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\def \Fix{{\text{Fix}\,}}
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\def\II{{\Cal I}}
\def\JJ{{\Cal J}}
\def\KK{{\Cal K}}
\def\LL{{\Cal L}}
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\def\NN{{\Cal N}}
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\topmatter
\title
Infinite kneading matrices \\
and weighted zeta functions \\
of interval maps\\
\endtitle
\author Viviane Baladi
\endauthor
\address
CNRS, UMR 128, UMPA, ENS Lyon, 46, all\'ee d'Italie, F-69364 Lyon Cedex 07,
France
\endaddress
\address
Present address (on leave from CNRS): ETH Z\"urich,
CH-8092 Z\"urich, Switzerland
\endaddress
\email
baladi\@math.ethz.ch \endemail
\date{January 1994 (revised version)}
\enddate
\subjclass
58F20 58F03 58F19 47B10
\endsubjclass
\abstract
{We consider a piecewise continuous, piecewise monotone interval map and
a weight of bounded variation,
constant on homtervals and continuous at
periodic points of the map. With these data we associate
a sequence of weighted Milnor-Thurston kneading matrices, converging to a
countable matrix with coefficients analytic functions. We
show that the determinants of these
matrices converge to the inverse of the
correspondingly
weighted zeta function for the map. As a corollary, we obtain
convergence of the discrete spectrum of
the Perron-Frobenius operators of piecewise linear approximations
of Markovian, piecewise expanding and piecewise $C^{1+BV}$ interval maps.}
\endabstract
\endtopmatter
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\document
\head Introduction
\endhead
Let $f$ be a transformation of a compact interval, say $I=[0,1]$.
Assume that $f$ is piecewise continuous, and
piecewise strictly monotone,
with a finite number $N$ of pieces
defined by turning points
$0 =a_0 < a_1 < a_2 < \ldots < a_{N-1} < a_N=1$. Let
$g :I \to \complex$ be a weight
(a natural choice is $g = 1/|f'|$, if $f$ is piecewise differentiable
and without critical points).
If each iterate $f^n$ has only finitely fixed points, we can formally define
the weighted Ruelle zeta function
$$
\zeta_g(t) = \exp \sum_{n \ge 1} {t^n \over n}
\sum_{ x \, : \, f^n x = x }
\prod_{k=0}^{n-1} g(f^k x) \, . \tag0.1
$$
In Section 1 we shall introduce a {\it reduced zeta function} which can
also be defined when some iterate(s) of $f$ have infinitely many
fixed points. We shall also denote it by $\zeta_g(t)$,
no confusion should arise since all our results are about the reduced
function.
If the weight $g$ is of bounded variation,
the analytic properties of $\zeta_g(t)$ in some domain
of the complex plane can often be studied with the help of the
transfer operator $\LL_g$ defined by
$$
\LL_g \varphi (x)
= \sum_{y \, : \, f y = x}
g(y) \cdot \varphi (y) \tag0.2
$$
acting on the Banach space $BV$ of functions $\varphi : I \to
\complex$ of bounded
variation
(see Baladi-Keller [1990] for the case
where the partition is generating, and
Ruelle [1993a, 1993b] for more general results). In particular, one shows
under suitable assumptions that the essential spectral radius of $\LL_g$
is not larger than $\rho$, and that
$\zeta_g(t)$ admits a meromorphic extension to the open disc of
radius $1/\rho$, where
$$
\rho = \rho(g) = \lim _{n \to \infty}
( \sup _x |\prod_{k=0}^{n-1} g(f^k x)|)^{1/n} \, .\tag0.3
$$
Furthermore, the poles of $\zeta_g(t)$ in this disc are exactly
the inverses of eigenvalues of $\LL_g$.
\medskip
Zeta functions may also be studied
via the approach initiated by Milnor and Thurston [1988]
who express a slightly different reduced
(unweighted) zeta function as the determinant of a finite
``kneading matrix,'' whose coefficients are
power series. In a previous
paper (Baladi-Ruelle [1993]), we defined weighted
kneading matrices and determinants,
and extended Milnor and Thurston's main result
to the case of reduced weighted zeta functions $\zeta_g$ {\it under the strong assumption that the map $g$ be
constant on each interval} $[a_{i-1},a_i]$. With this assumption,
the weighted kneading matrix is again a finite matrix.
In the present paper we extend this result to the case
where the weight $g$ is of bounded variation. (We need
technical assumptions on $g$: it should
be constant in homtervals,
and continuous at the periodic points of $f$.) In this case, one has
to consider infinite (countable) kneading matrices.
More precisely, we construct
a sequence of finite
weighted kneading matrices
$K^{(n)}(t)$, of strictly increasing sizes, corresponding
to better and better locally constant approximations $g^{(n)}$ of $g$.
Set $\rho^{(n)}=\rho(g^{(n)})
\le \rho$,
and write $\bar \rho = \lim_{n \to \infty} \rho^{(n)}$.
By the
previous result in Baladi-Ruelle [1993] the kneading determinant
$\Delta^{(n)}(t) = \det (K^{(n)}(t))$, viewed as a power series,
coincides with $1/\zeta_{g^{(n)}}(t)$ up to a trivial factor.
(By slightly changing the kneading matrices,
we are now able to replace this trivial factor by the constant $1$.)
Also, the series $\Delta^{(n)} (t)$ converges in the
disc of radius $1/{\rho^{(n)}}$.
Because of the assumptions on $g$,
the power series $\zeta_{g^{(n)}}(t)$ converges to $\zeta_g(t)$ as
$n$ goes to infinity, and the analytic function
defined by $\zeta_{g^{(n)}}(t)$ converges
to $\zeta_g(t)$ on some nonempty open disc. From this, it follows that
$\lim_{n \to \infty} \Delta^{(n)}(t)$ exists and defines an
analytic function $\Delta(t)$,
in this disc.
However, the properties of the limits of the meromorphic
functions $\zeta_{g^{(n)}}(t)$
and of the analytic functions $\Delta^{(n)} (t)$ on the (possibly larger)
disc of radius $1/\bar\rho$,
are not clear a priori:
we must show that
the analytic functions $\zeta_{g_n}^{-1}(t) = \Delta^{(n)}(t)$ are uniformly
bounded iby applying the classical Hadamard inequality
to matrices $M^{(n)}(t)=
1-L^{(n)}(t)$, where $1$ is the identity
matrix and $L^{(n)}(t)$ is obtained from
$K^{(n)}(t)$ by elementary row
operations. This yields
our main result: the two
functions $\Delta(t)$ and $1/\zeta_g(t)$
admit analytic extensions to the disk $D$ of radius $1/\bar \rho$,
where they coincide, and $\zeta_{g_n}(t)$ converges
to $\zeta_g(t)$ as a meromorphic function in $D$.
A precise statement is given in Section 1. Section 2 contains the proofs.
\smallskip
In the cases where the relationship between the spectrum
of the transfer operator and the properties of the zeta function have
been established (for example, when the partition into
intervals of monotonicity is generating, see Baladi--Keller [1990]), our main result
implies that in the domain
$\rho < |z|$ the spectrum of the transfer operator $\LL_{g^{(n)}} : BV
\to BV$ converges to that of $\LL_g:BV \to BV$.
{\it If we assume further that the map $f$ has a finite Markov partition}
(i.e., all the turning points are (pre)periodic), each coefficient
of the finite matrix $K^{(n)}(t)$ is by definition
a polynomial in $t$ multiplied
by a geometric series in (a power of) $t$, and the matrix
is therefore a ``finite'' object. Also,
the Markov property implies that the transfer operator
preserves a finite-dimensional vector space on which
it can be described by a finite matrix. In the Markov case,
our results hence yield two
procedures to approach the discrete spectrum of the transfer operator
acting on $BV$.
Under more restrictive assumptions
(the map $f$ is piecewise $\CC^2$, the weight is
$g= 1/|f'|$, and $\rho < 1$), Mori
[1990, 1992] has introduced ``Fredholm'' matrices $\Phi_n(t)$,
and has
related the limit of the determinants of these matrices to the
correspondtransfer operators (acting on $BV$) and weighted zeta functions.
Mori uses different techniques and does not relate his results
with the Milnor-Thurston theory. However, he is able to
define Banach spaces on which his limiting matrix $\Phi(t)$ defines
a bounded operator, and to show that if the limit of the determinants
has a zero at $t$, the countable matrix $\Phi(t)$ has a fixed point.
In our case, the limit $K^{(\infty)}(t)$ of the matrices $K^{(n)}(t)$
is also well defined, and we conjecture that $K^{(\infty)}(t)$ has
a fixed point in a suitable Banach space whenever $\Delta(t)$
vanishes.
\medskip
I am much indebted to H.H. Rugh who pointed out to me the key lemma
in Section ~2 (and how to prove it). I am grateful to B.~Schmitt and
S.~Isola who
mentioned to me the conjecture of S.~Isola that is solved
in the present paper. I am very thankful towards H.H.~Rugh and D.~Ruelle
for pointing out several mistakes
in a previous version of this text and suggesting improvements.
Part of this work was done at
the D\'epartement de Math\'ematiques de l'\'Ecole Polytechnique
F\'ed\'erale de Lausanne, the Niels Bohr Institute (Copenhagen), the
Instituto de Matem\'atica of the
Universidade Federal do Rio de Janeiro, the
Instituto de Matem\'atica Pura e Aplicada (Rio de Janeiro),
and the Landau Institute for Theoretical Physics (Moscow).
I am grateful to these
institutions for their hospitality and financial support.
\head 1. Definitions and statement of results
\endhead
\subhead The basic ingredients: the map $f$ and the weight $g$
\endsubhead
\smallskip
We start by fixing notation, which we have tried
to keep essentially consistent with Baladi-Ruelle [1993].
Let $1 \le N < \infty$, let $a_0 < a_1 < \ldots < a_N$, and
let $f_i : [a_{i-1}, a_i] \to
[a_0,a_N]$ be strictly monotone and continuous maps for $i=1, \ldots, N$.
(The intervals are not necessarily maximal intervals
of monotonicity or continuity.)
We write $f = (f_1, \ldots, f_N)$. By abuse of language we also denote
by $f$ the multivalued transformation of $[a_0,a_N]$
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, and
we define a
function $\epsilon$ on $[a_0, a_N]$ such that it has the constant
value $\epsilon_i$ on $(a_{i-1},a_i)$ and the value $0$
on $\{ a_0, a_1, \ldots, a_N\}$.
Let $g =(g_1, \ldots, g_N)$
where each $g_i : [a_{i-1}, a_i] \to \complex$ is a function
of bounded variation.
We also view $g$ as a multivalued function on $[a_0, a_N]$.
Denote by $\ZZ=\ZZ_1$
the ``partition'' of $I$ into $N$ closed intervals
given by the $a_i$. For $n \ge 2$, let
$\ZZ_n$ be the $n^{\text{th}}$ refinement of $\ZZ$ under $f$, i.e.,
the partition into intervals
$\eta_0 \cap f^{-1} \eta_1 \cap \ldots \cap f^{-n+1} \eta_{n-1}$ with
$\eta_j \in \ZZ$. Recall that the partition
$\ZZ$ is called {\it generating} if the maximal length of the intervals
in $\ZZ_n$ tends to zero as $n$ tends to infinity (this
is an intrinsic property of the map $f$ which does not vary
by considering a finer partition $\ZZ'$). We define
a {\it $\ZZ$-homterval} to be a maximal nontrivial
interval $J$ such that
for each $n \ge 1$ there is $\eta \in \ZZ_n$ with $J \subset \eta$.
(The partition is generating if and only if there are no $\ZZ$-homtervals.)
We make the additional assumption that $g$
is constant on each $\ZZ$-homterval of $f$.
\smallskip
\subhead The locally constant approximations $g^{(n)}$
\endsubhead
\smallskip
For $n \ge 1$,
and $\eta \in \ZZ_n$, let $\{ x_{\eta,m}, m \ge 1 \} \subset \eta$
be such that $\lim_{m \to \infty}
|g(x_{\eta,m)}| = \text{ess}\, \inf_\eta |g|$,
and such that $\lim_{m\to \infty}
g(x_{\eta,m})$ existsmultivalued function
whose restriction to each closed interval $\eta \in \ZZ_n$ is equal
to the constant $\lim_{m\to \infty} g(x_{\eta,m})$.
The constants $\rho=\rho(g)$ and
$\rho^{(n)}= \rho(g^{(n)})\le \rho$ are
defined as in Equation \thetag{0.3},
and we write $\bar \rho = \lim_{n \to \infty} \rho^{(n)}$.
If $g$ is (piecewise) continuous, $\bar \rho=\rho$.
\smallskip
\subhead The kneading matrices $K^{(n)}(t)$,
$L^{(n)}(t)$, the kneading determinants
$\Delta^{(n)}(t)$
\endsubhead
\smallskip
We introduce a sequence of finite kneading matrices $K^{(n)}(t)
=K^{(n)}(g)(t)$ of
sizes $(N_n+1) \times (N_n +1)$,
where $N_n+1$ is the number of endpoints
of the partition $\ZZ_n$. Let us start with $K^{(1)}(t)$, which
is essentially the matrix considered in Baladi-Ruelle [1993] (we
add two rows and two columns in order to suppress
the ``trivial factor''
mentioned in the introduction). We need some preliminary
definitions:
the {\it first address} of $x \in [a_0, a_N]$ is the vector
$$
\vec \alpha^{(1)} (x) = (\sgn (x-a_0), \ldots,
\sgn (x-a_{N})) \in \{-1,1,0\}^{N+1}\, ;
$$
the {\it first weighted invariant
coordinate} of $x\in [a_0,a_N]$
is the $(N+1)$-tuple of power series
$$
\vec \theta^{(1)} (x)=\vec \theta^{(1)}_f (g) (x) (t) =
\sum_{m=0}^\infty t^m\cdot
\bigl ( \prod_{k=0}^{m-1} (\epsilon g^{(1)}) (f^k x) \bigr )
\cdot \vec \alpha^{(1)}(f^ m x)
\in \complex [[t]]^{N+1} \, .
$$
Note that $\vec \theta^{(1)} (x)$ is single-valued as a function
of $x$ (because if
$f^k x \in \{a_0, \ldots, a_N\}$ for some $k \le m$,
then $\epsilon (f^m x)=0$).
Writing
$$
\phi(a\pm) = \lim \phi(x) \text{when } x \downarrow a \, ,
\text{ respectively } x \uparrow a\, ,
$$
and setting
$$
\vec \theta^{(1)} (a_0 -) = (-1, \ldots, -1) \, , \qquad
\vec \theta^{(1)} (a_N +) = (1, \ldots, 1) \, ,
$$
we define the rows of the
$$
\eqalign
{
\vec K_i^{(1)} (t)&=
{1\over 2}
\biggl [
\vec \theta^{(1)}(a_i +) -
\vec \theta^{(1)} (a_i -) \biggr ]\cr
&=
(K_{i,0} ^{(1)}, \ldots, K_{i,N}^{(1)} ) \, , \, \,
\text{ for } i=0, \ldots, N \, . \cr
}
$$
The determinant of $K^{(1)}(t)$
$$
\Delta ^{(1)}(t) =\Delta_f^{(1)} (g) (t)= \det
\bigl [ K_{ij}^{(1)} (t) \bigr ] \in
\complex[[t]]
$$
is called the {\it first kneading determinant}. Since
$K_{ij}^{(1)} (t)= \delta_{ij} + \text{higher order
in } t$, we
have $\Delta^{(1)} (t)= 1 + \text{higher order in } t$. By
the definition of $\rho^{(1)}$, the power
series $\Delta^{(1)}(t)$ converges
in the disc of radius $1/\rho^{(1)} \ge 1/\rho$.
We now give the general step for the construction
of the {\it $n^{\text{th}}$ kneading matrix}.
We first order the $N+1$ endpoints of the partition
$\ZZ_1$ according to $a_0 < \ldots < a_N$. For $n \ge 2$, we order
the $N_n+1$ endpoints
$a_0^{(n)} < \ldots < a_{N_n}^{(n)}$
of the partition $\ZZ_n$ inductively:
we decompose this set of endpoints into $A \cup B$,
where $A$ is the set of endpoints of the
partition $\ZZ_{n-1}$, we keep the order on $A$ given
by the preceding inductive step, order the points
in $B$ according to their position on the real line, and
impose $\max A \prec \min B$. Let $\pi$ denote the
permutation on $\{ 0, \ldots, N_n\}$ induced by this reordering
of the endpoints:
$$
a^{(n)}_{\pi(0)} \prec a^{(n)}_{\pi(1)} \prec \cdots \prec a^{(n)}_
{\pi(N_n)} \, .
$$
The permutation $\pi$ allows us to handle the bookkeeping of
rows and columns inductively.
We define the {\it $n^{\text{th}}$ sign function}
$\epsilon^{(n)} : I \to \{ -1,1,0\}$ which vanishes at the points
$a_i^{(n)}$ and otherwise coincides with $\epsilon$, and
the {\it $n^{\text{th}}$ address} of $x \in I$ by
$$
\vec \alpha^{(n)} (x) \sgn (x-a_{\pi(N_n)}^{(n)}) ) \in \{-1,1,0\}^{N_n+1}\, .
$$
Similarly
as above, using $\epsilon^{(n)}$,
$\vec \alpha^{(n)}$, and $g^{(n)}$, we define the
{\it $n^{\text{th}}$ weighted invariant coordinate}
$$
\vec \theta^{(n)} (x) \in \complex [[t]]^{N_n+1} \, ,
$$
and the rows of the
{\it $n^{\text{th}}$ $(N_n+1) \times (N_n+1)$
kneading matrix}, for $i=0, \ldots, N_n$:
$$
\vec K_i^{(n)} (t)=
{1\over 2}
\biggl [
\vec \theta^{(n)}(a_{\pi(i)} +) -
\vec \theta^{(n)} (a_{\pi(i)} -) \biggr ]\in
\complex [[t]]^{N_n+1} \, ,
$$
and finally the
{\it $n^{\text{th}}$ kneading determinant}
$$
\Delta ^{(n)}(t) = \det \bigl [ K_{ij} ^{(n)} (t) \bigr ] \in
\complex [[t]] \, .
$$
The power series $\Delta^{(n)}(t)$ converges in the disc of
radius $1/\rho^{(n)}$.
The permutation $\pi$
on the rows and columns has
no effect on the determinant of the matrix, and
it allows us to define a countable
matrix $K^{(\infty)}(t)$: for each finite index pair
$(i,j)$, we define the {\it infinite kneading matrix}
$$
K^{(\infty)}_{ij}(t)= \lim_{n \to \infty} K^{(n)}_{ij}(t)
$$
(if the partition is
generating, this limit exists because $g$ is of bounded variation;
when it is not generating, use also the fact that $g$ is
constant on $\ZZ$-homtervals).
In the proof of our main theorem (see \thetag{2.5,2.6,2.7}) we shall perform
elementary row operations on the sequence $K^{(n)}(t)$ to
obtain another sequence of kneading matrices
$L^{(n)} (t)$ (with $\det L^{(n)}(t)=\det K^{(n)}(t)$), converging to
a countable matrix $L^{(\infty)}(t)$.
Note that, although
the countable matrix
$M^{(\infty)}(t_0)= 1-
L^{(\infty)}(t_0)$, for $t_0$ in the disc of radius
$1/\bar \rho$,
defines a Fredholm operator in the sense of Grothendieck [1956]
on the Banach space $l^1(\natural)$, and hence the determinant of
$L^{(\infty)}(t_0)$ is a well-dobject (see the Appendix), this determinant does {\it not} coincide in general with
$\Delta(t_0)$ (we give a counter-example after the proof of the main
theorem in Section 2).
\smallskip
\subhead The reduced zeta function $\zeta_g(t)$
\endsubhead
\smallskip
For the convenience of the reader, we recall the definition
of the
weighted {\it reduced zeta function} associated with $f$,
introduced in Baladi-Ruelle [1993] for a locally
constant weight $g$.
Denote by $\Fix f^m$ the set of fixed points of $f^m$
which have an orbit
disjoint from the endpoints
$\{ a_0, \ldots, a_N\}$ of the partition
$\ZZ$. Since a periodic orbit is
disjoint from $\{ a_0, \ldots, a_N\}$ if and only if it is disjoint from the
endpoints of all partitions $\ZZ_n$ for $1 \le n$,
the set $\Fix f^m$ is invariant under the successive refinements of
the partition.
We first assume that
for each $m$ the set $\Fix f^m$ is finite (we shall see below
how to remove this assumption).
For $x \in \Fix f^m$, we introduce a Lefschetz index:
$$
L (x,f^m)=
\cases
0 & \text{if the graph
of $f^m$ doesn't cross the diagonal at $x$,}\cr
\lim_{y\to x} { \sgn(f^m
y-y) \over \sgn(x-y)} &\text{otherwise,} \cr
\endcases
$$
and we set
$$
\nu(x,f^m) = -L(x,f^m) \cdot \prod_{k=0}^{m-1}
\epsilon(f^k x) \in \{-1,1,0 \}\, .
$$
(If the graph of
$f^m$ crosses the diagonal at $x \in \Fix f^m$,
the point $x$ is either attracting or repelling, and
$\nu (x,f^m) = -1$ if and only if $f^m$ is increasing and
attracting at $x$.)
We extend now the set $\Fix f^m$ to a set $\Fixs f^m$ containing
{\it all} the periodic orbits,
obtained by adding
symbols $x *$ where $x \in [a_0, a_N]$, and $*$ is $+$ or $-$:
$$
\eqalign
{
\Fixs f^m &=
\Fix f^m \bigcup \{ x* \, : \, f^m(x*)= x \, , \cr
&\qquad\qquad \exists \, k, i \text{ with } f^k (x*) = a_i \text{ and }
\prod_{s=0}^{m-1} \epsilon(f^s(x}
$$
Again, $\Fixs f^m$ is invariant under refinements of $\ZZ$.
For $x* \in \Fixs f^m \setminus \Fix f^m$, let
$$
L(x*,f^m) = \cases
0 &\text{if $x*$ is (one-sided) repelling} \cr
1&\text{if $x*$ is (one-sided) attracting} \cr
\endcases
$$
and $\nu(x*,f^m) = -L(x*,f^m)$.
The {\it reduced zeta function} (Baladi-Ruelle [1993]) is:
$$
\zeta_g(t) =
\exp
\sum_{m=1}^\infty {t^m \over m}
\sum_{x \in \Fixs f^ m}
\nu(x,f^m) \cdot \prod_{k=0}^{m-1} g(f^k x) \, .
$$
If all periodic points are repelling, and if
$f^m a_i* \ne a_i$, for all $m \ge 1 $ and
$1 \le i \le N-1$, we recover
the usual (weighted) zeta function \thetag{0.1}.
\smallskip
Removing the assumption that $\# \Fix f^m< \infty$,
we introduce another
definition of the reduced zeta function
(again following Baladi-Ruelle [1993]).
For $m \ge 1$ we 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
\smallskip
When $\Fix f^m$ is finite we have thus
$$
\eqalign
{
\sum_{x \in \Fixs f^m}
\nu(x,f^m) &=
\sum_{\ell_1, \ldots, \ell_m} \,
\sum_{x \in \Fixs f_{\ell_m} \circ \cdots \circ f_{\ell_1}}
\nu(x,f_{\ell_m} \circ \cdots \circ f_{\ell_1})&=
-\sum_{\ell_1, \ldots, \ell_m} (\epsilon_{\ell_1} \cdots \epsilon_{\ell_m})
\cdot L(f_{\ell_m} \circ \cdots \circ f_{\ell_1}) \, ,\cr
}
$$
and it is natural to define:
$$
\tilde \zeta_g(t) =
\exp -
\sum_{m=1}^\infty {t^m \over m}
\biggl ( \sum_{\ell_1, \ldots, \ell_m} (\epsilon_{\ell_1} \cdots \epsilon_{\ell_m})
L(f_{\ell_m} \circ \cdots \circ f_{\ell_1})
\cdot g_{\ell_1, \ldots, \ell_m} \biggr ) \, ,
$$
where for any $\ell_1, \ldots, \ell_m$ such that
$L(f_{\ell_m} \circ \cdots \circ f_{\ell_1}) \ne 0$,
we set $g_{\ell_1, \ldots, \ell_m} = g_\eta$,
where $\eta \in \ZZ_m$ is the interval corresponding
to $\ell_1, \ldots, \ell_m$, and $g_\eta$ is
chosen arbitrarily if $\Fix f^m \cap \eta$ is empty,
and equal to the constant value
of $g$
on $\Fix f^m \cap \eta$ otherwise (if $\ZZ$ is generating
this set contains at most one point, otherwise use the fact that $g$
is constant on $\ZZ$-homtervals).
If $\Fix f^m$ is finite for all
$m \ge 1$, we have
$\tilde \zeta_g (t) = \zeta_g(t)$. We set
$\zeta_g(t) := \tilde \zeta_g(t)$ if $\Fix f^m$ is infinite for
some $m$.
\medskip
\subhead The results
\endsubhead
\smallskip
\noindent We can now state our main result.
\proclaim{Theorem} Let $f$ and $g$ be as defined in the
first paragraph of this section.
With the above notations we have:
\roster
\item
For each $n \ge 1$, the following equality
holds between analytic
functions in the open disc $D_n$ of radius $1/\rho^{(n)}$:
$$
\Delta ^{(n)}(t) ={1 \over \zeta_{g^{(n)}} (t)}\, .\tag1.1
$$
If the partition $\ZZ$ is generating, the zeroes of
$\Delta^{(n)}(t)$ in $D_n$
are exactly the inverses of the eigenvalues of the operator
$\LL_{g^{(n)}} : BV \to BV$
$$
(\LL_{g^{(n)}}\varphi ) (x) = \sum_{f y = x} \varphi (y) \cdot g^{(n)} (y) \, ,
$$
outside of the closed disc of radius $\rho^{(n)}$.
The order of a the corresponding eigenvalue.
\endroster
Assume additionally that $g$ is continuous at each point
of $\Fix f^m$, and one-sided continuous
at each point of $\Fixs f^m \setminus \Fix f^m$
for all $m \ge 1$.
Then:
\roster
\item[2]
The sequence $\Delta^{(n)} (t)$ of analytic
functions in $D_n$
converges
to a function $\Delta (t)$, analytic in the open disc
$D$ of radius
$1/\bar \rho$.
We have the
following identity between analytic functions in $D$:
$$
\Delta (t)={1 \over \zeta_g(t) } \, .
$$
In particular
$\zeta_g(t)$ is meromorphic in $D$, and
the set of poles of $\zeta_{g^{(n)}}(t)$ in $D$ converges to
the set of poles of $\zeta_g(t)$ in $D$.
If the partition $\ZZ$ is generating, the zeroes of
$\Delta(t)$ in the open disc of radius $1/\rho$
are exactly the inverses of the eigenvalues of the operator
$\LL_{g} : BV \to BV$ outside of the closed disc of radius $\rho$.
The order of a zero coincides with the multiplicity of
the corresponding eigenvalue.
\endroster
\endproclaim
\demo{Remarks}
\roster
\item
It may happen that both members
of \thetag{1.1} admit analytic extensions to larger
domains, possibly the whole complex plane,
in particular when the map $f$ has a finite Markov
partition.
\item
If $\ZZ$ is generating,
the fact that
the zeta function $\zeta_g(t)$ admits a meromorphic
extension to the
disc $D$ of radius $1/\rho$
was obtained by Baladi-Keller [1990]
(see Baladi-Ruelle [1994]
for the removal of the piecewise continuity assumption on $g$) by
relating the meromorphic properties of $\zeta_g(t)$
to the spectral properties of $\LL_g :BV \to BV$.
In particular, they prove
that the essential spectral radius of $\LL_g$ (or $\LL_{g^{(n)}}$) acting
on $BV$ is bounded above by $\rho$ (respectively $\rho^{(n)}$).
The claims of the above theorem
relating the zeroes of the determinantspectrum of $\LL_g$ when $\ZZ$
is generating follow essentially
from these previous results.
\item
When the partition $\ZZ$ is not
necessarily generating, Ruelle [1993a, 1993b] introduces
a different reduced zeta function, $\hat \zeta_g(t)$, using
the notion of a representative set of
periodic points. He relates the poles of
$\hat \zeta_g(t)$ to the discrete spectrum
of $\LL_g$. The two reduced zeta functions
$\zeta_g(t)$ and $\hat \zeta_g(t)$ in general do not coincide.
When the weight $g$ is locally constant,
Ruelle [1993c] obtains in some cases
a relationship between the spectrum of $\LL_g$
and the poles of $\zeta_g(t)$. It would be interesting to see
whether this holds also for weights $g$ of bounded variation,
with the aims of suppressing the assumption that $\ZZ$ is generating
in all statements of our theorem, and comparing the
zeta functions $\zeta_g(t)$ and $\hat \zeta_g(t)$.
\item
The assumption that $g$ is (one-sided) continuous at periodic points of
$f$ can be relaxed to the requirement that the intersection
of the set of discontinuities of $g$ with the periodic points
of $f$ is finite. The equation $\Delta (t) \cdot \zeta_g(t) =1$
must then be corrected
by a finite product.
\endroster
\enddemo
\noindent We have the following
positive answer to a conjecture of S. Isola:
\proclaim{Corollary}
Let $f$ and $g$ be as in the first paragraph of this section.
Assume further that each $g_i$ is continuous,
and that the partition $\ZZ$ is
generating and Markovian,
i.e., $f(a_i\pm) \in \{ a_0, \ldots, a_N \}$ for all $i$.
For $n \ge 1$, let $f^{(n)}:I\to I$ be a
(possibly $\ZZ_n$-multivalued) transformation
which is monotone on each interval of $\ZZ$,
and which coincides with $f$ at the endpoints
of the intervals of $\ZZ_n$.
Write $\zeta_g$, $\zeta_{g^{(n)}}$, and
$\zeta_{g^{(n)},functions of $f$ and $g$, of $f$ and $g^{(n)}$, and
of $f^{(n)}$ and $g^{(n)}$.
Write $\LL_g$,
$\LL_{g^{(n)}}$, and
$\LL_{g^{(n)},f^{(n)}}$ for the corresponding transfer operators
acting on $BV$.
Then:
\roster
\item
As power
series, $\zeta_{g^{(n)}} = \zeta_{g^{(n)}, f^{(n)}}$. Thus,
in the open disc $D$ of radius $1/ \rho$,
the poles of $\zeta_{g^{(n)}, f^{(n)}}$
and $\zeta_{g^{(n)}}$ converge to the
poles of $\zeta_g$.
\item
Let $L_n$ be the restriction
of $\LL_{g^{(n)},f^{(n)}}$ to the finite-dimensional vector space $E_n$
of functions $\varphi : I \to \complex$ which are constant
on the intervals of $\ZZ_n$.
Outside of the closed disc
of radius $\rho$ the following spectra coincide
(including the multiplicities) and
converge to the spectrum of
$\LL_g$ as $n \to \infty$:
\itemitem {a)} that of $L_n$;
\itemitem {b)} that of $\LL_{g^{(n)}, f^{(n)}}$;
\itemitem {c)} that of $\LL_{g^{(n)}}$.
\endroster
\endproclaim
The interest of the above corollary
is that we approach the discrete spectrum of $\LL_g$
by the spectra of the finite matrices
$L_n$, or by the zeroes of the determinants associated with $f$
(or $f^{(n)}$)
and $g^{(n)}$
(which, under the assumptions of the corollary, are ``finite''
objects, as pointed out in the introduction). Of course, this is only
really useful
when $\rho$ is strictly smaller than the spectral radius of $\LL_g$.
One case where this holds is when $f$ is piecewise $C^{1+BV}$,
the weight $g=1/|f'|$, and
$|f'| \ge \lambda > 1$.
One can then choose $f^{(n)}$ to be the piecewise
affine approximations of $f$ which coincide with
$f$ at the endpoints of $\ZZ^n$ (and are affine on these
intervals). In that case, since $g$ is piecewise
continuous, and ${f^{(n)}}'$ is piecewise constant,
one can set $g^{(n)}=1/{|f^{(n)}}'|$, by the mean value theorem.
Note that our result doesthe eigenspaces corresponding to the discrete spectrum of $\LL_g$.
\head 2. Proofs
\endhead
\noindent The main ingredient of our proof
of the theorem is the following lemma:
\proclaim{Key Lemma}
Let $M^{(n)}$, $n\ge 1$ be a sequence of $m_n \times m_n$
matrices ($1 \le m_n < m_{n+1} < \infty$)
with complex coefficients. Write
$1_{m_n}$ for the $m_n \times m_n$ identity
matrix. Assume that
there exist a constant $V > 0$, and
for each $n \ge 1$ an $m_n$-dimensional vector
$v^{(n)}= (v^{(n)}_1, \ldots, v^{(n)}_{m_n}) \in \real_+^{m_n}$ with
$$
\cases
\sum_{i=1}^{m_n} v^{(n)}_i \le V\, , & \cr
|M^{(n)}_{i j}| \le v^{(n)}_i \, , & \forall \, 1 \le i,j \le m_n \, .
\endcases \tag2.1
$$
Then for each $K > 0$, there exists
a constant $C(K,V)$ such that
the determinants $d^{(n)} (\lambda):= \det (1_{m_n} -
\lambda M^{(n)})$ satisfy $|d^{(n)} (\lambda)| \le C(K,V)$,
for all $\lambda$ in the disc of radius $K$ and $n \ge 1$.
In particular, $|d^{n} (1)|$ is bounded by an expression
depending
only on $V$.
\endproclaim
\demo{Proof of the lemma}
\roster
We follow the classical
Fredholm argument (see e.g.
Riesz-Sz.-Nagy [1955, page 172]).
The determinant can be developed as follows:
$$
\eqalign
{
\det (1_{m_n} -\lambda M^{(n)})&=
1 - \lambda \sum_{\ell=1}^{m_n} M^{(n)}_{\ell \ell} \cr
&+ {\lambda^2 \over 2!}
\sum_{1 \le \ell_1 , \ell_2 \le m_n}
\det \left (\matrix
M^{(n)}_{\ell_1 \ell_1} & M^{(n)}_{\ell_1 \ell_2} \\
M^{(n)}_{\ell_2 \ell_1} & M^{(n)}_{\ell_2 \ell_2} \\
\endmatrix \right ) - \cdots + \cr
+
(-1)^{m_n}
{\lambda^{m_n} \over {m_n}!}
&\sum_{1 \le \ell_1 , \ell_2 , \ldots , \ell_{m_n}\le m_n}
\det \left (\matrix
M^{(n)}_{\ell_1 \ell_1} & M^{(n)}_{\ell_1 \ell_2} & \cdots &
M^{(n)}_{\ell_1 \ell_{m_n}} \\
M^{(n)}_{\ell_2 \ell_1} & M^{(n)}_{\ell_2 \ell_2} & \cdots &
M^{(n)}_{\ell_2 \ell_{m_\vdots & \vdots & & \vdots \\
M^{(n)}_{\ell_{m_n} \ell_1} & M^{(n)}_{\ell_{m_n} \ell_2} &
\cdots& M^{(n)}_{\ell_{m_n} \ell_{m_n}} \\
\endmatrix \right ) \, .\cr
}\tag2.2
$$
We shall obtain the following bound on the
absolute value of the coefficient of $\lambda^k$
(for $1 \le k \le m_n$)
in the polynomial \thetag{2.2}:
$$
{1 \over {k}!}
\sum_{1 \le \ell_1 , \ell_2 , \ldots , \ell_k \le m_n}
\biggl | \det \left ( \matrix
M^{(n)}_{\ell_1 \ell_1} & M^{(n)}_{\ell_1 \ell_2} & \cdots &
M^{(n)}_{\ell_1 \ell_{k}} \\
M^{(n)}_{\ell_2 \ell_1} & M^{(n)}_{\ell_2 \ell_2} & \cdots &
M^{(n)}_{\ell_2 \ell_{k}}\\
\vdots & \vdots & & \vdots \\
M^{(n)}_{\ell_{k} \ell_1} & M^{(n)}_{\ell_{k} \ell_2}
&\cdots & M^{(n)}_{\ell_{k} \ell_{k}} \\
\endmatrix \right ) \biggr |
\le {1 \over k!}
\cdot V^k \cdot k^{k/2} \, .
\tag2.3
$$
Since the bound \thetag{2.3} is independent of $n$, the
lemma follows
from the Stirling formula.
To prove \thetag{2.3}, we apply the classical Hadamard inequality (see e.g. Riesz-Sz.-Nagy
[1955, page 176]), which says that the determinant $\det C $ of a
$k\times k$ matrix with coefficients $C_{i,j} \in \complex$
satisfies the bound
$$
|\det C| \le \|C_1\|_{2} \cdots \|C_k\|_{2} \, ,\tag2.4
$$
where $\| C_i\|_{2} = (\sum_{j=1}^k |C_{ij}|^2)^{2}$
is the euclidean length of the $i^{\text{th}}$ row of the matrix.
Indeed, let $1 \le \ell_1 , \ldots , \ell_k \le m_n$ and
set $C_{ij} = M^{(n)}_{\ell_i \ell_j} $.
The assumption on the matrices implies that
$\| C_i\|_{2}= (\sum_{j=1}^{k} |M^{(n)}_{\ell_i \ell_j}|^2)^{1/2}
\le k^{1/2} v^{(n)}_{\ell_i}$. By the Hadamard inequality
\thetag{2.4},
the determinant of the $k\times k$
minor $C_{ij}$ satisfies:
$$
|\det C| \le k^{k/2} \prod_{i=1}^k v^{(n)}_{\ell_i} \, .
$$
Summing over the $\ell_i$, it suffices to use the upper
bound $V$ in \thetag{2.1} to obtain \thetag{2.3}.
\qed
\endr\enddemo
\noindent We can now prove our main result:
\demo{Proof of the theorem}
\roster
\item
Equality \thetag{1.1} is a direct
application of Theorem 1.1 in Baladi-Ruelle [1993]. We leave to the reader
the verification that the two additional rows and columns in the kneading
matrices defined here lead to the suppression
of the factor
$1-{1 \over 2} (\epsilon_1 z_1 + \epsilon_N z_N)$ in
the previous paper
(it suffices to check that this factor disappears in
Lemma 2.1 of Baladi-Ruelle [1993]). Each
$\Delta^{(n)}(t)$ is analytic in the disc of
radius $1/ \rho^{(n)}$ because each coefficient of the matrix
$K^{(n)}(t)$ converges in this disc.
As mentioned in Section 1, the relationship between the zeroes of
$\Delta^{(n)}$ (or poles of $\zeta_g^{(n)}$) and the discrete spectrum
of $\LL_{g^{(n)}}$ follows from Hofbauer-Keller [1984] and
Baladi-Keller [1990]. The special treatment reserved for the periodic
orbits through the turning points does not affect the discrete spectrum
of the transfer operator.
\item
Since $g$ is continuous at periodic
points and constant on $\ZZ$-homtervals, for each fixed $m \ge 1$, the
expression $g^{(n)}_{\ell_1, \ldots, \ell_m}$
in the definition of $\tilde \zeta_{g^{(n)}}(t)=\zeta_{g^{(n)}}(t)$
converges to
$ g_{\ell_1, \ldots, \ell_m}$,
the corresponding expression in $\tilde \zeta_g(t)= \zeta_g(t)$,
as $n \to \infty$.
Also, if we define
$$
\zeta_{m,g}=
\sum_{\ell_1, \ldots, \ell_m}
|L(f_{\ell_m} \circ \cdots \circ f_{\ell_1}) |
\cdot |g_{\ell_1, \ldots, \ell_m}| \, ,
$$
(and analogously $\zeta_{m,g^{(n)}}$),
denoting by $R_n$ the radius of convergence
of $\zeta_{g^{(n)}}$,
and $R$ the radius of convergence
of $\zeta_g(t)$, we have $R_n \ge
\lim_{m \to \infty}
\zeta_{m,g^{(n)} }^{-1/m}\ge \lim_{m \to \infty}
\zeta_{m,g}^{-1/m}$.
Hence, since
$$
R_\infty := \lim_{m \to \infty}
\zeta_{m,g}^{-1/m} \in ($$
(recall that $g$ is bounded and the number of laps
of $f^n$ is bounded by $(N+1)^n$), the analytic
functions $\zeta_{g^{(n)}} (t)$ converge to $\zeta_g(t)$
in the disk of radius $R_\infty$. Therefore, if we can
show that the analytic functions
$\Delta^{(n)}(t)$ are uniformly
bounded in the (possibly larger) disc $\tilde D$ of radius
$1/\tilde \rho$ for each $\tilde \rho > \bar \rho$,
then we have proved \therosteritem{2}.
We shall perform elementary row operations on the
matrices $K^{(n)}(t)$ for $1 \le n$, obtaining new matrices
$L^{(n)}(t)$ with $\det K^{(n)}(t) = \det L^{(n)}(t)$.
For $1 \le n$,
write $m_n = N_n+1$ and define
$$
M^{(n)} (t) = 1_{m_n}-L^{(n)}(t) \, , \tag 2.5
$$
where $1$ is the $m_n \times m_n$ identity matrix.
We shall prove that for any fixed $t_0 \in \tilde D$
the conditions of the key lemma
are satisfied for the sequence of matrices $M^{(n)}(t_0)$ and
a constant $V$ independent of $t_0 \in \tilde D$. This yields the needed
uniform bound.
Let $t_0 \in \tilde D$ and $2 \le n < \infty$ be fixed.
We leave unchanged the rows which correspond
to the initial partition of the interval:
$$
L^{(n)}_i(t_0)=K^{(n)}_i(t_0)
\, , \text{ for } i=0, \ldots, N_1 \, . \tag2.6
$$
We now compute the corresponding bounds
$v^{(n)}_i$. For
any $0 \le i, j\le N_n$, we have
$M^{(n)}_{ij}(t_0) = \sum_{k=1}^\infty a_{ij}(k) t_0^k$ with the
$a_{ij} (k) \in
\complex$ satisfying:
$$
|a_{i j}(k)| \le \sup_{x} \prod_{s=0}^{k-1} |g^{(n)} (f^s x)|
\le \KK \cdot \bar\rho^k \, ,
$$
where $\KK > 0$ is a suitable constant.
Setting
$v^{(n)}_i = \KK/((\tilde\rho/\bar\rho) -1) $,
for $i=0, \ldots, N_1$, we have
$$
\sum_{i=0}^{N_1} v^{(n)}_i
\le (N_1+1) \cdot {\KK\over (\tilde\rho / \bar \rho) -1 }\, .
$$
We now use the recursive construction of the matrix to perform
the elementary operations
on the remaining rows of $K^{(n)}(t_0)$We start with the bottom rows and
fix $m_{n-1} \le i < m_n$. By definition
$$
\vec K^{(n)}_i (t_0)= {1\over 2}
\biggl [ \vec \theta^{(n)} (u_i +) -
\vec \theta^{(n)} (u_i-) \biggr ] \, ,
$$
where $u_i$ is an endpoint of the partition $\ZZ_n$ which is not an endpoint
of the partition $\ZZ_{n-1}$. We have $\epsilon (u_i+)
=\epsilon(u_i-)$, we denote the
common value $\epsilon_i$. By definition, if $\epsilon_i=+1$
(respectively $-1$)
$f(u_i-)$ and $f(u_i+)$
are limits from the left and from the right
(respectively right and left)
to an
endpoint $u_{\ell(i)}$
of the partition $\ZZ_{n-1}$
(which is not an endpoint of the partition $\ZZ_{n-2}$). The index
$\ell(i)$
thus corresponds to a {\it higher}
row $m_{n-2} \le \ell(i) < m_{n-1}$ of the
matrix $K^{(n)}(t_0)$. The crucial remark is:
$$
\eqalign
{
\vec K^{(n)}_i (t_0)&=(\delta_{ij}, j=0, \ldots, N_n) \cr
&\quad+\epsilon_i t_0 \cdot g^{(n)}(u_i+)
\cdot (\vec \theta^{(n)} (u_{\ell(i)}[\epsilon_i])
-\vec \theta^{(n)} (u_{\ell(i)}[-\epsilon_i])) / 2\cr
&\quad+\epsilon_i t_0 \cdot g^{(n)}(u_i+)
\cdot \vec \theta^{(n)} (u_{\ell(i)}[-\epsilon_i]) /2 \cr
&\quad-\epsilon_i t_0 \cdot g^{(n)}(u_i-)
\cdot \vec \theta^{(n)} (u_{\ell(i)}[-\epsilon_i]) /2 \cr
= &(\delta_{ij}, j=0, \ldots, N_n) \cr
&\quad +t_0 \cdot g^{(n)}(u_i+) \cdot K^{(n)}_{\ell(i)} (t_0)\cr
&\quad+
\epsilon_i t_0 \cdot {g^{(n)}(u_i+)-g^{(n)}(u_i-)
\over 2 } \cdot \vec \theta^{(n)} (u_{\ell(i)}[-\epsilon_i]) \, .\cr
}
$$
For $m_{n-1}\le i < m_n$, we define:
$$
\eqalign
{
\vec L^{(n)}_i (t_0)&=(\delta_{ij}, j=0, \ldots, N_n) \cr
&\quad + \epsilon_i t_0 \cdot {(g^{(n)}(u_i+)-g^{(n)}(u_i-)) \over 2 }\cdot
\vec \theta^{(n)} (u_{\ell(i)}[-\epsilon_i]) \, .
}
\tag2.7
$$
(If the weight $g$ is in fact locally constant on some
refinement $\ZZ_k$, the rows defined by \thetag{2.7} are
of the form $(\delta_{ij}, j=0, \ldots, N_n)$ $n \ge k$ and $i\ge m_k$.)
We proceed similarly for rows $m_{n-p} \le i < m_{n-p+1}$ with
$2 \le p \le n-1$. By the same analysis
as for the $N_1+1$ first rows, we have for
$N_1 < i < m_n$
$$
|\theta^{(n)}(u_i *)_j (t_0)| \le { \KK \over (\tilde\rho / \bar\rho) -1}\, ,
\text{ for } 0 \le j \le N_n \, .
$$
Thus for $N_1
+1 < i < m_n$
we can set
$$
v^{(n)}_i = |g^{(n)}(u_i +) - g^{(n)}(u_i -)| \cdot
{ \KK \cdot \tilde \rho /2 \over (\tilde\rho / \bar \rho) -1}\, .
$$
Since the points $u_i$ are all distinct, we have
$$
\sum_{i=N_1+1}^{N_n} v^{(n)}_i
\le \varr g^{(n)} \cdot { \KK \cdot \tilde \rho /2
\over (\tilde\rho / \bar \rho) -1}
\le \varr g \cdot { \KK \cdot \tilde \rho /2 \over (\tilde\rho / \bar \rho) -1} \, .
$$
The key lemma yields the needed
uniform bound on $\Delta^{(n)}(t)$ for $t \in \tilde D$.
The claim on the discrete spectrum of $\LL_g$ follows from
Baladi-Keller [1990]
(and Baladi-Ruelle [1994] to remove the assumption
that the $g_i$ are continuous).
\qed
\endroster
\enddemo
\smallskip
\subhead A counter-example to $\lim_{n \to \infty} \det(1-M^{(n)}(t))=
\det(1-M^{(\infty)}(t))$
\endsubhead
\smallskip
If $g$ is continuous, the vectors $v^{(n)}$
constructed in the proof of the theorem satisfy
$v_i^{(n)} \to 0$ as $n \to \infty$,
for each $i > N_1$. Hence each matrix row
$M^{(n)}_i(t_0)$, with $i > N_1$, converges to the countable zero
vector as $n \to \infty$. This allows us to construct a counter-example
which shows that, in general, $\det(1-M^{(\infty)}(t))\ne \Delta(t)$
(where $M^{(\infty)}(t)$ is viewed as a Fredholm operator
acting on $l^1(\natural)$ as described in the appendix).
Indeed, let $f:[0,1]\to[0,1]$ be the tent-map which has slope $2$ on
$[0,1/2]$ and $-2$ on $[1/2,1]$
(with turning points $a_0=0$, $a_1=1/2$, $a_2=1$),
and let $g$ be a strictly positive
continuous function of bounded variation with $i \ge 3$ in the matrix $M^{(\infty)}(t)$ vanish, the determinant
of $1-M^{(\infty)}(t)$ is equal to the determinant of the $3\times 3$ matrix
obtained by considering the first three rows and columns of
$1-M^{(\infty)}(t)$.
>From the definition of $M^{(n)}(t)$, we see that this determinant is a
rational fraction in $t$, which depends only on
(the orbit of $a_1$ under $f$ and) the three parameters
$g(0)$, $g(1/2)$ and $g(1)$. However, the zeta function
$\zeta_g(t)$ of $f$ (which is equal to the ordinary Artin-Mazur
zeta function multiplied by a factor $(1-g(0) t)$ to take into account
the fixed point at $0$), is not invariant when we consider different
weights having the same values at $0$,$1/2$ and $1$ (take for
example a strictly positive continuous weight $h$ of bounded variation
with $h(0)=g(0)$, $h(1/2)=g(1/2)$ and $h(1)=g(1)$, but with
$h(x)\ne g(x)$, where $1/2< x < 1$ is the other fixed point of $f$ ---
then the coefficients of order one in the Taylor series
of $\zeta_g(t)$ and $\zeta_h(t)$ are different).
Note that
when $g$ has finitely or countably many discontinuities, the
matrix $M^{(\infty)}(t)$ contains information not only on the
values of the weights at the turning points, but also on these
discontinuities.
\medskip
\demo
{Proof of the corollary}
\roster
\item
Since $f$ is Markovian, the kneading determinant $\Delta^{(n)} (t)$
only depends on $f$ evaluated at the endpoints
of $\ZZ_n$. Since $f$ and $f^{(n)}$ coincide on these points by construction,
the $n^{\text{th}}$ kneading determinant of $f^{(n)}$
coincides with $\Delta^{(n)} (t)$. The claimed equality thus follows from
\thetag{1.1}, and
the assertion about convergence of the poles follows from
\therosteritem{2} in the theorem.
\item
We first show that the spectra a) and b) are the same.
We follow the argument
of Pollicott [1986,since the spectrum of $\LL_{g^{(n)}, f^{(n)}}$ is a subset of the union of the
spectrum of $\LL_{g^{(n)}, f^{(n)}}$ restricted to $E_n$ and the spectrum of
the quotient operator $\LL_{g^{(n)}, f^{(n)}}/E_n : BV/E_n \to BV/E_n$,
it suffices to check that the spectral radius of
$\LL_{g^{(n)}, f^{(n)}}/E_n$ is not larger than $\rho^{(n)}$. We have
not been able to find a proof of this (certainly
well-known) fact in the literature, and
we include it
here. The induced norm on $BV/E_n$ is
$\| \varphi \|_{E_n} = \sum_{\eta \in \ZZ_n}
\varr_\eta (\varphi)$.
The key observation is that if $\eta \in \ZZ_n$ and
$m > n$, then for any $\varphi \in BV$,
$$
\eqalign{
\varr_\eta &\biggl (\sum_{\xi \in \ZZ_m}
\LL^m_{g^{(n)}, f^{(n)}} (\varphi \cdot \chi_\xi) \biggr )
= \cr
&\sum_{\xi \, : \, \text{ interior }
((f^{(n)})^m \xi \cap \eta ) \ne \emptyset}
\varr_\xi(\varphi \cdot g^{(n)}_m )\cr
&\qquad
\qquad\qquad\qquad+ \sum_{x \in \DD(\eta,m)} |(\varphi \cdot g^{(n)}_m(x+) - (\varphi \cdot g^{(n)}_m(x-)| \, ,\cr
}
$$
where the set $\DD(\eta,m)$ is the set of endpoints of the connected components
of ${f^{(n)}}^{-m}(\eta)$ which are not endpoints of $\eta$,
and where we have written
$$
g^{(n)} _m (x) = g^{(n)} (x) \cdot g^{(n)} (f^{(n)} (x))\cdots
\cdot g^{(n)} ({f^{(n)}}^{m-1} (x))\, .
$$
Thus for $m \ge n$:
$$
\align
\| \LL^m _{g^{(n)}, f^{(n)}} \varphi \|_{E_n}
&= \sum_{\eta \in \ZZ_n} \varr_\eta (\LL^m _{g^{(n)} ,f^{(n)}} \varphi)\cr
&= \sum_{\eta \in \ZZ_n}\varr_\eta \biggl (\LL^m _{g^{(n)} ,f^{(n)}}
(\sum_{\xi \in \ZZ_m} \varphi \cdot \chi_\xi) \biggr ) \cr
\allowdisplaybreak
&=\sum_{\eta \in \ZZ_n}
\varr_\eta (\sum_{\xi \in \ZZ_m}
\LL^m_{g^{(n)}, f^{(n)}} (\varphi \cdot \chi_\xi) )\cr
\allowdisplaybreak
&=\sum_{\xi \in \ZZ_m}\varr_\xi (\varphi \cdot g^{(n)}_m )\cr
&\qquad +
\sum_{\eta \in \ZZ_n}
\sum_{x \in \DD(\eta,m)} |(\varphi \cdot g^{(n)}_m\allowdisplaybreak
&\le \sup g^{(n)}_m \cdot
\sum_{\eta \in \ZZ_n} \varr_\eta (\varphi)
= \sup g^{(n)}_m \cdot \| \varphi \|_{E_n} \, . \cr
\endalign
$$
Hence,
$\lim_{m \to \infty} (\| \LL^m _{g^{(n)}, f^{(n)}} \|_{E_n})^{1/m}
\le \rho^{(n)}$,
as desired.
The spectra b) and c) are the same because of
part \therosteritem{1} of this corollary
combined with the second assertion
of claim \therosteritem{1} in the theorem.
Since the partition is generating,
outside of the closed disc of radius $\rho^{(n)}$,
the spectrum of $\LL_{g^{(n)}, f^{(n)}}$
acting on $BV$ consists
of the inverses of the poles of $\zeta_{g^{(n)},f^{(n)}}$.
Similarly, outside of the closed disk of radius $\rho$,
the spectrum of $\LL_g$
acting on $BV$ consists
of the inverses of the poles of $\zeta_g$.
It hence suffices to apply part \therosteritem{1} of this corollary
again to get the convergence.
\qed
\endroster
\enddemo
\head Appendix: Viewing the countable matrix $M^{(\infty)}(t)$
as a Fredholm operator
\endhead
In this appendix, we will indicate
how the theory of Grothendieck yields:
\proclaim {Proposition}
Let $f$ and $g$ be as in Section 1. Assume that
$g$ is continuous at each point
of $\Fix f^m$, and one-sided continuous
at each point of $\Fixs f^m \setminus \Fix f^m$,
for all $m \ge 1$.
Let $M^{(\infty)}(t)=\lim_{n \to \infty}
M^{(n)} (t)$ be the countable matrix defined by \thetag{2.5},
\thetag{2.6} and \thetag{2.7}.
Then for each $t_0 \in D$,
the open disc of radius $1/\bar\rho$, the matrix $M^{(\infty)}(t_0)$,
when viewed as a linear operator $u=u(t_0)$:
$$
u : l^1(\natural) \to l^1(\natural) \, , \qquad
u : (x_j)_j \mapsto
(\sum_j M^{(\infty)}_{ij} (t_0) x_j)_i
$$
is a Fredholm operator (in the sense of Grothendieck [1956])
with trace
norm
$\| u\|_1 = \| M^{(\infty)} (t_0) \| = \sum_{i=0}^\infty v_i$,
wher$v_i := \sup_j | M^{(\infty)}_{ij} (t_0 )|$.
In particular,
for each $t_0 \in D$, the
operator $u$ has a Fredholm determinant
$d(t_0) (\lambda) = \det(1-\lambda u)$
which is an entire function of the complex variable $\lambda$,
and an entire
function of the operator $u \in {l^1(\natural) }'
\hat \otimes l^1(\natural)$. It follows that
$d(t_0) (\lambda)$ is analytic in $t_0$ for $t_0 \in D$.
\endproclaim
\smallskip
\demo{Proof of the proposition}
Let $v_i^{(n)}$, for $n \ge 1$ be the sequence of vectors introduced in
the proof of \therosteritem{2} in the main theorem
(for some fixed $\tilde \rho > \bar \rho$). Note that
$v_i^{(n)}$ does not depend on $n$, for $i \le N_1$, and that
$v_i^{(n)}$ converges to $|g(u_i+) -g(u_i-)| \cdot \KK\cdot \tilde \rho/
(2 \cdot ((\tilde\rho/\bar\rho)-1))$, as $n \to \infty$,
for $i > N_1$. Writing
$v_i^{(\infty)} := \lim _{n\to \infty} v_i^{(n)}$, we have
$\sum_{i=0}^\infty v_i ^{(\infty)}< \infty$. Also, the numbers $v_i$
introduced
in the statement of this proposition satisfy $v_i \le v_i^{(\infty)}$.
(Indeed, for any fixed $i,j$ we have, just as in the
proof of the theorem in Section 2, $M^{(\infty)}_{ij}(t_0)
=\sum_k a_{ij} (k) t_0^k$, with $|a_{ij} (k)| \le \KK \cdot \bar \rho^k$
for $i \le N_1$, and $|a_{ij} (k)| \le
|g(u_i +)-g(u_i-)| \cdot \KK \cdot \tilde \rho\cdot \bar \rho^k/2$ for
$i > N_1$.)
We now wish to show that a result in
Grothendieck [1956, III.1, Corollary 1]
can be applied. For the convenience of the reader we give the
statement below.
\smallskip
\item {}
{\it
Let $\II$ be a locally compact space endowed with a measure $\mu$ and
$E$ a Banach space; the Fredholm applications $u : E \to \LL^1(\mu)$
are those defined by an integrable map $h : X \to E'$ by
$(u (x) )(i) = < x, h(i) > $.
The trace norm of $u$ is the norm of
$h$ in $\LL^1_{E'}(\md |\mu| (i)$.}
\smallskip
We apply the result above to the case $\II=\natural$,
$\mu$ the discrete measure, $E= \LL^1(\mu)=l^1(\natural)$,
and $h (i) = M^{(\infty)} _{i \cdot}
(t_0)$. The row $M^{(\infty)}_{i \cdot}(t_0)$
acts on $l^1(\natural)$
by $ = \sum_j M_{ij} (t_0) x_j$.
We have
$$
\eqalign{
\| h(i) \| &= \| M^{(\infty)} _{i \cdot} (t_0) \|
= \sup_{x \in \l^1(\natural), \|x\|=1}
\sum_j |M^{(\infty)} _{i j} (t_0) \cdot x_j | \cr
&= \sup_j |M^{(\infty)}_{ij} (t_0 )|
= \sum_{i=0}^\infty v_i < \infty\, , \cr
}\tag{A.1}
$$
which shows that $h(i)$ is of integrable norm.
The chain of equalities \thetag{A.1} also gives
the formula for the trace norm.
\qed
\enddemo
\smallskip
We note that for each $t_0 \in D$, the
transposition of the matrix $M^{(\infty)}(t_0)$,
when viewed as a linear operator $u'=u'(t_0)$:
$$
u' : l^\infty(\natural) \to l^\infty(\natural) \, , \qquad
u' : (y_i)_i \mapsto
(\sum_i M^{(\infty)}_{ij} (t_0) y_i)_j
$$
is a Fredholm operator with trace
norm
$\| u'\|_1 = \| M^{(\infty)} (t_0) \| = \sum_{i=1}^\infty v_i$.
(Apply Proposition 1 in
Grothendieck [1956, III.1] to the measure space $\natural$,
the Banach space
$E=F=l^\infty(\natural)$, and the functions
$k \in \LL^\infty_{E'_s}(\natural)=l^\infty_{E'_s}(\natural)$
and $h \in \LL^1_E(\mu)=l^1_E(\natural)$:
$$
\eqalign{
k(i) : l^\infty(\natural) \to \complex \, , \quad i \in \natural \, ,
&\qquad
k(i) : (y_j)_j \mapsto y_i \, , \cr
h: \natural \to l^\infty(\natural) \, , &\qquad
h : i \mapsto M^{(\infty)}_{i \cdot} (t_0) \, , \cr
}
$$
using again \thetag{A.1} to see that the sequence $\|h(i)\|$ is summable.
The operator $u' : E \to E$ is the one noted
$\int k(i) \otimes h(i)\, d\mu(i) = \sum_i k(i) \otimes h(i)$
by Grothendieck, to whom we refer for details.)
The expressions by Grothendieck
[1956, III.1 (9)] (see also
Grothendieck [1956, III.2 (4)]) for the coefficients
$\alpha_n$ in the Taylor series of $d(t_0) (\lambda)$
and $d'(t_0) (\lambda)$ (see also Grothendieck [1956, II.2. Proposition 1])
coincide.
Hence the Fredholm determinants $\det (1-\lambda \cdot u)$
and $\det (1-\lambda \cdot u')$ coincide as analytic functions
in $t_0 \in D$ and $\lambda \in \complex$.
\smallskip
Note that the fact that a matrix such that $\sum_i \sup_j
|M^{(\infty)}_{ij}| < \infty$, acting to the right
on $l^1(\natural)$ or to the left on $l^\infty(\natural)$, defines
a {\it bounded} operator is an easy exercise
(see, e.g. Kato [1984, III.2.1 Ex. 2.3-2.4,
III.3.1 Ex 3.1]).
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\enddocument