In this paper we study the
ergodic theory of a class of symbolic dynamical
systems
where
is the
left shift transformation on
and
m
is a
s
-finite
T-invariant
measure having the property that one can find a
real number
d so that
but
for all
where
t is the first
passage time function in the reference state 1.
In particular, we shall consider invariant
measures
m arising
from a potential
V which is uniformly
continuous but not of summable variation. If
then
m can be
normalized to give the unique non-atomic
equilibrium probability 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 property
(decay of correlations) for observables of
polynomial variation. If instead
then
m 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.
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