188 research outputs found
On systems with finite ergodic degree
In this paper we study the ergodic theory of a class of symbolic dynamical
systems (\O, T, \mu) where T:{\O}\to \O the left shift transformation on
\O=\prod_0^\infty\{0,1\} and is a \s-finite -invariant measure
having the property that one can find a real number so that
but ,
where is the first passage time function in the reference state 1. In
particular we shall consider invariant measures arising from a potential
which is uniformly continuous but not of summable variation. If then
can be normalized to give the unique non-atomic equilibrium probability
measure of for which we compute the (asymptotically) exact mixing rate, of
order . We also establish the weak-Bernoulli property and a polynomial
cluster property (decay of correlations) for observables of polynomial
variation. If instead then is an infinite measure with scaling
rate of order . Moreover, the analytic properties of the weighted
dynamical zeta function and those of the Fourier transform of correlation
functions are shown to be related to one another via the spectral properties of
an operator-valued power series which naturally arises from a standard inducing
procedure. A detailed control of the singular behaviour of these functions in
the vicinity of their non-polar singularity at is achieved through an
approximation scheme which uses generating functions of a suitable renewal
process. In the perspective of differentiable dynamics, these are statements
about the unique absolutely continuous invariant measure of a class of
piecewise smooth interval maps with an indifferent fixed point.Comment: 42 page
On the rate of convergence to equilibrium for countable ergodic Markov chains
Using elementary methods, we prove that for a countable Markov chain of
ergodic degree the rate of convergence towards the stationary
distribution is subgeometric of order , provided the initial
distribution satisfies certain conditions of asymptotic decay. An example,
modelling a renewal process and providing a markovian approximation scheme in
dynamical system theory, is worked out in detail, illustrating the
relationships between convergence behaviour, analytic properties of the
generating functions associated to transition probabilities and spectral
properties of the Markov operator on the Banach space . Explicit
conditions allowing to obtain the actual asymptotics for the rate of
convergence are also discussed.Comment: 31 pages. to appear in Markov Processes and Related Field
On a set of numbers arising in the dynamics of unimodal maps
In this paper we initiate the study of the arithmetical properties of a set
numbers which encode the dynamics of unimodal maps in a universal way along
with that of the corresponding topological zeta function. Here we are concerned
in particular with the Feigenbaum bifurcation.Comment: 12 page
On the generic triangle group
We introduce the concept of a generic Euclidean triangle and study the
group generated by the reflection across the edges of . In
particular, we prove that the subgroup of all translations in
is free abelian of infinite rank, while the index 2 subgroup of all
orientation preserving transformations in is free metabelian of rank
2, with as the commutator subgroup. As a consequence, the group
cannot be finitely presented and we provide explicit minimal infinite
presentations of both and . This answers in the affirmative
the problem of the existence of a minimal presentation for the free metabelian
group of rank 2. Moreover, we discuss some examples of non-trivial relations in
holding for given non-generic triangles .Comment: 21 pages, 6 figure
A renormalization approach to irrational rotations
We introduce a renormalization procedure which allows us to study in a
unified and concise way different properties of the irrational rotations on the
unit circle , \alpha \in \R\setminus \Q. In
particular we obtain sharp results for the diffusion of the walk on
generated by the location of points of the sequence on a
binary partition of the unit interval. Finally we give some applications of our
method.Comment: 27 page
Series expansions for Maass forms on the full modular group from the Farey transfer operators
We deepen the study of the relations previously established by Mayer, Lewis
and Zagier, and the authors, among the eigenfunctions of the transfer operators
of the Gauss and the Farey maps, the solutions of the Lewis-Zagier three-term
functional equation and the Maass forms on the modular surface
PSL(2,\Z)\backslash \HH. In particular we introduce an "inverse" of the
integral transform studied by Lewis and Zagier, and use it to obtain new series
expansions for the Maass cusp forms and the non-holomorphic Eisenstein series
restricted to the imaginary axis. As corollaries we obtain further information
on the Fourier coefficients of the forms, including a new series expansion for
the divisor function.Comment: 35 page
Spectral analysis of transfer operators associated to Farey fractions
The spectrum of a one-parameter family of signed transfer operators
associated to the Farey map is studied in detail. We show that when acting on a
suitable Hilbert space of analytic functions they are self-adjoint and exhibit
absolutely continuous spectrum and no non-zero point spectrum. Polynomial
eigenfunctions when the parameter is a negative half-integer are also
discussed.Comment: 28 pages, 1 figur
Mean field behaviour of spin systems with orthogonal interaction matrix
For the long-range deterministic spin models with glassy behaviour of
Marinari, Parisi and Ritort we prove weighted factorization properties of the
correlation functions which represent the natural generalization of the
factorization rules valid for the Curie-Weiss case.Comment: Improved exposition, few typos and a combinatorial mistake corrected.
(To appear in Journal of Statistical Physics
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