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 $\mu$ is a $\s$-finite $T$-invariant measure
having the property that one can find a real number $d$ 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 non-atomic equilibrium probability
measure of $V$ for which we compute the (asymptotically) exact mixing rate, of
order $n^{-d}$. We also establish the weak-Bernoulli 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 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 $z=1$ 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.