \le m \le M$, $1 \le i \le N-1$, $1 \le j \le N-1$; \item"(c)" if $\lambda \in G$, there is a single $(a_i*)$ and a single $j$ such that $(g^\lambda)^{\circ p} (a_i*) = a_j$, for some $p \in \{1, \ldots, M\}$; \item"(d)" if $\lambda \in G$, $\Fix (g^\lambda)^{\circ m}$ is finite for $m \le M$ and the points $x \in \Fix (g^\lambda)^{\circ m}$ are not neutral. \endroster To prove the theorem it suffices now (under the assumptions (a),(b),(c),(d)) to show that $1/ \zeta_{g^\lambda}$ and $\Delta_{g^\lambda}$ are multiplied by the same factor $(\modd \II_{M+1}$) when $\lambda$ crosses a point in $G$. This is done in the following lemma. \qed \enddemo \smallskip \proclaim{2.4. Lemma} Let $M \ge 1$ and let $f \in P^1$ be such that the sets $\Fix f^{\circ m}$ are finite for $1 \le m \le M$ and such that there is a single $(a_i*)$ and a single $j$ for which $$ f^{\circ p}(a_i*) = a_j $$ with $1 \le p \le M$. If $j=i$, we take the smallest permissible $p$ and write $$ \hat\epsilon = \prod_{k=0}^{p-1} \epsilon(f^{\circ k}(a_i*)) \,\,\, , \, \, \, Z(\gamma) = \prod_{k=0}^{p-1} z(f^{\circ k}(a_i*)) \, . $$ We assume that the $f$-periodic points with period at most $M$ are not neutral. Then, given $g,h \in P^1_M$ sufficiently close to $f$ and such that $g^{\circ p}(a_i*) > a_j$, $h^{\circ p}(a_i*) < a_j$ we have ($\modd \II_{M+1}$) $$ \eqalign { \zeta_g = \zeta_h \, , \phantom{ \cdot (1-\hat \epsilon Z(\gamma))^{*1} } \,\,\, &\Delta_g = \Delta_h \qquad \qquad\qquad\phantom{if }\, \text{ if }\, j \ne i \cr \zeta_g = \zeta_h \cdot (1-\hat \epsilon Z(\gamma))^{*1} \, , \,\,\, &\Delta_g = \Delta_h / (1-\hat \epsilon Z(\gamma))^{*1} \quad \text{ if }\, j = i\, . \cr } $$ \endproclaim \figure {} fig2.eps {\it Graph of $f^{\circ p}$:} (Lemma 2.4, case $i=j$) The graph of $g^{\circ p}$ (resp. $h^{\circ p}$) is obtained by pushing the graph of $f^{\circ p}$ upwards (resp. downwards). \cr \newpage \smallskip We first discuss the easy proof of the formulas for the zeta function. If $j \ne i$, $\zeta$ ($\modd \II_{M+1}$) is locally constant at $f$ (Lemma 2.2), hence $\zeta_g = \zeta_h$. Let $j=i$. If $*=+$, the orbit $(a_i+, f(a_i+), \ldots, f^{\circ p-1} (a_i+))$ for $f$ turns into an attracting period $p$ orbit $\gamma$ for $g$, absent for $h$ (see the figure). If $*=-$ the attracting orbit corresponding to $(a_i-, f(a_i-), \ldots, f^{\circ p-1} (a_i-))$ occurs for $h$ and is absent for $g$. In both cases, the orbit $\gamma$ is of type (2) if $\hat \epsilon = 1$, of type (3) if $\hat \epsilon =-1$. Apart from the new attracting orbit $\gamma$ just described, $f,g,h$ have corresponding orbits $\gamma'$ with the same weight $Z(\gamma')$ up to order $\ge M+1$ if they are close enough with respect to the distance $d_1$. Therefore $\zeta_g = \zeta_h \cdot (1-\hat \epsilon Z(\gamma))^{*1}$ as announced. Let us now consider the changes for $\Delta$. The ``jump'' at $f$ of the invariant coordinate is ($\modd \II_{M+1}$) $$ \eqalign { \delta\vec \theta&(a_i*,Z)\cr &:=\vec \theta_{g} (a_i*,Z) -\vec \theta_{h} (a_i*,Z) \cr &= \sum_{n=0}^\infty \bigl [ \prod_{k=0}^{n-1} (\epsilon z) (g^{\circ k}(a_i *)) \vec \alpha(g^{\circ n}(a_i*) ) -\prod_{k=0}^{n-1} (\epsilon z) (h^{\circ k}(a_i *)) \vec \alpha(h^{\circ n}(a_i*) ) \bigr ] \cr &= \prod_{s=0}^{p-1} (\epsilon z) (f^{\circ s}(a_i*)) \cr & \quad\cdot\sum_{\ell=0}^\infty \bigl [ \prod_{k=0}^{\ell-1} (\epsilon z) (g^{\circ p+k}(a_i*)) \vec \alpha(g^{\circ p+\ell}(a_i*) ) -\prod_{k=0}^{\ell-1} (\epsilon z) (h^{\circ p+k}(a_i *)) \vec \alpha(h^{\circ p+\ell}(a_i*) ) \bigr ] \, . \cr } $$ We shall denote by $\Phi$ a function which is constant on each interval $(a_{i-1},a_i)$ like $\epsilon, z, \vec \alpha$. If $j \ne i$, we have $\Phi(g^{\circ p+k} (a_i*)) = \Phi(f^{\circ k}(a_j+))$, $\Phi(h^{\circ p+k} (a_i*)) = \Phi(f^{\circ k}(a_j -))$ when $g$ and $h$ are sufficiently close to $f$ and $p+k \le M$, hence $$ \delta\vec \theta(a_i*,Z) = \prod_{s=0}^{p-1} (\epsilon z) (f^{\circ s}(a_i*)) \cdot \bigl [ \vec \theta_f (a_j+,Z)-\vec \theta_f(a_j-,Z)\bigr ] \, , $$ which is proportional to $\vec K_j(Z)$, hence $\Delta_g =\Delta_h$. If $j=i$ we may write $$ \prod_{s=0}^{p-1} (\epsilon z) (f^{\circ s}(a_i*))=\hat \epsilon \cdot Z(\gamma) \, , $$ and we shall take for simplicity $*=+$. In what follows we shall always assume $g$ and $h$ sufficiently close to $f$ in $P^1$. When $p+k \le M$, we have $$ \Phi(g^{\circ p+k} (a_i+)) = \Phi(g^{\circ k}(a_i+))\, . $$ [Using the figure, this is readily verified when $k$ is a multiple of $p$ and then in general.] Also, when $p+k \le M$ $$ \Phi(h^{\circ p+k} (a_i+)) = \Phi(f^{\circ k} (a_i-)) =\Phi(g^{\circ k} (a_i -)) $$ hence ($\modd \II_{M+1}$) $$ \eqalign { \delta \vec \theta(a_i+, Z) &:= \vec \theta_g(a_i+, Z) -\vec \theta_h (a_i+, Z)\cr &= \hat \epsilon Z(\gamma) \cdot \bigl [ \vec \theta_g (a_i+, Z) -\vec \theta_g (a_i -, Z) \bigr ]\cr &= \hat \epsilon Z(\gamma)\cdot 2 \vec K_i^{(g)} (Z) \, , } $$ where $\vec K^{(g)}_i$ denotes $\vec K_i$ computed with $g$ instead of $f$, so that $$ \eqalign { \delta \Delta(Z) &= \Delta_g (Z) - \Delta_h(Z) \cr &=\hat \epsilon Z(\gamma) \cdot \Delta_g(Z) \, , \cr } $$ hence $$ (1-\hat \epsilon Z(\gamma)) \Delta_g = \Delta_h $$ and finally $$ \Delta_g = \Delta_h /(1-\hat \epsilon Z(\gamma)) \, . $$ The proofs for $*=-$ would be similar. \qed \smallskip \remark{Remarks on Lemma 2.4} \roster \item The assumptions that $\Fix f^{\circ m}$ is finite for $1 \le m \le M$ and that no periodic point of period at most $M$ is neutral have been inserted for convenience. By using the definition of the zeta function with the $L(f_{\ell_m} \circ \cdots \circ f_{\ell_1})$ given after Lemma 2.2, one could do without them. \item By making use of Lemma 11.5 in Milnor-Thurston [1988], we may prove a version of Lemma 2.4 for $M=\infty$. If one wishes to construct a homotopy satisfying the assumptions of Lemma 2.4 in the case $M=\infty$, the method used in the proof of Theorem 1.1 can be replaced by the more abstract arguments of Milnor-Thurston [1988, Lemmas 11.7, 11.8]. \item Except to prove the case $i=j$ when $M=\infty$ (which we do not require in our proof of Theorem 1.1), where the above-mentioned Lemma 11.5 in Milnor-Thurston [1988] is crucial, we do not really need the $\CC^1$ topology (the hypothesis that the periodic orbits $f^{\circ p} (a_i*)=a_i$ are one-sided attracting is a topological one). (This does not contradict the remark on page 533 of Milnor-Thurston [1988], which is precisely concerned with the case $i=j$ and $M=\infty$.) \endroster \endremark \bigskip \Refs \ref \no 1 \by V. Baladi and G. Keller \paper Zeta functions and transfer operators for piecewise monotone transformations \jour Comm. Math. Phys. \yr 1990 \vol 127 \pages 459--477 \endref \ref \no 2 \by F. Hofbauer and G. Keller \paper Zeta-functions and transfer-operators for piecewise linear transformations \jour J. reine angew. Math. \vol 352 \pages 100--113 \yr 1984 \endref \ref \no 3 \by G. Keller and T. Nowicki \paper Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps \jour Comm. Math. Phys. \yr 1992 \vol 149 \pages 31--69 \endref \ref\no 4 \by D.H. Mayer \book The Ruelle-Araki transfer operator in classical statistical mechanics \bookinfo Lecture Notes in Physics Vol. 123 \publ Springer-Verlag \publaddr Berlin \yr 1980 \endref \ref \no 5 \by J. Milnor and W. Thurston \paper Iterated maps of the interval \inbook Dynamical Systems (Maryland 1986-87) \bookinfo Lecture Notes in Math. Vol. 1342 \ed J.C. Alexander \publ Springer-Verlag \publaddr Berlin Heidelberg New York \yr 1988 \endref \ref \no 6 \by M. Mori \paper Fredholm determinant for piecewise linear transformations \jour Osaka J. Math \vol 27 \yr 1990 \pages 81--116 \endref \ref \no 7 \by M. Mori \paper Fredholm matrices and zeta functions for piecewise monotonic transformations \inbook Dynamical systems and related topics (Nagoya 1990) \bookinfo Adv. Ser. Dyn. Syst., 9 \publ World Sci. Publishing \publaddr River Edge, NJ \yr 1991 \pages 388--400 \endref \ref \no 8 \by C. Preston \paper What you need to know to knead \jour Adv. Math. \vol 78 \yr 1989 \pages 192--252 \endref \ref \no 9 \by D. Rand \paper The topological class of Lorenz attractors \jour Math. Proc. Cambridge Philos. Soc. \vol 83 \yr 1978 \pages 451-460 \endref \ref \no 10 \by D. Ruelle \paper Analytic completion for dynamical zeta functions \paperinfo IHES preprint (1992) \endref \ref \no 11 \by R.F. Williams \paper The structure of Lorenz attractors \yr 1979 \pages 73--100 \jour Inst. Hautes Etudes Sci. Publ. Math. \vol 50 \endref \endRefs \enddocument %!PS-Adobe-2.0 EPSF-1.2 %%Creator: MATLAB, The Mathworks, Inc. %%Title: MATLAB graph %%CreationDate: 03/16/93 14:44:07 %%DocumentFonts: Times-Roman %%DocumentNeededFonts: Times-Roman %%DocumentProcessColors: Cyan Magenta Yellow Black %%Pages: 1 %%BoundingBox: 018 180 594 612 %%EndComments %%BeginProlog % MathWorks dictionary /MathWorks 120 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc /setrgbcolor ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % bounding box in default coordinates %initialize variables used in binded macros /llx 0 def /lly 0 def /urx 0 def /ury 0 def /bbox {/ury xdef /urx xdef /lly xdef /llx xdef} bdef % orientation switch /portraitMode (op) def /landscapeMode (ol) def /Orientation portraitMode def /portrait {/Orientation portraitMode def} bdef /landscape {/Orientation landscapeMode def} bdef % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize dpi2point mul 0 0 FontSize dpi2point mul neg 0 0] makefont def }bdef /setPortrait { 1 dpi2point div -1 dpi2point div scale llx ury neg translate } bdef /setLandscape { 1 dpi2point div -1 dpi2point div scale urx ury neg translate 90 rotate } bdef /csm {Orientation portraitMode eq {setPortrait} {setLandscape} ifelse} bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { % LineTo lineto stroke } bdef /MP { % MakePoly 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { % AddPoly {rlineto} repeat } bdef /PP { % PaintPoly closepath fill } bdef /DP { % DrawPoly closepath stroke } bdef /MR { % MakeRect 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { % FrameRect MR stroke } bdef /PR { % PaintRect MR fill } bdef /L1i { % Level 1 Image { currentfile picstr readhexstring pop } image } bdef /half_width 0 def /half_height 0 def /MakeOval { newpath /ury xstore /urx xstore /lly xstore /llx xstore /half_width urx llx sub 2 div store /half_height ury lly sub 2 div store llx half_width add lly half_height add translate half_width half_height scale .5 half_width div setlinewidth 0 0 1 0 360 arc } bdef /FO { gsave MakeOval stroke grestore } bdef /PO { gsave MakeOval fill grestore } bdef /PD { 2 copy moveto lineto stroke } bdef currentdict end def %dictionary %%EndProlog %%BeginSetup MathWorks begin % fonts for text, standard numbers and exponents %%IncludeFont: Times-Roman %line width, line cap, and joint spec 1 setlinecap 1 setlinejoin end %%EndSetup %%Page: 1 1 %%BeginPageSetup %%PageBoundingBox: 018 180 594 612 MathWorks begin bpage %%EndPageSetup %%BeginObject: graph1 1 bplot /dpi2point 12 def 0216 2160 7128 7344 bbox portrait csm /TR8 /Times-Roman 8 FMS /TR12 /Times-Roman 12 FMS 0 0 6912 5184 MR c np 6.00 setlinewidth % Colortable Dictionary /colortable 76 dict begin /c0 { 0 0 0 sc} bdef /c1 { 1 1 1 sc} bdef /c2 { 1 0 0 sc} bdef /c3 { 0 1 0 sc} bdef /c4 { 0 0 1 sc} bdef /c5 { 1 1 0 sc} bdef /c6 { 1 0 1 sc} bdef /c7 { 0 1 1 sc} bdef currentdict end def % Colortable colortable begin TR12 setfont c1 0 0 6912 5184 PR c0 26 0 26 1 26 1 26 2 26 2 26 2 26 3 25 4 26 4 26 4 26 5 26 6 26 5 26 7 26 6 26 8 26 7 26 9 26 8 26 9 26 10 26 10 26 10 25 11 26 12 26 12 26 12 26 13 26 13 26 14 26 14 26 15 26 15 26 16 26 16 26 16 26 17 26 18 26 18 25 18 26 19 26 20 26 19 26 21 26 20 26 22 26 21 26 23 26 22 26 23 26 24 26 24 26 24 26 25 26 26 25 26 26 26 26 27 26 27 26 28 26 28 26 29 26 29 26 30 26 30 26 30 26 31 26 32 26 32 26 32 26 33 25 33 26 34 26 35 26 34 26 36 26 35 26 36 26 37 26 37 1036 840 81 MP stroke 26 0 26 -1 26 -1 26 -2 26 -2 26 -2 26 -3 25 -4 26 -4 26 -4 26 -5 26 -6 26 -5 26 -7 26 -6 26 -8 26 -7 26 -9 26 -8 26 -9 26 -10 26 -10 26 -10 25 -11 26 -12 26 -12 26 -12 26 -13 26 -13 26 -14 26 -14 26 -15 26 -15 26 -16 26 -16 26 -16 26 -17 26 -18 26 -18 25 -18 26 -19 26 -20 26 -19 26 -21 26 -20 26 -22 26 -21 26 -23 26 -22 26 -23 26 -24 26 -24 26 -24 26 -25 26 -26 25 -26 26 -26 26 -27 26 -27 26 -28 26 -28 26 -29 26 -29 26 -30 26 -30 26 -30 26 -31 26 -32 26 -32 26 -32 26 -33 25 -33 26 -34 26 -35 26 -34 26 -36 26 -35 26 -36 26 -37 26 -37 1036 3826 81 MP stroke 1037 -933 1038 -933 1036 4199 3 MP stroke 1037 0 1038 0 1036 2333 3 MP stroke 0 -2333 0 -2333 3111 4666 3 MP stroke 26 -23 26 -24 26 -23 26 -23 1036 2426 5 MP stroke 26 23 26 24 26 23 26 23 1036 2240 5 MP stroke 26 -117 26 -116 3059 233 3 MP stroke 26 116 26 117 3111 0 3 MP stroke 1049 2614 mt (x) s 1036 4481 mt (diagonal) s 1088 631 mt (\() s 1244 631 mt (=-1\)) s 1166 655 mt (c) s 1166 596 mt (c) s 1166 550 mt (^) s 1088 2964 mt (\() s 1244 2964 mt (=+1\)) s 1166 2988 mt (c) s 1166 2929 mt (c) s 1166 2883 mt (^) s 2696 281 mt (f) s 2878 281 mt (\(x\)) s 2748 141 mt (o p) s 3163 2381 mt (a) s 3241 2474 mt (i) s 3137 2848 mt (\() s 3293 2848 mt (=-\)) s 3202 2871 mt (*) s 5166 5025 mt ( ) s 3823 2351 mt -90 rotate( ) s 90 rotate 26 -37 26 -37 26 -36 26 -35 26 -36 26 -34 25 -35 26 -34 26 -33 26 -33 26 -32 26 -32 26 -32 26 -31 26 -30 26 -30 26 -30 26 -29 26 -29 25 -28 26 -28 26 -27 26 -27 26 -26 26 -26 26 -26 26 -25 26 -24 26 -24 26 -24 26 -23 26 -22 26 -23 25 -21 26 -22 26 -20 26 -21 26 -19 26 -20 26 -19 26 -18 26 -18 26 -18 26 -17 26 -16 26 -16 25 -16 26 -15 26 -15 26 -14 26 -14 26 -13 26 -13 26 -12 26 -12 26 -12 26 -11 26 -10 26 -10 25 -10 26 -9 26 -8 26 -9 26 -7 26 -8 26 -6 26 -7 26 -5 26 -6 26 -5 26 -4 26 -4 26 -4 25 -3 26 -2 26 -2 26 -2 26 -1 26 -1 26 0 4147 2333 81 MP stroke 26 37 26 37 26 36 26 35 26 36 26 34 25 35 26 34 26 33 26 33 26 32 26 32 26 32 26 31 26 30 26 30 26 30 26 29 26 29 25 28 26 28 26 27 26 27 26 26 26 26 26 26 26 25 26 24 26 24 26 24 26 23 26 22 26 23 25 21 26 22 26 20 26 21 26 19 26 20 26 19 26 18 26 18 26 18 26 17 26 16 26 16 25 16 26 15 26 15 26 14 26 14 26 13 26 13 26 12 26 12 26 12 26 11 26 10 26 10 25 10 26 9 26 8 26 9 26 7 26 8 26 6 26 7 26 5 26 6 26 5 26 4 26 4 26 4 25 3 26 2 26 2 26 2 26 1 26 1 26 0 4147 2333 81 MP stroke 1037 -933 1037 -933 4147 2333 3 MP stroke 1037 0 1037 0 4147 2333 3 MP stroke 0 -2333 0 -2333 4147 4666 3 MP stroke 26 23 26 24 26 23 26 23 6117 2240 5 MP stroke 26 -23 26 -24 26 -23 26 -23 6117 2426 5 MP stroke 26 -117 26 -116 4095 233 3 MP stroke 26 116 26 117 4147 0 3 MP stroke 6143 2614 mt (x) s 5703 281 mt (diagonal) s 5703 1798 mt (\() s 5858 1798 mt (=+1\)) s 5780 1821 mt (c) s 5780 1763 mt (c) s 5780 1716 mt (^) s 5703 4131 mt (\() s 5858 4131 mt (=-1\)) s 5780 4154 mt (c) s 5780 4096 mt (c) s 5780 4049 mt (^) s 4199 281 mt (f) s 4380 281 mt (\(x\)) s 4251 141 mt (o p) s 3991 2381 mt (a) s 4069 2474 mt (i) s 3758 2848 mt (\() s 3914 2848 mt (=+\)) s 3836 2871 mt (*) s end % pop colortable dictionary eplot %%EndObject graph 1 epage end showpage %%Trailer %%EOF ENDBODY