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\baselineskip=18pt
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\def\di{\displaystyle}
\def\R{I\!\!R}
\def\C{I\!\!\!\!C}
\def\N{I\!\!N}
\def\T{I\!\!\!T}
\def\Q{I\!\!\!\!Q}
\def\Z{Z}
\def\u{\upsilon}
\def\O{\Omega}
\def\o{\omega}
\def\t{\theta}
\def\z{\zeta}
\def\fa{{2\pi i\over p}}
\def\fp{{2\pi i\over p}}
\def\s{\sigma}
\def\a{\alpha}
\def\b{\beta}
\def\k{\kappa}
\def\g{\gamma}
\def\pp{\pi^{\prime}}
\def\eps{\epsilon}
\def\A{{\cal A}}
\def\U{{\cal U}}
\def\F{I\!\!F}
\def\QT{{\hat T}}
\def\Qf{{\hat f}}
\def\l{\ell}
\def\lj{\lambda_j}
\def\I1{{\cal I}_1}
\def\I2{{\cal I}_2}
\def\aj{\alpha_j}
\def\P{{\cal P}}
\def\vj{v_j}
\def\st{\sum_{s=0}^{T-1}}
\def\sx{\sum_{x\in \Z_p^*}}
\def\bk{\beta_k}
\def\lk{\lambda_k}
\def\L{I\!\!L}
\def\t{{\rm T}}
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\centerline{\bf DISTRIBUTION OF CLOSED ORBITS}
\centerline{\bf FOR LINEAR AUTOMORPHISMS OF TORI}
\vglue 0.2cm
\vglue 1.0cm
\centerline{ Mirko Degli Esposti and Stefano Isola}
\vglue 0.5cm
\centerline{\it Dipartimento di Matematica, Universit\'a degli Studi di
Bologna,}
\centerline{\it piazza di Porta S.Donato 5, I-40127 Bologna, Italy}
\vskip 0.5cm
\centerline{\it e-mail : desposti@dm.unibo.it, isola@dm.unibo.it}
\vskip 2cm
{\bf Abstract.} In this paper we study the distribution properties of
periodic orbits for the linear hyperbolic automorphisms
of the $d$-torus. We first obtain an explicit expression of the dynamical zeta
function and prove general equidistribution results similar to those
obtained for Axiom A flows.
We then study in detail some families of periodic
orbits living on invariant prime lattices:
they have the property that the integral of any
character along any single orbit can be
reduced to a number theoretic exponential sum over a finite
field. This fact enables us to obtain
explicit estimates on their asymptotic
distributional properties.
\vfill \eject
{\bf 1. Introduction.}
\vskip 0.5cm
A number of interesting results on the distribution of periodic orbits
for hyperbolic dynamical systems have recently been obtained pursuing
strong analogies with general results of number theory and probability
theory.
Indeed, using the Ruelle zeta function [18] for a hyperbolic flow in the same
way as the Riemann zeta function is used to prove the prime number theorem
(that is through Wiener-Ikehara Tauberian theorem, see, e.g., [11]), Parry and
Pollicott [15] proved
a `prime orbit theorem', saying that the number $\pi (x)$ of periodic orbits
$\gamma$ with (minimal) period $p(\gamma) \leq x$
is asymptotic to $e^{hx}/ hx$ as $x\to \infty$, where $h$ is the
topological entropy of the flow. Moreover, if $\int_{\gamma} \, f$ denotes
the (normalized) integral of the continuous function $f$ along $\gamma$,
then $(1 /\pi (x)) \, \sum_{p(\gamma)\leq x} \int_{\gamma} \, f
\rightarrow \int
\, f\, d\mu$ as $x\to \infty$, where $\mu$ is the invariant
probability measure of maximal entropy (see [16] and references therein).
Other types of theorems yield the equidistribution of periodic orbits
for hyperbolic flows in the sense of probability theory [12], [10]. For
instance, under certain conditions and for any $\eps >0$,
${\rm Prob}\{ |\int_{\gamma} \, f - \int \, f\, d\mu | > \eps \} \to 0$
as $p(\gamma )\to \infty$, which of course improves the former result.
This means that, except for a small (tending to zero) fraction, long
periodic orbits are nearly uniformly distributed according to $\mu$.
In what follows we shall study the distribution properties of
periodic orbits for a particular example: the linear hyperbolic
automorphism of the $d$-torus. Besides proving general results as
those mentioned above (Section 2), we shall concentrate on some particular
families of periodic orbits living on invariant prime lattices (Section 3).
These enjoy the nice
properties that the integrals of
characters over any single orbit can be reduced
to (some combinations of) number theoretic exponential
sums, like generalized Gauss sums or Kloosterman sums, over
a finite field (see [20]). This makes possible to obtain explicit estimates on
their asymptotic distributional properties.
We point out that the statistical properties of the periodic orbits turn out
to be crucial in the investigation of the quantum counterpart of this
dynamical system [8], [9]. In particular there is an intriguing
connection between the spatial distribution of certain families of closed
orbits (see
Section 3) and the localization properties of the eigenfunctions of the quantum
propagator in the classical limit [4], [5].
\vskip 1cm
{\bf 2. The algebraic hyperbolic automorphisms of the $d$-torus.}
\vskip 0.5cm
We consider the discrete time dynamical system
$F\, :\, M\to M$ where $M$ is the
$d-$torus $\T^d=\R^d /\Z^d$ (points on $\T^d$ are denoted by
$x=(x_1,\dots ,x_d)\in [0,1]^d$) and $F$ is the hyperbolic
automorphism of $\T^d$ generated by the $d\times d$ matrix
$$
A = \pmatrix{ a_{11} & a_{12} & \ldots & a_{1d}\cr
a_{21} & a_{22} & \ldots & a_{2d}\cr
\vdots & \vdots & \ddots & \vdots \cr
a_{d1} & a_{d2} & \ldots & a_{dd}\cr }
\eqno(2.1)
$$
where $a_{ij}\in \Z$, $i,j =1\ldots d$, and $A$ has real
spectrum $\Lambda=\{\lambda_1, \ldots , \lambda_d\}$ such that
$\lambda_1 \lambda_2\dots \lambda_d = 1$.
The Lebesgue measure $\mu$ is invariant because $\det A=1$.
The above condition on the eigenvalues makes this dynamical
system an Anosov one and hence, in particular, ergodic and mixing
with respect to $\mu$.
Now, an orthonormal basis in $L^2 (\T^d,d\mu)$ is given by the set
$$
\{T(n) = e^{2\pi i}|n\in \Z^dJ\}\eqno(2.2)
$$
and $A$ acts on points
$x=(x_1,\dots ,x_d)$ and on suitably smooth functions
$$
f(x)=\sum_{n\in\Z^d}\, f_n\, T(n)
$$
respectively as:
$$
\cases{ Ax =\left( (a_{11}x_1 +\dots a_{1d}x_d)({\rm mod}\,1),
\ldots , (a_{d1}x_1 +\dots a_{dd}x_d)({\rm mod}\,1) \right) \cr
f(Ax) = \sum_{n\in\Z^d}\, f_n\, T(A^tn) \cr }
\eqno(2.3)
$$
where $A^t$ is the transposed matrix of $A$ (notice that we have
used the same symbol
$A$ to denote the matrix $A$ and the map $F$; this will be
repeatedly done in what follows without fear of confusion).
Now, if $A$ is a continuous map of a compact metric space
$X$ then $h(A)$, the topological entropy, satisfies the
restricted variational principle:
$$
h(A) = \sup_{\mu \in {\cal M}_A(X)}h_{\mu}(A) \eqno(2.4)
$$
where ${\cal M}_A(X)$ the set of the probability measures on
the Borel
$\sigma$-algebra of $X$ which are $A$-invariant and $h_{\mu}(A)$ is
the measure theoretic entropy (see, e.g., [13]). Moreover, $A$ is
{\it intrinsically ergodic} if there exist a unique $\mu \in {\cal
M}_A(X)$ such that
$h(A)=h_{\mu}(A)$. In this case $\mu$ is called the
intrinsic measure of $A$, or maximal entropy measure of $A$.
Thus, for a linear automorphism of the torus the Haar measure,
i.e. the Lebesgue one,
is actually an intrinsic measure. We have the
\vskip 0.2cm
{\bf Theorem 2.1 (Sinai)} {\sl Let $A:\T^d\to \T^d$ be a linear automorphism
with eigenvalues $\lambda_1, \ldots \lambda_d$. Let $\mu$ be the
Haar measure of $T^d$. Then}
$$
h(A)=h_{\mu}(A)=\log \prod_{i\, : \, |\lambda_i| >1}|\lambda_i|
\eqno(2.5)
$$
\vskip 0.2cm
{\it Proof.} See, e.g., [13] p. 265.
\vskip 0.2cm
Consider again the general situation of
a continuous map $A$ of a compact metric space
$X$.
We say that $x\in X$ is a {\it periodic
point} of $A$, of period $n$, if it is a {\sl fixed point} of $A^n$, i.e.
$A^nx=x$. We denote by $Fix_n$ the set of such points.
It is easy to see that for linear automorphism of the torus the set of periodic points of
$A$ is dense in $\T^d$, because it coincides with the subset of $\T^d$
formed by all points having rational coordinates.
More generally we have the
\vskip 0.2cm
{\bf Theorem 2.2 (Bowen-Sinai)} {\it Every
topologically mixing hyperbolic homeomorphism
$A:X\to X$ is intrinsically ergodic.
If $\mu$ denotes its intrinsic measure then for
every continuous map $f :X\to \R$ :
$$
\int_Xf d\mu = \lim_{n\to \infty}{1\over \# Fix_n}
\sum_{x\in Fix_n}f (x)\eqno(2.6)
$$
and $A$ has topological entropy}
$$
h= \lim_{n\to \infty}{1\over n} \log{\# Fix_n}\eqno(2.7)
$$
{\it Proof.} See [13], p.254.
\vskip 0.2cm
For the linear automorphism of the $d$-torus we have the additional result:
\vskip 0.2cm
{\bf Proposition 2.1.}
$$
\# Fix_n = \left|\, \det(A^n-1)\, \right| = \left|\,
\prod_{i=1}^d(\lambda_i^n-1) \, \right|
\eqno(2.8)
$$
\vskip 0.2cm
{\bf Remark.} Notice that from (2.7) and (2.8) one immediately recovers (2.5).
\vskip 0.2cm
{\it Proof.}
The periodic points of period $n$ of the map $A:\T^d\to \T^d$ are given by the
solution of the linear system
$$
\left( A^n - 1\right)\, x \, = \, k, \qquad x\in \R^d, \; k\in \Z^d\eqno(2.9)
$$
It is easy to realize that for $k$ varying in $\Z^d$ the above system
determines a regular lattice $L_n$ in the covering space $\R^d$ so that
the density of periodic points belonging to $Fix_n$ is given by the inverse
of the volume of the fundamental domain $\Delta_n$ of $L_n$. On the other
hand, if $\{e_1, \ldots , e_d\}$ denotes the basis of the lattice $L_n$
and each vector $e_i$ is expressed as
$e_i = (r_{1i}, \ldots , r_{di})$ then the volume
$\u (\Delta_n)$ of $\Delta_n$ is given by
$$
\u (\Delta_n) = \left| \det r_{ij} \right| \eqno(2.10)
$$
Now, a basis $\{e_1, \ldots , e_d\}$ of the lattice $L_n$ can be easily
obtained by solving eq. (2.9) for $d$ distinct and linearly independent
vectors $k_1= (k_{11}, \dots , k_{1d}), \ldots , k_d= (k_{d1}, \dots ,
k_{dd})$, $k_{ij}\in \Z$, such that
$$
\det K \equiv \det \left| \, \matrix{ k_{11} & \ldots & k_{1d}\cr
\vdots & \ddots & \vdots \cr
k_{d1} & \ldots & k_{dd}\cr }\, \right|
=1 \eqno(2.11)
$$
This yields
$$
\u (\Delta_n) = {1\over \Gamma^d}
\det \left| \, \matrix{ D_{11} & \ldots & D_{1d}\cr
\vdots & \ddots & \vdots \cr
D_{d1} & \ldots & D_{dd}\cr }\, \right| \eqno(2.12)
$$
where $\Gamma = \left| \, \det \left( A^n-1\right)\, \right|$ and
$D_{ij}$ is the determinant of the $d\times d$ matrix obtained by
inserting the vector $k_i$ in the place of the $j$-th column of the matrix
$A^n-1$ (Cramer's rule). Finally, a tedious but trivial calculation where
the basic properties of the determinant are exploited gives
$$
\u (\Delta_n) = {\Gamma^{d-1}\det K \over \Gamma^d} \eqno(2.13)
$$
Hence, from (2.11) and (2.13) one obtains
$$
\# Fix_n = {1\over \u (\Delta_n)} = \left| \, \det \left( A^n-1\right)\,
\right| \eqno(2.14)
$$
Q.E.D.
Let us now restrict ourselves to symplectic matrices, i.e. $d$ is even, say
$d=2r$, and
$A^tJA=J$, where $J=\left(\matrix{ 0 & E \cr -E & 0 \cr }\right)$, and $E$ denotes
the $r\times r$ identity matrix, so that if $\lambda \in
\Lambda$ then also
$1/\lambda \in \Lambda$. The following result extends an earlier result for $d=2$
[I].
\vskip 0.2cm {\bf Corollary 2.1.} {\it Let $A$ be symplectic and
$\Lambda =
\{\lambda_1,\ldots , \lambda_r, 1/\lambda_1, \ldots , 1/\lambda_r \}$ where
$\lambda_i >1$, $i=1,\dots ,r$. Then,}
$$
\# Fix_n = \prod_{i=1}^r\left( \lambda_i^n + \lambda_i^{-n} -2
\right) \eqno(2.15)
$$
\vskip 0.2cm
{\it Proof.}
$$
\left| \prod_{i=1}^d\left( \lambda_i^n -1 \right) \right| =
\prod_{i=1}^r\left( \lambda_i^n -1 \right) \,
\prod_{i=1}^r\left( 1- \lambda_i^{-n} \right) =\prod_{i=1}^r\left(
\lambda_i^n + \lambda_i^{-n} -2 \right)
$$
Q.E.D.
We now consider the following dynamical zeta function [21]:
$$
\zeta (z) = \exp \sum_{n=1}^{\infty}{z^n\over n} \# Fix_n \eqno(2.16)
$$
By virtue of Theorem 2.2. the series converges for $|z|< e^{-h}$.
Moreover, from Proposition 2.1 it easily follows that for any
algebraic hyperbolic automorphisms of the $d$-torus $\zeta(z)$ is a rational
function with real zeroes and poles. In the simpler case of an orientation
preserving symplectomorphism, from Corollary 2.1 one obtains the
following expression:
\vskip 0.2cm
{\bf Proposition 2.2.} {\it Under the assumptions of Corollary 2.1 we have
$$
\zeta (z) = \prod_{\scriptstyle \di i_1\not= \dots \not= i_{r_1}
\not= j_1\not= \dots \not=j_{r_2}
\atop \scriptstyle \di r_1 + r_2+r_3 = r }
{1\over
\left(\di 1 - \lambda_{\di i_1}\dots
\lambda_{\di i_{r_1}}\lambda_{\di j_1}^{-1}\dots
\lambda_{\di j_{r_2}}^{-1} \, \cdot z \right)^{(\di -2)^{\di r_3} } }
\eqno(2.17)
$$
where the product is over distinct $(r_1+r_2)$-tuples.}
\vskip 0.2cm
Observe that the above zeta function has $2^r$ simple real poles located at
$z = \left(\lambda_{i_1}\dots \lambda_{i_{r_1}}\lambda_{j_1}^{-1}\dots
\lambda_{j_{r_2}}^{-1}\right)^{-1}$ where $i_1\not= \dots \not= i_{r_1}
\not= j_1\not= \dots \not=j_{r_2}$ and $r_1+r_2=r$. In particular, for
$r_2=0$ one gets the first pole
$z=\left(\lambda_1\dots \lambda_r\right)^{-1} = e^{-h}$.
All the other
poles and zeroes have multiplicities larger than one. In particular,
for $r_1=r_2=0$ one gets the most degenerate zero (or pole, depending on the
parity of $r$) located at $z=1$, with multiplicity $2^r$.
\vskip 0.2cm
{\bf Remarks.}
\vskip 0.2cm
1) For the case $d=2$ ($r=1$) one finds the expression
$$
\zeta (z) = { (1-z)^2 \over (1-\lambda z)(1-z/\lambda ) }
$$
where $\lambda$ is the largest eigenvalue of $A$ [7].
\vskip 0.2cm
2) If instead of the zeta function (2.16) one considers the
following Fredholm determinant:
$$
d(z) = \exp - \sum_{n=1}^{\infty}{z^n\over n} \sum_{x\in Fix_n}
{1\over |\det (A^n(x)-1)|}
$$
then using Proposition 2.1 one gets the expression (valid for any algebraic
hyperbolic automorphism $A$ of the $d$-torus): $\, d(z)=1-z$. In particular,
this implies the otherwise easily proven result that correlation functions
for analytic observables decay faster than any exponential (see [19]).
\vskip 0.2cm
We now deduce some consequences of the above results. For simplicity' sake
we shall give the arguments for
the case of the hyperbolic symplectomorphism of the $d$-torus, with
$d=2r$, but all the results given in the rest of this Section
can be easily
extended to the general setting.
Denote by $P$ the set of all primitive (i.e., non repeated) periodic orbits of
$A$ and by $\pp (n)$ and $\pi (x)$ the number of them whose
period is $n$ and less or equal to $x$ respectively, i.e.
$$
\pp (n) = \# \{\g \in P \, : \, p(\g ) = n \} \qquad
\pi (x) = \# \{\g \in P \, : \, p(\g )\leq x\} \eqno(2.18)
$$
\vskip 0.2cm
{\bf Proposition 2.3.} {\it
$$
\pp (n) \sim { e^{nh} \over n}\eqno(2.19)
$$
and
$$
\pi (x) \sim { e^{h} \over e^{h} -1}\cdot { e^{xh}\over
x}\eqno(2.20)
$$
where
$f(t)\sim g(t)$ means that $f(t)/g(t)\rightarrow 1$ when $t\to \infty$.}
\vskip 0.2cm
{\bf Remark.} Proposition 2.2 is the analogue of the {\it prime orbit theorem}
proved in the context of Axiom A flows (see [16]).
\vskip 0.2cm
{\it Proof.}
The strategy is to gain insight on the distribution of closed orbits out of the
meromorphy domain of $\zeta (z)$.
>From Proposition 2.2 we have that
$$
{\zeta^{\prime} (z)\over \zeta (z)}={e^h \over 1-e^hz} +g(z)
$$
where $g(z)$ is analytic in $\{z\, : \, |z|0$.
The rest of the proof makes use of the Wiener-Ikehara Tauberian theorem
and proceeds exactly in the same way as in the proof for subshifts of finite
type ([16], p.100).
Q.E.D.
\vskip 0.2cm
We now prove that closed orbits exhibit a regularity in a spatial sense.
In particular, we show that they are equidistributed on the average
with respect to the Lebesgue measure $\mu$.
Let $\mu_{\g}$ be the measure defined by
$$
\mu_{\g}={1\over p(\g)}\sum_{k=0}^{p(\g)-1}\delta_{A^k(x)}\qquad x\in \g
\eqno(2.24)
$$
and set $\int_{\gamma}f =\int_{\T^d}f d\mu_{\gamma}$.
Then we have the
\vskip 0.2cm
{\bf Proposition 2.4.} {\it For every continuous map $f :\T^d \to \R$ :}
$$
{1\over \pp (n)} \sum_{p(\gamma )=n}\int_{\gamma}f \longrightarrow
\int_{\T^d}f d\mu \quad\hbox{as}\quad x\to \infty \eqno(2.25)
$$
\vskip 0.2cm
{\bf Remark.} This result is an easy consequence of more
general results proved in [16]. Nevertheless, for the convenience of the
reader, we shall give a simple direct proof based on Theorem 2.2 and
Proposition 2.1.
\vskip 0.2cm
{\it Proof.} We
first write the number of fixed points of period $n$ as
$$
\# Fix_n = \sum_{l|n} l\cdot \pp (l)\eqno(2.26)
$$
where $l|n$ means that $l$ divides $n$.
Form Proposition 2.1 we then have
$$
\sum_{l|n} l\cdot \pp (l) \sim e^{nh}
\eqno(2.27)
$$
On the other hand
$$
\pp (n) = {1\over n}\left(
\sum_{l|n} l\cdot \pp (l) - \sum_{l|n, \; l< n} l\cdot \pp (l) \right)
$$
which, together with (2.27), gives
$$
\pp (n) \sim {e^{nh}\over n} (1 - C_n) \eqno(2.28)
$$
where $C_n$ is of order strictly smaller than one.
Analogously,
$$
\sum_{x\in Fix_n} f(x) = \sum_{l|n} l\cdot \sum_{p(\gamma )=l}\int_{\gamma}f
\eqno(2.29)
$$
>From Theorem 2.2 we then have
$$
\sum_{l|n} l\cdot \sum_{p(\gamma )=l}\int_{\gamma}f \sim e^{nh}\int_{\T^d}f
d\mu
\eqno(2.30)
$$
On the other hand
$$
\sum_{p(\gamma )=n}\int_{\gamma}f = {1\over n}\biggl(
\sum_{l|n} l\cdot \sum_{p(\gamma )=l}\int_{\gamma}f -
\sum_{l|n, \; l0$ and for any continuous
function $f$}
$$
\lim_{n\to \infty }\biggl( {\#\{\g\, : \, p(\g )=n,\;\; |\int_{\g}f -
\int_{\T^d}f d\mu |> \epsilon \}\over \pp (n)}\biggr) = 0
\eqno(2.32)
$$
{\it Proof.} For the sake of simplicity we shall use the notation $
\int_{\T^d}f d\mu = {\bar f}$. For any $k\in \Z_+$ set
$$
m_kf(x) = {1\over k}\sum_{l=0}^{k-1}f(A^lx)\eqno(2.33)
$$
Now, given $\delta >0$, introduce the sets
$$
\eqalign{ S_{\delta ,n} &= \{\gamma \, : \, p(\gamma )=n,\;\;
\left|\int_{\gamma}f - {\bar f}\right|\leq \sqrt{\delta} \} \cr
R_{\delta,n,k} &= \{\gamma \, : \, p(\gamma )=n,\;\;
\int_{\gamma}|m_kf - {\bar f}|\leq \sqrt{\delta} \} \cr } \eqno(2.34)
$$
We have
$$
\left|\int_{\gamma}f - {\bar f}\right|\leq \left|\int_{\gamma}(f - m_kf)\right|+
\int_{\gamma}|m_kf - {\bar f}|
$$
On the other hand it is obvious that $\int_{\gamma}(f - m_kf)=0$,
for any $k\in \Z_+$; and thus
$$
S_{\delta ,n}\supset R_{\delta ,n,k}\Longrightarrow
R_{\delta ,n,k}^c \supset S_{\delta ,n}^c
\quad\hbox{for any}\quad
\delta >0, \; k\in \Z_+
\eqno(2.35)
$$
The ergodicity of $A$ implies that, for $\mu$-almost every $x\in \T^d$,
$$
\lim_{k\to \infty} m_kf(x) = {\bar f}\eqno(2.36)
$$
Hence, for any continuous $f$, by the Lebesgue dominated convergence theorem
we can find a $k_0>0$ such that
$$
\int_{\T^d}|m_{k_0}f(x) - {\bar f}|d\mu \leq {\delta \over 2}\eqno(2.37)
$$
Set
$$
\int_{\T^d} f d\mu_n :=
{1\over \pp (n)} \sum_{p(\gamma )=n}\int_{\gamma}f
$$
Then Proposition 2.4 entails that $d\mu_n$ converges vaguely to the
Lebesgue measure. This implies the existence of $n_0(\delta )>0$ such that
for any $n>n_0$
$$
\int_{\T^d}|m_{k_0}f(x) - {\bar f}|d\mu_n \leq \delta \eqno(2.38)
$$
Hence by the Chebychev inequality we obtain
$$
{\# \{\gamma \, : \, p(\gamma )=n,\;\;
\int_{\gamma}|m_{k_0}f - {\bar f}|\geq \sqrt{\delta} \}\over \pp (n)}
\leq \sqrt{\delta}\eqno(2.39)
$$
and therefore, by the second of (2.34):
$$
{\# R_{\delta ,n,k_0}^c\over \pp (n)}
\leq \sqrt{\delta},\qquad n>n_0\eqno(2.40)
$$
If we now put $\delta = 1/k^2$ then from (2.35) we find
$$
{ \# \{\g\, : \, p(\g )=n,\;\; |\int_{\g}f -
\int_{\T^d}f d\mu |> {1\over k} \} \over \pp (n)}
\leq {1\over k},\qquad n>n_0
\eqno(2.41)
$$
so that, given $\eps >0$, we have for $k > 1/\eps$ and $n > n_0(k)$,
$$
{ \# \{\g\, : \, p(\g )=n,\;\; |\int_{\g}f -
\int_{\T^d}f d\mu |> \eps \} \over \pp (n)}
\leq {1\over k}\eqno(2.42)
$$
and the assertion follows by taking $k \to \infty$ and consequently
$n\to \infty$.
Q.E.D.
\vskip 1cm
{\bf 3. Periodic orbits living on prime lattices}
\vskip 0.5cm
We now turn to study the asymptotic properties of the probability measures
$\mu_{\gamma}$ (cf. (2.24))
supported on {\sl single} periodic orbits $\gamma$'s of $A$ when
$p(\gamma) \to\infty$.
To this end we first remark the following fact.
Consider a sequence $\{ \g_k\}$ of periodic orbits of $A:\T^d \to \T^d$
ordered with non-decreasing periods.
Then $\{\mu_{\gamma_k}\}$ converges weakly to $\mu$ as $k\to \infty$
if and only if [1]
$$
\int_{\gamma_k} e^{2\pi i} \rightarrow \int \, e^{2\pi i}\, d\mu
\eqno(3.1)
$$
for any $n\in \Z^d$. Thus, it suffices to consider integrals of
characters.
\vskip 0.2cm
Let $A\in GL(d,\Z)$ with det$\,A=1$, we denote with $f(x)\in\Z[x]$ the
corresponding characteristic polynomial.
Given an integer $p$, we denote by $\L_p$ the invariant subset of
$\T^d$ given by all the rational points with denominator $p$, i.e.
$$
\L_p=\{x\in\T^d\,\,\vert p x\in\Z^d\,\,\}\eqno(3.2)
$$
$\L_p$ is invariant under the action of $A$, so that any point in $\L_p$
is periodic with period $\leq p^d$.
This means that $\L_p$ splits into periodic orbits (which in general may have
different periods) of $A$.
As it will become more clear in the sequel, to study the structure of
periodic orbits living on $\L_p$ amounts to perform with a modulo
$p$ arithmetic.
In what follows we shall only consider lattices $\L_p$ for primes
$p$ such that $f(x)$ completely
splits over $\Z_p$, that is $f(x)=\prod_{j=1}^{d} (x-\lj)$, $\lj\in\Z_p$,
and call such a prime {\it completely splitting}.
As a consequence of Chebotarev's theorem [3], the set of such
primes $p$ as positive density among all primes. One can actually
give an estimate from below of such a density in terms of the order of the
Galois group associated to the polynomial $f(x)$ [3].
Given a completely splitting prime $p$, we denote by
$$
\Lambda =\{\lambda_1,\cdots,\lambda_d\}\eqno(3.3)
$$
the set of the eigenvalues of $A$, which in general are not all distinct and satisfy
$\lambda_1\lambda_2....\lambda_d=1$.
In the case $d=2r$ and $A\in SL(r,\Z)$, i.e. $A$ symplectic,
$\lj\in\Lambda$ implies $\lj^{-1}\in \Lambda$.
For any $j=1,...,d$, we denote by $\t_j$ the order of $\lj$ in the
multiplicative group $\Z_p^*$,
that is $\lj^{\t_j}=1$ and $\lj^l\neq 1,\,\, 01$) such that $A^{\t(x)}x=x$.
Clearly, if $x=\sum_{j\in \D} \aj\vj$ then
$$
\t(x)=l.c.m.\{\, \t_j\,: \,
j\in \D \}\eqno(3.8)
$$
The set $\P$ of all {\it admissible periods} is then given by
$$
{\P} =\{\, \t \, = \, l.c.m.\left( \t_{j_1},\t_{j_2},....,\t_{j_k}\,: \,
(j_1,....,j_k)\subset (1,.....,d)\right)\, \}\eqno(3.9)
$$
$\t_1$ is the smallest period, whereas $\t_{max}={\rm l.c.m.}\{\t_1,\dots,\t_d\}
\leq (p-1)^{d}$ is the largest one among the admissible periods.
Clearly, the natural ordering of $\P$ induces a filtration of the lattice $L_p$
in subsets of points with common period.
\vskip 0.2cm
{\bf Remark.}
In the two dimensional case, it is known ([14], [4]) that all points
in $\L_p\setminus \{0\}$ share the same period: $\t (x) = \t = (p-1)/m$
(for inert primes $p$, $\t (x) = \t = (p+1)/m$, [14]), so that
$\P$ is made out of a single point.
The number $m={(p-1)/ \t}$ corresponds exactly to the degeneracy of the
eigenvalues of the associated quantum propagator [4].
\vskip 0.2cm
In order to study the spatial distribution of a given periodic orbit,
we are concerned in evaluating the average of an arbitrary (non
trivial) character.
Given $p$ as above, $x\in \Z_p^d$ and $n\in \Z^d$, we define
$$
I_p(x,n)={{1}\over \t(x)}\sum_{s=0}^{\t(x)-1} \,e^{\fp}\eqno(3.10)
$$
or, equivalently,
$$
I_p(x,n)={{1}\over \t(x)}\sum_{s=0}^{\t(x)-1} \,e^{\fp\sum_{j=1}^{d}\beta_j
\lambda_j^s}\eqno(3.11)
$$
where $\beta_j=\aj\in\Z_p$.
We shall reduce the above expression to (some combinations of)
generalized sums over finite fields of the type:
$$
\sum_{x\in \Z_p} \chi (f(x)) \psi(g(x))
$$
where $\chi$ is a non trivial multiplicative character of order $d|(p-1)$ of
$\Z_p^*$
(i.e. $\chi^d$ is equal to the trivial character $\chi_0$),
$\psi$ a non trivial additive character of $\Z_p$ and
$f(x), g(x)\in \Z_p [x]$ are given algebraic functions, for instance polynomials, over
$\Z_p$.
The reference result is then the following:
\vskip 0.2cm
{\bf Theorem 3.1.} {\it Let $\chi$,$\psi$ be a multiplicative character $\neq
{\chi}_0$ of order $d$ with $d\vert (p-1)$, and a non trivial additive character,
respectively, of $\Z_p$. Let $f(x)\in \Z_p [x]$ admit $m$ distinct roots,
and let $g(x)\in \Z_p [x]$ have degree
$n$. Suppose that either $(d,deg(f))=(n,p)=1$, or, more generally, that the polynomials
$y^d-f(x)$ and $z^p-z-g(x)$ are absolutely irreducible (i.e. irreducible over any
finite algebraic extension of $\Z_p$). Then}
$$
\vert \sum_{x\in\Z_p} \chi (f(x)) \psi(g(x))\vert\leq (m+n-1) p^{1/2}
$$
{Proof.} See, e.g., [20] page 45.
\vskip 0.2cm
This result has been obtained by A. Weil as a consequence of the validity of the
Riemann hypothesis for curves over finite fields and important extensions
to cases where
$f,g$ are rational functions over arbitrary finite fields
has been achieved by P. Deligne [6].
{\bf Examples.}
\item{1)} For $f(x)=g(x)=x$ we have the
generalized {\it Gauss sum} $g(\psi ,\chi )$,
and, by direct application of Theorem 3.1, $|g(\psi,\chi)|\leq
\sqrt{p}$.
\vskip 0.2cm
\item{2)} Setting
$f(x)=x^2-4ab$, $g(x)=x$, $\chi$ the (non trivial) character of order two (Legendre
symbol) and
$\psi =\exp{\left({2\pi ix\over p}\right)}$, we find the {\it Kloosterman sum}
$$
Kl(p,a,b)=
\sum_{x\in \Z_p}\left({x^2-4ab \over p}\right)\exp{ \fa x}=
\sum_{x\in \Z_p^*}\exp{ {2\pi i \over p} (ax+bx^{-1})}
$$
Again, from Theorem 3.1, one has the estimate
$\vert Kl(p,a,b)\vert \leq 2\sqrt{p}$.
\vskip 0.2cm
The next result is useful
in reducing to the previous case some sums over cyclic
subgroups.
\vskip 0.2cm
{\bf Lemma 3.1.} {\it
$\forall \lambda\in\Z_p^*$, denote
$\Lambda_{\lambda}=<\lambda>$ the cyclic subgroup generated by $\lambda$. Let
$\#\Lambda_{\lambda}={p-1\over m}=\t$. If
$f:\Z_p\longrightarrow\C$ is any complex valued function, then
$$
\sum_{s=0}^{\t-1} f({\lambda}^s) ={1\over m}\sum_{j=0}^{m-1}
\sum_{x\in{\Z_p}^*} {\chi}_j(x)f(x)
$$
where $\{{\chi}_0,\cdots,{\chi}_{m-1}\}$ is a set of multiplicative characters of
order $m$.}
\vskip 0.2cm
{\it Proof.} See the Appendix in [4].
\vskip 0.5cm
{\sl 3.1. Ideal lines and ideal orbits.}
\vskip 0.2cm
As a first step toward the understanding of the distribution
of a generic periodic orbit,
we now consider those orbits which lay entirely on
invariant sublattices of the form $\G_j=\Z_p\vj=\{u\vj\,\,\vert\,u\in\Z_p\}
\subset \L_p$ for $j=1,\dots ,d$, where $v_j\in \Z_p^d$ is an eigenvector of $A$.
In analogy with the two dimensional case, we call $\G_j$ and $\gamma \subset \G_j$
{\it ideal line} and {\it ideal orbits}, respectively ([2],[4]). Notice that any
ideal line is almost completely equidistributed, in the following sense:
for any $n\in \Z^d$ such that $\not= 0\, {\rm mod}\, p$, we have
$$
\sum_{x\in \G_j} e^{\fp } = \sum_{u\in \Z_p} e^{\fp u}=0\eqno(3.12)
$$
Define
$$
K_j \equiv K_j(p) = \{n\in \Z^d\setminus \{0\}\, : \, =0\, {\rm mod}\, p\}
$$
for $j=1,\dots ,d$, and
$$
k_j = \inf_{n\in K_j}|n|
$$
where $|n|$ denotes the euclidean length. We then have the following,
\vskip 0.2cm
{\bf Lemma 3.2.} {\it Let $A$ be irreducible.
$$
k(p)=\inf_j k_j \rightarrow \infty \quad\hbox{as}\quad p \to \infty
$$
where the limit is taken over an arbitrary increasing sequence of completely
splitting primes.}
\vskip 0.2cm
{\it Proof.} Assume that there exists an infinite sequence of primes
$\{p_l\}$ such that
$k(p_l)$ remains bounded as $l\to \infty$.
By eventually restricting to a subsequence, we can assume that there exist a
fixed
$n\in\Z^d$ such that for each $l$ there exist an eigenvector
$v_{j}\in \Z^d_{p_l}$ such that $=0\,\, {\rm mod}\, p_l$.
Given $l\in \Z_+$, let us denote by $M_l\subset \Z_{p_l}^d$ the
$\Z_{p_l}$-module generated by $\{B^sn\}_{s\in \Z}$, where $B=A^t$.
$M_l$ is a non trivial submodule of dimension $1\leq r_l\leq d-1$. In
particular, there exist a submodule $N_l\supset M_l$ with the same dimension
$r_l$ and another non trivial submodule $R_l$ such that
$Z^d_{p_l}=N_l\bigoplus R_l$.
Since this decomposition holds for an infinite number of primes and since
$n$ has been chosen independent of $l$, a simple argument shows in fact the
existence of a non trivial $B$-invariant $\Z$-module $M\subset \Z^d$, in
contradiction with the irreducibility hypothesis. Q.E.D.
\vskip 0.2cm
{\bf Proposition 3.1.} {\it
For any completely splitting prime $p$ and for any $x\in \G_j$ ($x\neq
\{0\}$) }
$$
\left| I_p(x,n)\right| = \cases{\leq {m_j\over\sqrt{p}}, &if $n\notin K_j$; \cr
1, &otherwise \cr }
$$
\vskip 0.2cm
{\bf Remark.}
\item{1)} If $A$ is irreducible, then Lemma 3.2 and the above
result implies that any sequence
$\gamma_k\subset
\L_{p_k}$ of periodic ideal orbits, where $\{p_k\}$ is any increasing sequence
of completely splitting primes such that
$m_k/\sqrt{p_k}\to 0$ as $k\to \infty$,
becomes equidistributed according to
$\mu$ as $k\to \infty$.
\item {2)} If $A$ is reducible, e.g. $A=\left( \matrix{
B & 0 \cr 0 & C \cr}\right)$, where $B\in GL(r,\Z)$, $C\in GL(s,\Z)$ with
$r+s=d$, then there is an invariant decomposition $\T^d=\T^r\times\T^s$. In
this case, one can also find families of periodic orbits which are confined
to invariant tori of dimension $< d$.
The same technique can then be used to describe the equidistribution of
the such periodic orbits with respect the Haar measure supported on these
invariant tori.
\vskip 0.2cm
{\it Proof.}
Let $x\in \G_j$, i.e. $x=\alpha v_j$, $\alpha \in\Z_p^*$.
If $n\in K_j$ then, clearly, $I_p(x,n)=1$.
On the other
hand, if $\beta_j=\alpha \neq 0\, {\rm mod}\, p$, using Lemma 3.1 we
find
$$\eqalign{
&I_p(x,n)={1\over \t_j}\sum_{s=0}^{\t_j-1} e^{{2\pi i\over p}\beta_j\lambda_j^s}=
{1\over \t_j m_j}\sum_{x\in\Z_{p}^*}e^{{2\pi i\over p}\beta_j x^{m_j}}\cr
&= {1\over \t_j\, m_j}\sum_{k=0}^{m_j-1}\sum_{x\in\Z_{p}^*}\chi_k(x)e^{{2\pi i\over
p}\beta_j x}= {1\over \t_j\, m_j}\sum_{k=0}^{m_j-1} g(\psi_{\beta_j},\chi_j)\cr }
\eqno(3.16)
$$
where we have used the notation
$\psi_a(x) = \exp{\left(\di {2\pi iax \over p}\right)}$, $a\in \Z_p$.
The first statement then follows from the estimate on the
absolute value of an arbitrary Gauss sum. Q.E.D.
\vskip 0.5cm
{\sl 3.2. Generic orbits.}
\vskip 0.2cm
Having proved the equidistribution of ideal orbits,
we now turn to the generic situation.
The simplest cases to treat
are those corresponding to a {\it completely non resonant} situation
and/or a {\it completely resonant} situation.
Let $x\in \L_p$ and write again
$x=\sum_{j\in \D} \alpha_j v_j$, where $\D \subseteq \{1,\dots ,d\}$ and
$0 \not= \alpha_j\in Z_p$ for any $j\in \D$.
\vskip 0.2cm
{\bf Definition 3.1.} {\it A point $x\in\L_p$ as above is called
{\rm completely non
resonant} if
$(\t_i,\t_j)=1$ for all $i,j\i \D$, i.e. all the cyclic subgroups generated by
$\lambda_j$'s with $j\in \D$ are distinct with trivial intersections.}
\vskip 0.2cm
{\bf Proposition 3.2.} {\it Let $x\in\L_p$ be completely non resonant and
$n\in \Z^d\setminus \cap_{j\in \D}K_j$, then
$$
\vert I_p(x,n)\vert\leq {1\over \sqrt{p}}\min_{j\in \D'}m_j
$$
where $\D' \subseteq \D$ is such that $\beta_j=\alpha_j \not= 0$ for
$j\in \D'$}.
\vskip 0.2cm
{\it Proof.}
Set $k=\# \D$. Let $\{\lambda_1,
\dots , \lambda_k\}$ be the eigenvalues corresponding to $v_j$, $j\in \D$,
and let $\{\t_1,\dots ,\t_k\}$, $\{m_1, \dots ,m_k\}$ be the corresponding
periods and degeneracies, respectively. We have $\t(x)=\t=\t_1\t_2...\t_k$.
Set moreover $\t_0=\t_1 \t_2 \dots\t_{k-1}$.
Note that $(\t_0,\t_k)=1$
and therefore
$\{\lambda_k^{l\t_0}\}_{l=0}^{\t_k-1}=\{\lambda_k^s\}_{s=0}^{\t_k-1}$.
We then get
$$\eqalign{
I_p(x,n)&=\ti\st e^{\fp[\beta_1\lambda_1^s+\dots\bk\lk^s]}\cr
&=\ti\sum_{s=0}^{\t_0-1}\sum_{l=0}^{\t_k-1}
e^{\fp[\beta_1\lambda_1^{(s+l\t_0)}+\dots+\bk\lk^{(s+l\t_0)}]}\cr
&=\ti\sum_{s=0}^{\t_0-1}e^{\fp(\beta_1\lambda_1^s+\dots+\beta_{k-1}\lambda_{k-1}^s)}\,
\sum_{l=0}^{\t_k-1} e^{\fp\bk\lk^{s}\lk^{l\t_0}} \cr
&=\ti\sum_{s=0}^{\t_0-1}e^{\fp(\beta_1\lambda_1^s+\dots+\beta_{k-1}\lambda_{k-1}^s)}\,
{{1}\over{m_k}}\sum_{l=0}^{m_k-1} g(\psi_{\bk\lk^s},\chi_l) \cr } \eqno(3.17)
$$
where, again, $g(\psi_a,\chi)=\sum_{x\in\Z_p^*} \chi(x) e^{\fp ax}$
denotes the Gauss sum.
If now $\beta_k \not= 0$ we obtain
$$
\vert I_p(x,n)\vert\leq {{\t_0}\over{\t}}\sqrt{p}\leq
{{m_k}\over{\sqrt{p}}}\eqno(3.18)
$$
which immediately yields the result. Q.E.D.
\vskip 0.2cm
Let us now consider the opposite case.
\vskip 0.2cm
{\bf Definition 3.2.} {\it A point $x\in\L_p$ as above is called
{\rm completely resonant} if, for all $i,j\i \D$, $<\lambda_i>=
<\lambda_j>=<\lambda>$}.
\vskip 0.2cm
This definition implies that $\t(x)=\t=(p-1)/m$. Moreover, for any $i,j\i \D$
there exist two integers $s_i$, $s_j$ such that $\lambda_i^{s_i}=\lambda_j^{s_j}$.
Set again $k=\# \D$. For any $j\in \D$ there exist
$\eta_j\in\{-{{\t-1}\over{2}},...,-1,0,1,....,{{\t-1}\over{2}}\}$ such that
$\lambda_j=\lambda^{\eta_j}$. We may choose $\lambda$
and the $\eta_j$ in such a way that
$\eta =2\, \max_{j\in \D} \vert\eta_j\vert$ is as small as possible.
\vskip 0.2cm
{\bf Proposition 3.3.} {\it Let $x\in\L_p$ be completely resonant and
$n\in \Z^d\setminus \cap_{j\in \D}K_j$,}
then
$$
\vert I_p(x,n)\vert\leq {m\over \sqrt{p}}\, \eta
$$
\vskip 0.2cm
{\it Proof.}
We have
$$\eqalign{
I_p(x,n)&={1\over\t}\sum_{s=0}^{\t-1} e^{\fp[\beta_1\lambda_1^{m\eta_1}+\dots
\beta_k\lambda_k^{m\eta_k}]}\cr
&={1\over \t
m}\sum_{x\in\Z_p^*}\, e^{\fp[\beta_1x^{m\eta_1}+...\beta_kx^{m\eta_k}]}\cr
& ={1\over \t \,
m}\sum_{j=0}^{m-1}\sum_{x\in\Z_p^*}\,\chi_j(x)\,e^{\fp[\beta_1x^{\eta_1}+\dots+\beta_kx^{
\eta_k}]}\cr } \eqno(3.19)
$$
and the result follows by direct application of Theorem 3.1. Q.E.D.
\vskip 0.2cm
{\bf Remark.} In the case $d=2$ one has $\Lambda =\{\lambda, \lambda^{-1}\}$.
Moreover, since any point in $\L_p\setminus \{0\}$ is completely resonant
(with $\t(x) = (p-1)/m$) one finds the expression
$$\eqalign{
I_p(x,n)&={1\over\t}\sum_{s=0}^{\t-1}
e^{\fp[\beta_1\lambda^s+ \beta_2\lambda^{-s}]} \cr
&={1\over \t m}\sum_{x\in\Z_p^*}\,
e^{\fp[\beta_1x^{m}+\beta_2x^{-m}]}\cr
& ={1\over \t \,
m}\sum_{j=0}^{m-1}\sum_{x\in\Z_p^*}\,\chi_j(x)\,
e^{\fp[\beta_1x+\beta_2x^{-1}]}\cr } \eqno(3.20)
$$
and the estimate on the Kloosterman sums can be used to give (see Example
2 above)
$$
\left| I_p(x,n)\right| \leq {2 m\over \sqrt{p}}
$$
so that any sequence
$\gamma_k\subset
\L_{p_k}$ of periodic orbits, where $\{p_k\}$ is any increasing sequence
of splitting primes such that
$m(p_k)/\sqrt{p_k}\to 0$ as $k\to \infty$, becomes equidistributed according to
$\mu$ as $k\to \infty$ [4],[5].
\vskip 0.2cm
To conclude, we briefly discuss the more general case,
corresponding to {\it partially resonant} situations.
Given an arbitrary point $x\in \L_p\setminus \{0\}$ let us write it again
in the form $x=\sum_{j\in \D} \alpha_j v_j$.
For any fixed $\lambda =\lambda_k$, $k\in \D$,
we decompose $\D$ as follows
$$
\D = \D_1 \cup \D_2 \cup \D_3
$$
where, if $\t=\t_k$ denotes the order of $<\lambda>$:
\item{1)} for any $j\in \D_1$: $(\t,\t_j)=1$;
\item{2)} for any $j\in \D_2$: $<\lambda_j>=<\lambda>$, ($\t_j=\t$);
\item{3)} ({\it partially resonant}) for any $j\in \D_3$:
$<\lambda_j>\not=<\lambda>$ and $<\lambda_j>\cap<\lambda>\not= \emptyset$.
In particular, if $j\in \D_3$ there exist $s_1=s_1(j)$ and $s_2=s_2(j)$
such that $\lambda^{s_1} = \lambda_j^{s_2}$.
\vskip 0.2cm
Now, if one can choose $\lambda$ in such a way that the corresponding
set $\D_3$ is empty, then the integral $I_p(n,x)$ can be estimated
by suitably combining the previous two cases. More precisely, one can
first isolate the non resonant eigenvalues as in (3.17) and then
proceed as in (3.19).
Finally, the case where the presence of partial resonances cannot be
avoided seems to be more involved. Nevertheless the same sort of
techniques might be used to reduce $I_p(n,x)$ to suitable exponential
sums over $\Z_p$ and thus produce, at least in principle, explicit estimates.
\vfill \eject
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\end