- 00-466 Stefano Isola
- ON SYSTEMS WITH FINITE ERGODIC DEGREE
Nov 22, 00
(auto. generated ps),
of related papers
Abstract. In this paper we study the ergodic theory of a class of symbolic
$(\O, T, \mu)$ where $T:\O \to \O$ the the left shift transformation on
and $\mu$ is a $\s$-finite $T$-invariant measure having the property
there is real number $d>-1$ so that $\mu(\tau^d)=\infty$ but
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 non-atomic
equilibrium measure of $V$
for which we compute the (asymptotically) exact mixing rate, of order
We also establish the weak-Bernoulli property and a polynomial cluster
(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 operator-valued power series which
from a standard inducing procedure. A detailed control of the singular
of these functions in the vicinity of their non-polar singularity at
is achieved through an approximation scheme which uses generating
a suitable renewal process. In the perspective of differentiable
these are statements about the unique absolutely continuous invariant
a class of piecewise smooth interval maps with an indifferent fixed