This papers shows that nonlinear filter in the case of deterministic dynamics
is stable with respect to the initial conditions under the conditions that
observations are sufficiently rich, both in the context of continuous and
discrete time filters. Earlier works on the stability of the nonlinear filters
are in the context of stochastic dynamics and assume conditions like compact
state space or time independent observation model, whereas we prove filter
stability for deterministic dynamics with more general assumptions on the state
space and observation process. We give several examples of systems that satisfy
these assumptions. We also show that the asymptotic structure of the filtering
distribution is related to the dynamical properties of the signal.Comment: 24 pages, 2 figures. In V3, few subsections are added and several
typos are correcte