 00466 Stefano Isola
 ON SYSTEMS WITH FINITE ERGODIC DEGREE
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Nov 22, 00

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Abstract. In this paper we study the ergodic theory of a class of symbolic
dynamical systems
$(\O, T, \mu)$ where $T:\O \to \O$ the the left shift transformation on
$\O=\prod_0^\infty\{0,1\}$
and $\mu$ is a $\s$finite $T$invariant measure having the property
that
there is real number $d>1$ so that $\mu(\tau^d)=\infty$ but
$\mu(\tau^{d\epsilon})<\infty$
for all $\epsilon >0$, where $\tau$ is the first passage time function
in the reference state $1$.
In particular we shall consider invariant measures $\mu$ arising from a
potential $V$ which
is uniformly continuous but not of summable variation.
If $d>0$ then $\mu$ can be normalized to give the unique nonatomic
equilibrium measure of $V$
for which we compute the (asymptotically) exact mixing rate, of order
$n^{d}$.
We also establish the weakBernoulli property and a polynomial cluster
property
(decay of correlations) for observables of polynomial variation.
If instead $d\leq 0$ then $\mu$ is an infinite measure with scaling rate
of order $n^d$.
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 operatorvalued 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 nonpolar singularity at
$z=1$
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.
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