  The problem of stability of the optimal filter is revisited. The optimal
filter (or filtering process) is the conditional probability of the current
state of some stochastic process (the signal process), given both present and
past values of another process (the observation process). Typically the
filtering process satisfies a dynamical equation, and the question investigated
here concerns the stability of this dynamics. In contrast to previous work,
signal processes given by the iterations of a deterministic mapping $f$ are
considered, with only the initial condition being random. While the stability
of the filter may emerge from strong randomness of the signal processes,
different and more dynamical effects will be exploited in the present work.
More specifically, we consider uniformly hyperbolic $f$ with strong
instabilities providing the necessary mixing. This however requires that the
filtering process is initialised with densities exhibiting already a certain
level of smoothness. Furthermore, $f$ may also have stable directions along
which the filtering process will eventually not have a density, a major
technical difficulty. Further results show that the filtering process is
asymptotically concentrated on the attractor and furthermore will have
densities with respect to the invariant (SRB)~measure along instable manifolds
of $f$.
