  We investigate state estimation of linear systems over channels having a
finite state not known by the transmitter or receiver. We show that similar to
memoryless channels, zero-error capacity is the right figure of merit for
achieving bounded estimation errors. We then consider finite-state, worst-case
versions of the common erasure and additive noise channels models, in which the
noise is governed by a finite-state machine without any statistical structure.
Upper and lower bounds on their zero-error capacities are derived, revealing a
connection with the {\em topological entropy} of the channel dynamics. Separate
necessary and sufficient conditions for bounded linear state estimation errors
via such channels are obtained. These estimation conditions bring together the
topological entropies of the linear system and the discrete channel.
