Stefano Isola
Dynamical zeta functions and correlation functions
for non-uniformly hyperbolic transformations.
(63K, Plain-Tex)
ABSTRACT. We consider a class of maps $f$ of $[0,1]$
which are expanding everywhere but at a fixed point,
which we allow to be neutral.
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 a suitable operator-valued power series
associated to an induced version $g$ of the map $f$.
One result is that, when the fixed point is neutral,
these functions are holomorphic in the unit disk and have
a non-polar singularity (branch point) in $z=1$.
Moreover, they have a power series expansion
in a neighbourhood of the singular point which can be
determined from the behaviour of the map near the fixed point.
The decay rate of correlations of a $\sigma$-finite absolutely
continuous invariant measure $\nu$, when the latter is finite,
as well as other relevant ergodic quantities when $\nu$ is infinite,
are then obtained straightforwardly from these expansions.
(This is an extended version of a previously archived paper).