MPEJ Volume 7, No. 1, 18pp.
Received: May 31, 2001. Accepted: Sep 26, 2001.

Mayer D., Neunhaeuserer J.
An isomorphism between polynomial eigenfunctions of the transfer
operator and the Eichler cohomology for modular groups

ABSTRACT: For the group $PSL(2,\Z)$ it is known that there is an
isomorphism between polynomial eigenfunctions of the transfer operator
for the geodetic flow and the Eichler cohomology in number theory, see
\cite{[CM3]}, \cite{[LW]}, \cite{[LZ]}. In \cite{[CM3]} it is
indicated that such an isomorphism exists as well for the subgroups
$\Gamma(2)$ and $\Gamma_{0}(2)$ of $PSL(2,\Z)$. We will prove this and
provide some evidence by computer aided algebraic calculations that
such an isomorphism exists for all principal congruence subgroups
$\Gamma(N)$ and all congruence subgroups of Hecke type
$\Gamma_{0}(N)$.
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MPEJ Volume 7, No. 2, 19 pp.
Received: Jun 13, 2000. Revised: Oct 3, 2001. Accepted: Oct 9, 2001.

M. Abadi:
Exponential approximation for hitting times in mixing processes

ABSTRACT: We present bounds for the error of the exponential
approximation of the first occurrence time of a rare event in a
stationary stochastic process with a finite alphabet with the
$\alpha$-mixing property with a summable function $\alpha$ or with a
general $\phi$-mixing property. We prove a lower bound for this error
in terms of the measure of the cylinder and an upper bound as a
function of the measure of the cylinder plus the decay of correlation
of the process.
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MPEJ Volume 7, No. 3, 22 pp.
Received: Oct 23, 2001. Revised: Nov 14, 2001. Accepted: Nov 17, 2001.

G. Benettin, L. Rondoni:
A new model for the transport of particles in a thermostatted system

ABSTRACT: We introduce an elementary microscopic model for the free
expansion of a gas from a smaller to a larger reservoir; a ``pump''
takes the particles back to the original reservoir, so as to maintain
a stationary situation.  The system is dissipative and thermostatted
(with an elementary non Gaussian reversible thermostat). We consider
here only the case of noninteracting particles. We identify a
parameter in the system, playing the role of the field, and
investigate numerically the usual transport properties, among them the
proportionality of the current to the field for small fields, and the
proportionality of the volume contraction rate to the product of field
and current, in the same conditions. In spite of the simplicity of the
model (and of the difficulty in the thermodynamical interpretation,
the model being non interacting) we find apparently normal transport
laws. We also make a particularly accurate test of the
Gallavotti--Cohen fluctuation formula, which turns out to be always
well satisfied (though fluctuations, in some conditions, are rather
far from Gaussian). A leading idea of the paper is to look for the
minimal requirements that a particle system should satisfy in order to
give rise, at least formally, to normal transport.
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MPEJ Volume 7, No. 4, 59 pp.
Received: Mar 15, 2001. Revised: Oct 17, 2001. Accepted: Dec 12, 2001.

F. Besnard:
Number Operator Algebras

ABSTRACT: Under some hypotheses (symmetry, confluence), we enumerate
all quadratically presented algebras, generated by creation and
destruction operators, in which number operators exist. We show that
these are algebras of bosons, fermions, their immediate
generalizations that we call pseudo-bosons and pseudo-fermions, and
also matrix algebras, in the finitely generated case. We then recover
$q$-bosons (and pseudo-$q$-bosons) by a completion operation.