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\begin{document}
\noindent DESY 92-012 \hfill ISSN 0418-9833\\
January 1992
\vspace*{2cm}
\begin{center}
{\LARGE\bf Crossing the Entropy Barrier of} \\
\vspace*{3mm}
{\LARGE\bf Dynamical Zeta Functions} \\
\vspace*{2.5cm}
\footnotetext[1]{Supported by Deutsche Forschungsgemeinschaft under
contract No. DFG--Ste 241/4-3}
{\large R. Aurich\footnotemark[1],
\addtocounter{footnote}{1}
J. Bolte\footnote{Supported by Doktorandenstipendium der
Universit\"at Hamburg},
\addtocounter{footnote}{-1}
C. Matthies, \\ \vspace*{5mm}
M. Sieber\footnotemark[1]
and F. Steiner} \\ \vspace*{1.5cm}
II. Institut f\"ur Theoretische
Physik\\ Universit\"at Hamburg\\ Luruper Chaussee 149, 2000
Hamburg 50\\ Fed. Rep. Germany
\end{center}
\vfill
\begin{abstract}
Dynamical zeta functions are an important tool to quantize chaotic
dynamical systems. The basic quantization rules require the
computation of the zeta functions on the real energy axis, where their
Euler product representations running over the classical periodic
orbits usually do not converge due to the
existence of the so--called entropy barrier determined by the
topological entropy of the classical system. We show that the
convergence properties of the dynamical zeta functions rewritten as
Dirichlet series are governed not only by the well--known topological
and metric entropy, but depend crucially on subtle statistical
properties of the Maslov indices and of the multiplicities of
the periodic orbits that are measured by a new parameter for which
we introduce the notion of a {\it third entropy}. If and only if
the third entropy is nonvanishing, one can cross the entropy
barrier; if it exceeds a certain value, one can even
compute the zeta function in the physical region by means of a
convergent Dirichlet series. A simple statistical model is
presented which allows to compute the third entropy.
Four examples of
chaotic systems are studied in detail to test the model numerically.
\end{abstract}
\newpage
%
\section{Introduction}
During the last years much effort has been undertaken to find
semiclassical quantization rules for classically chaotic
Hamiltonian systems as a counterpart to the WKB-- and
EBK--quantization for classically integrable systems.
A major breakthrough has been achieved by Gutzwiller
\cite{Gutz1,Buch}, when he derived his {\it periodic--orbit formula},
which expresses the trace of the Green's function in a semiclassical
approximation as a sum over all classical periodic orbits.
One problem that goes with this trace formula is that the
periodic--orbit sum considered as a function on the complex energy
plane does in general not converge on the real axis, i.e.
in the physical region. Gutzwiller himself \cite{Gutz2} discovered
that his trace formula is a special version of an
exact mathematical identity, known as {\it Selberg's trace formula}
\cite{Sel}, for a specific dynamical system --the free motion of
a particle on a surface with constant negative Gaussian curvature,
i.e. the motion on a surface endowed with a hyperbolic metric.
The Selberg trace formula provides a continuous variety
of convergent periodic--orbit sum rules \cite{neu}, as it is an
identity for the traces of certain functions of the Hamiltonian,
which is, in suitable units,
the negative of the hyperbolic Laplacian in this case.
The idea of ``smearing'' the Hamiltonian with an appropriate
test function can be carried over to the general case of
Gutzwiller's trace formula to yield also then a variety of
absolutely convergent periodic--orbit sum rules \cite{Sie1}.
The price to pay for the convergence of these sums is that
the smeared Green's functions do not exhibit poles at the
(semiclassical) quantum energies, but only show peaks of finite
widths. One such possible regularization is the Gaussian
smearing that has proven useful for several systems \cite{Aur}.
But knowing only a finite part of the length spectrum of
primitive periodic orbits permits only to resolve a finite part
--the lower end-- of the energy spectrum with finite accuracy.
The case of the free motion on hyperbolic surfaces shows that
there exists an alternative to using the (Selberg) trace formula
directly, since the regularized trace of the resolvent of the
Hamiltonian
can be expressed by the logarithmic derivative of a meromorphic
function of a complex variable directly related to the complex
momentum. It turns out that the
quantum energies are exactly given by the non--trivial zeroes of
this {\it Selberg zeta function} \cite{Sel} on the critical
line. The task of quantizing this system is therefore reduced to the
computation of these non--trivial zeroes. It would appear that the
identification of
the quantum energies as zeroes of an oscillating function
can be done with much higher precision than to identify them from
peaks of some
smeared Green's function. However, the representation of the
zeta function as an Euler product, as derived from the trace formula,
does in general not converge on the critical line and therefore cannot
be used directly
to calculate the zeta function in the region of interest.
Within the semiclassical approximation, all this holds also for
general dynamical systems to which Gutzwiller's trace formula applies.
The role played by Selberg's zeta function is now played by
so--called {\it dynamical zeta functions}.
The fact that the Euler product representing a given dynamical zeta
function does not converge on the critical line, but rather on a
half--plane not containing the physical region, is known
as the problem of the {\it entropy barrier}, since the
abscissa of absolute convergence of the Euler product is given by
the topological entropy of the classical system. These considerations
have led to the belief that it is impossible to find a quantization
rule for chaotic systems using dynamical zeta functions \cite{Vor}.
To some extend one can view the Riemann zeta function as a model for
a dynamical zeta function of
an unknown dynamical system. Again there is an entropy
barrier, since both the Dirichlet series and the
Euler product representing the Riemann zeta function do not
converge in the critical strip, where the famous Riemann--zeroes
are located. But in this case it is known how to compute these
non--trivial zeroes using the {\it Riemann--Siegel formula}
\cite{Siegel}. This formula allows a controlled
numerical computation on the critical line. It would therefore
be highly desirable to derive an analogue of the Riemann--Siegel
formula for general dynamical zeta functions. A Riemann--Siegel
lookalike formula for general chaotic systems has been proposed,
but attempts to prove this formula
have to cope with serious convergence
problems \cite{Berry}.
In this article we investigate the
convergence properties of certain representations of dynamical
zeta functions as Dirichlet series and show that the convergence
is determined by a new (classical) parameter called {\it third
entropy}. A model is presented that allows to compute the third
entropy and thus the region of convergence for these Dirichlet series
using a simple random walk model
for the coefficients of the series.
Our paper is organized as
follows. In sec. 2 we review the theory of dynamical zeta functions
relevant to the semiclassical quantization of chaotic systems and
describe how the zeta functions may be represented as Dirichlet
series. We also introduce the notion of a third entropy. A statistical
model for the third entropy
is developed in sec. 3. Numerical checks
of our model are presented in sec. 4 for four chaotic systems:
the hyperbola billiard, the geodesic flows on two hyperbolic
octagons and Artin's billiard. Finally, our results are summarized
in sec. 5.
%
\section{Dynamical Zeta Functions and the Third Entropy}
To be definite, we concentrate in this article on some
simple dynamical systems, which nevertheless show the typical behaviour
of classically chaotic Hamiltonian systems.
The systems under
consideration will be either plane billiards
or geodesic flows (free motion) on hyperbolic surfaces.
Chaotic
plane billiards consist of domains $D\subset \rz^2 $ with piecewise
smooth boundaries $\partial D$, such that the motion of a particle
sliding freely on $D$ and being elastically reflected on
$\partial D$ is chaotic. The quantum Hamiltonian is $H= -\Delta$
(we always choose units in which $\hbar =1 =2m$)
and the wave functions are required to vanish on $\partial D$
(Dirichlet boundary--value problem). We also require $H$ to have a
purely discrete spectrum, $00\}$ with Poincar\'e metric $ds^2 = y^{-2}
(dx^2 +dy^2 )$. $PSL(2,\rz)$ operates on $\cH$ via fractional
linear transformations. The wave functions are required to be periodic
with respect to
$\Ga$--transformations ($\Ga$--automorphic functions)
and again the spectrum should be discrete,
$0=E_0 \ +\ \sum_{\ga}\sum_{k=1}^{\infty}\frac{l_{\ga}\chi_{\ga}^k
g(kl_{\ga})}{e^{k\la_{\ga}l_{\ga}/2}-\si_{\ga}^k e^{-k\la_{\ga}
l_{\ga}/2}}\ .
\eeq
Here $h(p)$ is an even function, holomorphic in the strip
$|Im\,p|\leq \tau -\frac{\lb}{2}+\ve$, $\ve >0$, that decreases
faster than $|p|^{-2}$ for $|p|\rightarrow \infty$; $g(x)=
\int_{-\infty}^{+\infty}\frac{dp}{2\pi} e^{ipx}h(p)$ is its
Fourier--transform. $$ is the mean energy density, expressed
as a function of momentum $p$. The sum on the r.h.s. of (1)
runs over all primitive periodic orbits $\ga$ with lengths
$l_{\ga}$. $\chi_{\ga}\in \{\pm 1\}$ is a {\it character} attached to
$\ga$, where it is assumed that the {\it Maslov index} of $\ga$ is even,
and $\si_{\ga}$ is the sign of the trace of the monodromy matrix.
$\la_{\ga}$ is the (scaled) Lyapunov exponent of $\ga$ and $\lb$ is the
asymptotic average of all these exponents, which is also called the
{\it metric entropy}, because it measures the mean rate at which phase
space gets distorted in the neighbourhood of a periodic orbit
\cite{Buch}. The {\it topological entropy} $\tau
>0$ measures the exponential proliferation of periodic orbits,
\beq
N(l):= \# \{\ga |l_{\ga}\leq l\}\sim \frac{e^{\tau l}}
{\tau l}\ ,\ \ l\rightarrow \infty\ .
\eeq
To treat the systems considered here on the same footing, we
introduce the complex variable $s:= \frac{\lb}{2}-ip$. For the
motion on hyperbolic surfaces one knows that both the metric as
well as the topological entropy have the value one, since all
Lyapunov exponents are equal to one. In the case of plane billiards the
two entropies have to be calculated numerically. Notice that for
billiards with $\mbox{area}(D)<\infty$ one expects under rather
general assumptions Pesin's theorem
\cite{Pesin} $\tau =\lb$ to hold, whereas for non--compact systems
like the hyperbola billiard and some scattering systems
one finds $\lb > \tau$ \cite{Buch}.
>From the trace formula (1) one can derive \cite{Steiner} the trace
of the regularized resolvent in the following form
\beq
\sum_{n=1}^{\infty}\left[ \frac{1}{E_n -E(s)}-\frac{1}{E_n}
\right] \sim B-\phi (s)+\frac{1}{2s-\lb}\frac{Z'(s)}{Z(s)}\ .
\eeq
($B$ is a constant, which is irrelevant for our discussion in this
paper.) Notice that (1) and (3) become exact relations in the case of
the free motion on a hyperbolic surface. $\phi (s)$ is a function
with known analytic properties, $Im\,\phi (\frac{\lb}{2}\pm ip)
=\pm \pi $, $p\in \rz$. In (3) the {\it dynamical zeta function}
$Z(s)$ appears, which is defined by the Euler product
\beq
Z(s):=\prod_{\ga}\prod_{m=0}^{\infty}\left( 1-\chi_{\ga}\si_{\ga}^m
e^{-(s+m\la_{\ga}+\frac{1}{2}(\la_{\ga}-\lb))l_{\ga}}\right)\ ,\ Re\,
s\,>\tau\ .
\eeq
The {\it critical line}, on which in the semiclassical limit the
non--trivial zeroes of $Z(s)$ are located, is the line $s=\frac{\lb}
{2}-ip$, $p\in \rz$, i.e. $Re\,s\, =\frac{\lb}{2}$.
For the following discussion it is convenient to deal with a simpler
function, which contains, however, the same information on the
non--trivial zeroes as $Z(s)$.
This is the {\it Ruelle--type zeta function}
\beq
R (s):=\prod_{\ga}\left( 1-\hat \chi_{\ga}e^{-sl_{\ga}}\right)\ ,
\ Re\,s\,>\tau\ ,
\eeq
where $\hat \chi_{\ga} :=\chi_{\ga} \, e^{-\frac{1}{2}(\la_{
\ga} -\lb ) l_{\ga}}$.
Expanding the product over the primitive periodic orbits in (5)
transforms the Euler product into a generalized {\it Dirichlet series}
\beq
R (s)=\sum_{\ro}A_{\ro}e^{-sL_{\ro}}\ ,\ Re\,s\,>\tau\ .
\eeq
Here the sum runs over all {\it Dirichlet--orbits}, or briefly
{\it D--orbits}, defined by $\ro =
\ga_1 \oplus \dots \oplus \ga_n $, i.e. over all formal combinations of
primitive periodic orbits with {\it Dirichlet--lengths}, or briefly
{\it D--lengths}, $L_{\ro}:=
l_{\ga_1}+\dots +l_{\ga_n}$. The coefficients in the series are
determined by the quantities $\hat \chi_{\ga}$
attached to the primitive orbits that constitute a
D--orbit, $A_{\ro}:=\prod_{i=1}^n (-\hat \chi_{\ga_i})$. Since the
Euler product (5) converges absolutely for $Re\,s\,>\tau$, this
also holds for the Dirichlet series (6). Dirichlet
series such as (6) converge in right
half--planes $Re\,s\,>\si_c$ and converge absolutely in right
half--planes $Re\,s\,>\si_a$, $\si_a \geq \si_c$. Therefore, the series
(6) converges in the strip $\si_c From eqs. (9) and (7) and the definition of the coefficients $A_n$
one sees that the parameter $\de$ is a measure of the statistical
properties of the Maslov indices and of the multiplicities of the
periodic orbits, as will be discussed in more detail below.
$\de $ contains information beyond the topological
and metric entropy, and thus the name ``third entropy'' seems
to be appropriate. The need for a third entropy has clearly been
foreseen by Gutzwiller \cite{Buch}, but nothing has been done
as yet to develop this idea and to give a precise definition of it.
Obviously, the third entropy
determines whether the entropy barrier at $Re\,s\,=\si_a =
\tau$ can be crossed using the Dirichlet series representation.
Four different cases for the value of the third entropy
have to be distinguished:
\begin{enumerate}
\item $\de =0$: the entropy barrier is impenetrable;
\item $0<\de <\tau -\frac{\lb}{2}$: the entropy barrier is
transparent, but the critical line cannot be reached;
\item $\de =\tau -\frac{\lb}{2}$: the entropy barrier is transparent,
but it is not known in general whether
(6) converges on the critical line;
\item $\tau \geq \de >\tau -\frac{\lb}{2}$: the entropy barrier is
transparent and the Dirichlet series (6) converges conditionally on
the critical line.
\end{enumerate}
The fourth case is the most desirable one, but a priori either
case can occur. The first case is realized for the Riemann zeta
function, which therefore cannot be viewed as generic in this
respect.
We want to present in the following a rather simple random walk
model for the coefficients of the Dirichlet series, that allows
to predict the third entropy from a few input parameters. For a
given chaotic dynamical system it should then be possible to
decide from the input parameters, whether and how far the entropy
barrier can be crossed. For some systems $\si_c$ has been previously
calculated directly from (7), and the Dirichlet series has
been evaluated numerically on the critical line \cite{Diri}
rather successfully. For a different method to calculate the
zeroes on the critical line, see \cite{Tanner}.
As already mentioned, the advantage of this method is that the
quantum energies can be read of as zeroes of an oscillating function,
which is a much more accurate method than identifying them as
peaks of finite widths of some smearing function. Using the same
input, i.e. the same set of primitive lengths, therefore allows
to resolve considerably more quantum energies, when one
calculates the zeta function on the critical line.
%
\section{A Statistical Model for the Third Entropy}
Let us first consider an idealized situation where the Ruelle--type
zeta function is given by the generalized Dirichlet series ($B_0 =1$)
\beq
R_M (s) := \sum_{n=0}^{\infty}B_n e^{-s L_n }\ ,\ Re\,s\,>\tau_M \ ,
\eeq
and where the D--lengths $0=L_0 0)$ are randomly distributed
such that the value of $B_k $ is independent of the value of
$B_n$ for $k\neq n$. The coefficients $B_n$ should all be
distributed according to the same probability density $p(B)$
with mean $**=0$ and variance $\si_B^2 $. Then, according to the
central limit theorem for the distribution of sums of independent
random variables, the (partial) sums
\beq
S_N :=\sum_{n=1}^N B_n
\eeq
obey, in the limit $N\rightarrow \infty$, a normal distribution
with mean $=N****=0$ and variance $\si_S^2 =N\si_B^2 $.
Under these assumptions we can evaluate approximately formula (7) for
the abscissa $\si_c $ of conditional convergence for $R_M (s)$ by
replacing $|S_N |$ by
$^{1/2}=\sqrt{\si_S^2 }=\sqrt{N\si_B^2 }$:
\beqa
\si_c =\lsup \frac{\log |S_N |}{L_N}&\cong& \tau_M \lim_{N\rightarrow
\infty}\frac{\log \sqrt{N\si_B^2 }}{\log N}\nonumber \\
&=& \frac{1}{2}\tau_M \ .
\eeqa
This simple statistical model thus provides us with a prediction
for the location of the abscissa of conditional convergence of
the zeta function as well as for the third entropy, i.e. $\de
\simeq \frac{1}{2}\tau_M $.
In the following we want to argue in favour of a slight variation
of this model applicable to
real dynamical systems.
We will call a chaotic dynamical system of the type introduced
in the preceding section {\it ideal}, if the length spectrum of
primitive periodic orbits is not degenerate. Assuming, furthermore,
an irregular distribution of primitive lengths then guarantees
that the length spectrum of D--orbits is not degenerate
either.
The signs of the coefficients $A_{\ro}$ in (6) crucially depend on
whether the number of primitive orbits
$\ga$ constituting the D--orbit $\ro$ with $\chi_{\ga}=+1$
is even or odd. Arranging the D--lengths in ascending order,
as in (10), means that a change in sign, when going from $A_n$ to
$A_{n+1}$, depends on whether the numbers of primitive orbits
with positive characters in $L_n $ and $L_{n+1}$ differ in parity.
The absence of multiplicities and the irregularity of the primitive
length spectrum makes this change, at least in the limit of
long D--orbits we are interested in, random. Therefore the signs
of the coefficients $A_n$ can be considered as a random walk process.
The probabilities for $A_n$ to be positive or negative are the same,
since there is an equal number of D--orbits with an even
number of primitive orbits with positive character, as there are
D--orbits with an odd number of such primitive orbits. For the
hyperbolic surfaces we have $\hat \chi_{\ga} =\chi_{\ga}\in \{
\pm 1 \}$ and hence $A_n \in \{ \pm 1\}$. In the case of a plane
billiard, however, $\hat \chi_{\ga} =C_{\ga} \chi_{\ga}$, where the
$C_{\ga}$'s are distributed about one. Therefore the $A_n$'s are
distributed about $\pm 1$ with zero mean. In both cases one has a
distribution of the coefficients with mean $=0$ and variance
$\si_A $. This is exactly a distribution of the type discussed
above.
For an ideal chaotic system with topological entropy $\tau >0$
we therefore predict for
the abscissa of conditional convergence $\si_c =\frac{1}{2}
\tau$ and for the third entropy $\de =\frac{1}{2}\tau$.
Notice that in this case the third entropy is
determined solely by the topological entropy and therefore is not an
independent quantity. This fact, however, is due to the
``idealness'' of the system and is not typical for generic systems.
Whether the metric entropy $\lb $ is smaller, equal, or larger
than the topological entropy determines in this ideal situation,
whether case 2,3, or 4 in the classification of $\de$ in the
preceding section is realized. Thus the Dirichlet series is
conditionally convergent on the critical line if $\lb >\tau$.
In the case $\lb =\tau$ \cite{Pesin} one may have conditional
convergence or not.
In reality most systems will not be of the
ideal type and there will indeed occur multiplicities of
primitive lengths and therefore of D--lengths, too. But
knowing the asymptotic behaviour of the multiplicities of
D--lengths allows to modify the statistical model so as to
be applicable to realistic systems. In this process a new
parameter is introduced that makes the third entropy an
independent quantity, which is determined by the topological entropy
and this new parameter.
In most cases as illustrated by the models that will be discussed
in the next section, the mean multiplicities $$ of primitive
lengths either asymptotically approach a constant $\bar g$
for $l\rightarrow \infty$ or proliferate exponentially,
$\sim \frac{r}{l}e^{l/2}$, $r =const.$, $l
\rightarrow \infty$. The multiplicity $g_D (L)$ of a D--length
$L=l_1 +\dots +l_n $ then reads $g_D (L)=\prod_{i=1}^n g (l_i )$
for the vast majority of D--orbits composed of primitive periodic
orbits of different lengths.
One can now argue, under a few reasonable assumptions, that the
mean multiplicity $$ of a D--length $L$ behaves
asymptotically like
\beq
\sim d\, e^{\al L}\ ,\ \ \ L\rightarrow \infty\ ,
\eeq
for some positive constants $\al$ and $d$. In the examples, for which
we shall test our model, we have checked (13) numerically and found good
agreement, see fig. 1.
In the systems under consideration, primitive orbits $\ga$ can only
be degenerate with respect to their lengths, if they share the same
character $\chi_{\ga}$ and Lyapunov exponent $\la_{\ga}$.
Therefore the coefficients $A_{\ro}$
of D--orbits $\ro$ with the same D--length $L$ are
all equal. In the Dirichlet series (6) the sum over all
D--orbits $\ro$ can thus be replaced by a sum over distinct
D--lengths $L_n $. We denote the common coefficient of
these degenerate D--orbits by $A_n$ $(A_0 =1=g_D (0))$. Thus
\beqa
R (s)&=& \sum_{n=0}^{\infty} A_n g_D (L_n )\, e^{-sL_n }\nonumber \\
&=&\sum_{n=0}^{\infty}A_n d \, e^{-(s-\al )L_n}\left[ \frac{g_D
(L_n )}{de^{\al L_n}}\right]\ ,\ \ Re\,s\,>\tau \ .
\eeqa
The location of the abscissa of convergence is determined by the
coefficients of the Dirichlet series for large $L_n$. In this
regime we can omit the factor in the bracket, as it approaches one
in the mean due to (13). Let us introduce
\beq
\hat R (t):=\sum_{n=0}^{\infty} A_n d\,e^{-tL_n }\ ,\ \
Re\,t\,>\tau -\al \ ,
\eeq
which is of the same type as
$R_M (s)$ in eq. (10) with $B_n :=A_n d$,
$****=0$, $\si^2_B =d^2 \si_A^2 $, $\tau_M =\tau -\al$. The abscissa of
conditional convergence $\hat \si_c $ of $\hat R (t)$ thus is
according to eq. (12)
$\hat \si_c =\frac{1}{2}\tau_M =\frac{1}{2}(\tau -\al )$.
$\hat R (t)$ differs from $R (s)$ by a constant shift in
the argument, $s=t+\al $, and by the omission of the factor
$\frac{1}{d}e^{-\al L_n}g_D (L_n )$ in each term, which
is always positive and bounded as $L_n \rightarrow \infty$.
Furthermore, this factor approaches one in the mean. One therefore
expects that its omission does not influence the value of $\si_c$. We
thus conclude $\si_c =\hat \si_c +\al$, i.e.
\beq
\si_c =\frac{1}{2}(\tau +\al )\ .
\eeq
Our model then yields for the third entropy the value
\beq
\de =\frac{1}{2}(\tau -\al )\ .
\eeq
Since $0<\al <\tau$, the third entropy is bounded by $0<\de
<\frac{1}{2}\tau $. The question whether the Dirichlet series
representing $R (s)$ can be evaluated on the critical line can
now be answered once one knows the three entropies of the system,
i.e. the values of the three quantities $\tau$, $\lb $ and $\al$.
The condition to be fulfilled
is $\de >\tau -\frac{\lb}{2}$, i. e.
$$\al < \lb -\tau\ .$$
This condition means in particular that $\lb > \tau$ has to be
realized for the system. If $\al <\lb -\tau$ holds, the Dirichlet series
(6) will converge on the critical line $s=\frac{\lb}{2}-ip$,
$p\in \rz $, and it can be calculated numerically with the
available part of the length spectrum as an input. Such a computation
has been carried out rather successfully for some
systems in \cite{Diri}.
Finally, we want to remark on the conditionality of the convergence
for $\si_c \si_a $ the
Dirichlet series may be summed in any order of its terms; the
value of the sum does not depend on this order. But for $\si_c <
Re\,s\, \leq \si_a $ this is no longer the case. For any $s$
there is an ordering of the terms, such that the conditionally
convergent sum may take any desired value. Our point, however,
is the following: Define $R (s)$ for $Re\,s\,>\si_a $ (by the
Euler product for instance). Then form the Dirichlet series (6)
and arrange the terms in ascending order of the D--lengths.
For this ordering we concluded that $\si_c =\frac{1}{2}(\tau +\al )$.
Keeping the ordering fixed one can continue the Dirichlet series
beyond the entropy barrier up to $Re\,s\,>\si_c $. The ordering
played an essential role in our statistical model for the
coefficients. We argued that the choice of the signs going from
$A_n $ to $A_{n+1}$ would be random, since the parities of the
numbers of primitive orbits with positive characters in $L_n $
and $L_{n+1}$ are random. If one introduced any regularity in
the ordering of terms, such as e.g. first summing all the terms
with positive $A_n $'s and then the negative ones, the random
walk hypothesis would break down. Therefore, the value of
$\si_c $, and thus also the validity of our model, strongly
depends on the chosen order of terms in the Dirichlet series.
The conditionality of the convergence does not, however, touch the
analytic properties of this representation of the zeta function, as the
Dirichlet series still converges uniformly. Hence the holomorphy
of the zeta function is not destroyed for $Re\,s\,>\si_c$, when
it is being represented by an ``only'' conditionally
convergent series.
%
\section{Application to Four Chaotic Systems}
In this section we shall present a test of our statistical model for
the third entropy, i.e. for the location of the abscissa of
convergence, by investigating four specific chaotic
dynamical systems that have already previously been studied
in quite some detail: the hyperbola billiard,
two different hyperbolic octagons (Hadamard--Gutzwiller model), and
Artin's billiard. All these systems are of the type described in
sec. 2 and will be explained in more detail below.
For each system we shall plot the sequence $L_N^{-1} \log
|\sum_{n=1}^N A_n |$ obtained from the numerical data against $L_N$,
the lim sup of which yields $\si_c$. We then
compare these sequences with our theoretical value $\frac{1}{2}
(\tau +\al )$.
%
\subsection{The Hyperbola Billiard}
Our first example will be a plane billiard whose domain $D_0
\subset \rz^2$ is given by $D_0 =\{ (x,y)\in \rz^2 | x\geq 0,\
y\geq 0,\ x\cdot y \leq 1 \}$. Although the area of $D_0$,
measured with the usual Euclidean metric on $\rz^2$, is infinite,
the spectrum of the Laplacian $\De =\partial_x^2 +\partial_y^2 $
is discrete. This billiard has been previously studied in
\cite{Hyp,PhD}. The primitive length spectrum has
been completely determined up to $l=25$ and the quantum energies
have been computed up to $E=1,500$.
>From the calculated length spectrum the metric and topological
entropies have been determined to be $\lb =0.705\dots $ and
$\tau =0.592\dots $.
This system possesses a reflection symmetry across the $(x=y)$--axis
in $D_0$. Dividing out this symmetry and thus considering only
a desymmetrized system results in studying a billiard on the
domain $D=\{ (x,y)\in D_0 |\ x\geq y \}$. The stationary
Schr\"odinger equation then is the eigenvalue equation for
$-\De $ with Dirichlet boundary conditions on $\partial D$.
In the following we will always deal with this desymmetrized
hyperbola billiard. It is found that the metric and the topological
entropies remain unchanged after desymmetrization. Notice that
$\lb > \tau$.
>From Gutzwiller's trace formula (1) applied to the full billiard
domain $D_0$ one can derive the trace formula for the
desymmetrized system, for which (1) does not apply directly,
since on $D$ there exists the primitive periodic orbit $\ga_0$
running along the $(x=y)$--axis, which has to be treated
separately. One then finds \cite{PhD} that the Euler product for
the dynamical zeta function reads for $Re\,s>\tau$
\beqa
Z(s)=& &\prod_{n=0}^{\infty}\left( 1-\chi_{\ga_0}\si_{\ga_0}^{2n+1}\,
e^{-(s+(2n+1)\la_{\ga_0}+\frac{1}{2}(\la_{\ga_0}-\lb))l_{\ga_0}}
\right)\nonumber \\
&\cdot &\prod_{\ga \neq \ga_0}\prod_{m=0}^{\infty}\left( 1-
\chi_{\ga} \si_{\ga}^m \, e^{-(s+m\la_{\ga}+\frac{1}{2}(\la_{\ga}-
\lb ))l_{\ga}}\right) \ \ .
\eeqa
$(l_{\ga_0}=2\sqrt{2}$, $\si_{\ga_0}=\chi_{\ga_0}=-1$, $\la_{\ga_0}=
\frac{1}{2\sqrt{2}}\log (3+2\sqrt{2}))$.
The character for a primitive orbit $\ga$ is $\si_{\ga}=
\chi_{\ga}=(-1)^{
n_{\ga}}$, where $n_{\ga}$ denotes the number of reflections from
$\partial D$ when traversing the orbit once. The Ruelle--type
zeta function has the Euler product representation
\beq
R (s)=\prod_{\ga}\left( 1-\hat \chi_{\ga}\, e^{-sl_{\ga}}
\right)\ ,\ \ \ \ Re\,s>\tau\ ,
\eeq
where $\hat \chi_{\ga}=\chi_{\ga}\,e^{-\frac{1}{2}(\la_{\ga}-\lb)
l_{\ga}}$ for $\ga \neq \ga_0 $ and
$\hat \chi_{\ga_0}= 0$.
This function has the Dirichlet series representation
\beq
R (s)=\sum_{\ro} A_{\ro}\,e^{-sL_{\ro}}\ ,
\eeq
which converges absolutely for $Re\,s >\tau$. Here the coefficients
read $A_{\ro}=\prod_{\ga \in \ro}(-\hat \chi_{\ga})$.
Since the system possesses a time reversal invariance, most
primitive lengths are twofold degenerate. The only exceptions
are those orbits that are reflected into themselves from
the boundary $\partial D$. Their lengths occur without
multiplicities. But as their number becomes negligible when
going to higher and higher lengths, the mean multiplicity
$$ of primitive lengths approaches two for $l
\rightarrow \infty$. This leads to an exponential growth
of the mean multiplicity $$ of D--lengths. In fig. 1.a
we show a plot of $$ as calculated from the known length
spectrum. We use all 195,113 primitive orbits up to $l=25$,
among which we find 101,265 different lengths. Out of these 806,028
D--orbits of 459,204 different D--lengths with $L\leq 25$
can be formed. A fit with the exponential expression (13)
yields $d=0.958\dots$ and $\al =
0.026\dots$. From these parameters we predict according to our model
for the abscissa of conditional convergence $\si_c =\frac{1}{2}(\tau
+\al )=0.309\dots$. In fig. 2.a this prediction is shown as the
dotted line and is compared with the numerical approximation
to the exact definition of $\si_c$ according to eq. (7).
It is seen that our model is completely consistent with the
true values, at least in the limited $L$--range available to us.
>From our model we can also derive the third entropy for the
hyperbola billiard. Its value is $\de =\si_a -\si_c =\frac{1}{2}
(\tau -\al )=0.283\dots$. Since $\tau -\frac{\lb}{2}=0.239\dots$,
one sees that $\de > \tau -\frac{\lb}{2}$, which implies according
to the classification of sec. 2
that the Dirichlet series converges conditionally
on the critical line. Thus (20) may be used to compute the zeta
function on $Re\,s\,=\frac{\lb}{2}$ and to find thereby its non--trivial
zeroes there, which determine the quantum mechanical energies. In
the first reference of
\cite{Diri} such a calculation has been carried out for the
hyperbola billiard.
%
\subsection{Two Hyperbolic Octagons}
The two dynamical systems to be discussed next
will be provided by two different hyperbolic octagons.
These correspond to compact Riemann surfaces $M$ of genus $g=2$,
realized as fundamental domains of Fuchsian groups $\Ga$ on
the Poincar\'e upper half--plane $\cH =\{ z=x+iy |\,y>0 \}$
with hyperbolic metric $ds^2 = y^{-2} (dx^2 + dy^2 )$. The
surfaces $M$ are represented as $M=\Ga \backslash \cH$, where
$\Ga$ is a discrete, torsion--free subgroup of $PSL(2,\rz )$
isomorphic to the fundamental group $\pi_1 (M)$. $\Ga$ operates
on $\cH$ via fractional linear transformations,
\beq
z\in \cH\ ,\ \ \ga=\left( \begin{array}{cc} a & b \\ c & d
\end{array} \right) \ :\ \ \ \ \ \ \ \ \ \ga z =\frac{az+b}{cz+d}\ .
\eeq
The fundamental domain $\cF$ of $\Ga$ may be realized as a domain
in $\cH$ bounded by a $4g$--gon (i.e. an octagon for $g=2$).
The first of the octagons considered in this section is the so--called
{\it regular octagon} \cite{Aurich,Bogo}, which represents the
most symmetric Riemann surface of genus two. The second one is some
arbitrarily chosen {\it asymmetric octagon}.
The Hamiltonian for the free motion of a particle on $M$ is $H=
-\De$, where $\De = y^2 (\partial_x^2 +\partial_y^2 )$ is the
hyperbolic Laplacian on $\cH$. The eigenfunctions of $H$ are
realized as functions on $\cH$ which are invariant under the operation
of $\Ga$, so--called $\Ga$--automorphic functions: $\psi (\ga z)=
\psi (z)$, for all $\ga \in \Ga$. Then $H$ has a discrete spectrum
$0=E_0 1\ .
\eeq
>From this one infers that the Dirichlet series for the corresponding
Ruelle--type zeta function is given by (compare (6))
\beq
R (s)=\sum_{\ro} (-1)^{|L_{\ro}|}\,e^{-sL_{\ro}}\ ,\ \ \ Re\,s>1\ ,
\eeq
where $|L_{\ro}|$ denotes the number of primitive periodic
orbits constituting the D--orbit $\ro$.
The two octagons we are looking at in this section differ in one
point: the regular octagon has a huge symmetry group (of order 96),
whereas the asymmetric octagon possesses only one symmetry operation.
The latter
symmetry is always present for Riemann surfaces of genus two,
since these are all hyperelliptic and thus have the hyperelliptic
involution as a symmetry. The classical dynamics on the regular
octagon excels by a large degeneracy of the length
spectrum of primitive periodic orbits; the mean multiplicity
grows exponentially, $\sim \frac{8\sqrt{2}}{l} e^{l/2}$,
$l\rightarrow \infty$, see \cite{Aurich,Bogo}. On the other hand, in
the case of the asymmetric octagon the mean multiplicity approaches the
constant value of four. This is due to the time reversal invariance
and to the hyperelliptic symmetry. In any case the mean multiplicity
of D--lengths grows exponentially, $\sim d\,e^{\al L}$,
$L\rightarrow \infty$. One sees that a higher symmetry of the system
leads to a faster growth of the degeneracy in the length spectrum,
i.e. to a larger value of $\al$.
This in turn results in a higher value for the abscissa of
conditional convergence as can be seen from our formula $\si_c =
\frac{1}{2}(\tau +\al )$.
For the regular octagon the complete primitive length spectrum up
to $l_{max}=18.092025\dots$ has been determined in \cite{Bogo}.
It consists of 4,232,092 orbits of 1,500 different lengths. The
D--length spectrum up to $L=l_{max}$ then consists of
26,469,856 D--orbits with 2,336 different D--lengths.
In fig. 1.b we show a fit of $d\,e^{\al L}$ to $$.
The fit parameters are $d=4.5631\dots $ and
$\al = 0.4658\dots $. Thus our prediction is $\si_c =\frac{1}{2}
(1+\al)=0.7329\dots $. We compare this value in fig. 2.b with a
numerical evaluation of (7). Reasonably good agreement is found.
In the case of the asymmetric octagon we used the generator method
described in \cite{Aurich} to determine the primitive length
spectrum. We formed words in the group generators of lengths up
to 12 and truncated the length spectrum at $l=13$. Due to this method
the spectrum is, however, not complete, but there are some orbits
missing of lengths larger than about $l=12$. We got 36,336 primitive
orbits with 9,758 different lengths and generated 173,775 D--orbits
with 29,062 different D--lengths up to $L=13$. In fig. 1.c we show
a fit to
$$, from which we obtain the parameters $d=1.5311\dots $
and $\al = 0.1148\dots $. Hence we predict $\si_c =\frac{1}{2}(1+
\al )=0.5574\dots $ which is compared in fig. 2.c with the
numerical evaluation of (7). Again we find reasonably good agreement.
%
\subsection{Artin's Billiard}
Our final example of a chaotic dynamical system will be a
billiard system on the Poincar\'e upper half--plane $\cH$
constructed as follows. Let $\cF_0$ be a fundamental domain
on $\cH$ for the modular group $\Ga = PSL(2,\gz)$. It may be
chosen as $\cF_0 =\{ z\in \cH \,|\,|z|>1\ \mbox{for}\ \ -\frac{1}{2}
1\ .
\eeq
As in the previous examples, the Ruelle--type zeta function can
be derived from $Z(s)$ and a Dirichlet series can be found for it
\beq
R (s)=\sum_{\ro}A_{\ro}\,e^{-sL_{\ro}}\ ,\ \ \ Re\,s>1\ .
\eeq
A D--orbit $\ro$ now consists of $n_s$ primitive
periodic orbits of the symmetric type with negative characters
and of $n_u$ orbits of the other type with positive characters.
Therefore the coefficients of the Dirichlet series are $A_{\ro}=
\prod_{\ga \in \ro}(-\chi_{\ga})=(-1)^{n_u}$. Thus $A_{\ro}$
only depends on the parity of $n_u$.
>From the length spectrum of primitive periodic orbits, which has
been calculated in \cite{Schleicher,Mat}, one knows that
$\sim\frac{1}{l}e^{l/2}$, $l\rightarrow \infty$. Again
the mean multiplicity of the D--lengths grows like
$\sim d\,e^{\al L}$, $L\rightarrow \infty$. For our numerical
computations we take 166,319 primitive orbits with 3,000 different
lengths into account. These make up the full length spectrum up to
$l=14.6$. To determine the spectrum of D--orbits up to
$L=14.6$ completely, we have to form 722,226 D--orbits with 50,587
different D--lengths. In fig. 1.d we show a numerical fit of
the exponential law to $$ with parameters $d= 0.336\dots$
and $\al = 0.279\dots$.
This leads to the prediction $\si_c =\frac{1}{2}(1+\al )=0.639
\dots$ for the abscissa of conditional convergence of the Dirichlet
series (27). However, a comparison with the numerical value for $\si_c =
\lsup \frac{1}{L_N}\log |\sum_{n=1}^N A_n |$ in fig. 2.d shows a
mismatch between the predicted and the actual value. Therefore our
statistical model presented in sec. 3 appears not to be applicable
to Artin's billiard.
This specific system is known, however, to be an exception
among chaotic dynamical systems in another respect as well. It
is a general belief that the energy--level spacings
of classically integrable systems obey Poisson statistics,
whereas those of classically chaotic systems are distributed
like the eigenvalues of hermitian matrices in a Gaussian
orthogonal or unitary ensemble (GOE or GUE statistics).
For Artin's billiard a numerical computation of
the quantum energies shows \cite{Steil} that the level spacing
distribution is Poissonian, in contrast to what is expected.
This fact hints towards unexpected statistical properties of the
energy spectrum. Since by the trace formula (25) the energy and
length spectra are closely related, one would therefore expect
possible correlations in the length spectrum.
To check this hypothesis we have investigated
numerically the random walk hypothesis on which our statistical model
is built upon. In fig. 3 we plot the probability for the coefficient
$A_{N+1}$ in the Dirichlet series (14) for Artin's billiard
to have the same sign as the
preceding coefficient $A_N$. According to our statistical hypothesis
this probability is expected to be one half. This indeed is the case
in the first three examples we considered above.
For Artin's billiard we find, however, a
numerical value of about $0.41$. This shows that there are
correlations present in the length spectrum which lead to the fact
that the signs of the coefficients in the Dirichlet series are
not randomly distributed, in contrast to what is expected for
a generic chaotic system.
We have thus found another indication that Artin's
billiard is exceptional among chaotic systems.
Notice that from fig. 2d) one reads off a value of $\si_c \approx
0.45$, and thus the Dirichlet series (27) for Artin's billiard
converges on the critical line. In ref. \cite{Mat} this result
has been used to compute the quantal energies in the low energy region.
%
\section{Summary}
In this article we studied the convergence properties of dynamical
zeta functions for a class of classically chaotic dynamical systems.
In order to use the zeta function for the quantization of a chaotic
dynamical system, it is mandatory to know in which part of the complex
energy plane a given representation of the zeta function
converges. If it happens that a representation is available that
converges on the real energy axis (the critical line), then
one can use it to find the zeroes of the zeta function on this
critical line. These zeroes in turn give the semiclassical
energies of the system.
The derivation of the zeta function from Gutzwiller's trace
formula yields the zeta function as an Euler product.
Rewriting this Euler product as a
Dirichlet series does not alter the region of absolute convergence.
But in contrast to the case of Euler products there do exist
definite statements on the regions of conditional convergence of
Dirichlet series.
These may be --and in all the examples considered
by us they are-- larger than the region of absolute convergence.
Our aim was to present a model that describes to what extent
the conditional convergence of the Dirichlet series is
better than the absolute convergence. Knowing the abscissa
of conditional convergence one also
knows whether the Dirichlet series converges on the critical line
and thus whether it can be used to compute the zeta function there.
It turned out that a central role is played by a new parameter
called the third entropy, which measures statistical properties
of a given dynamical system beyond its topological and metric
entropy.
Making an assumption on the randomness of the length spectrum of
primitive periodic orbits of a dynamical system allowed us to set up a
statistical model for the third entropy and thus for the location of the
abscissa of conditional convergence. We found that the shift of this
abscissa away from the abscissa of absolute convergence is
given by the topological entropy that describes the strength
of the exponential proliferation of primitive periodic orbits,
and by the exponent $\al$ of the exponential increase of the
multiplicity of D--lengths: the less these multiplicities grow
the larger is the third entropy and
the more the conditional convergence gets improved. We
hence conclude that this single new parameter $\al$ or,
equivalently the third entropy $\de$, has to be calculated
from the length spectrum in addition to the topological and
metric entropy in order to get full information about the
convergence properties of the dynamical zeta function.
In sec. 4 we have tested our model in four specific chaotic systems:
the hyperbola billiard, the geodesic flows on two different
hyperbolic octagons, and Artin's billiard. In the first three
examples we found good agreement between the prediction obtained from
our model and the directly calculated value for the abscissa of
conditional convergence. The model, however, fails to explain
the situation in Artin's billiard. We argued that this failure is
due to correlations in the length spectrum that violate the
randomness hypothesis of our model. We claim that these
correlations are related to the observed unexpected statistical
properties of
the energy spectrum of this system. In this respect we
view Artin's billiard as an exceptional case and do not consider
it as generic.
%
\section*{Acknowledgements}
M.S. would like to thank Jon Keating for discussions.
F.S. wants to thank Martin Gutzwiller for a discussion on the
third entropy. We would like to thank the Deutsche
Forschungsgemeinschaft for financial support and the HLRZ at
J\"ulich for the access to the CRAY Y-MP 832 computer.
%
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%
\section*{Figurecaptions}
{\bf Figure 1:}
The mean mulitiplicity $$ is shown together with the fit
curve $d\,e^{\al L}$ for a) the hyperbola billiard, b) the regular
octagon, c) the asymmetric octagon and d) Artin's billiard \\
\noindent {\bf Figure 2:}
A numerical calculation of the sequences occuring in eq. (7) is shown
as a function of the D--length $L_N$
for a) the hyperbola billiard, b) the regular octagon, c) the
asymmetric octagon and d) Artin's billiard.
The upper curves correspond to $\si_a$, the lower ones to $\si_c$. The
full horizontal
lines indicate the critical lines, whereas the dotted lines show
$\si_c$ as derived from our model for the third entropy.
The dashed line in a) corresponds
to $\si_a =\tau$; $\si_a =\tau =1$ in b)--d).\\
\noindent {\bf Figure 3:}
The probability for the coefficient $A_{N+1}$ to have the same
sign as $A_N$ in the Dirichlet series (14) for Artin's billiard
is shown
\end{document}
ENDBODY
**