In this paper, we develop a communication protocol for the observation of discrete time, possibly unstable, dynamical systems over communication channels with limited communication capacity. We develop an observer based on the upper box dimension for one-way communication channels that leads to a certain type of observability. This communication scheme preserves observability under communication losses which makes the communication scheme robust towards communication losses without feedback in the communication channel. Using Lyapunov-like techniques, we provide bounds on the minimum communication rate required to implement this observer. We also use the Lyapunov dimension to provide analytical upper bounds on the communication rate. We compute an analytical upper bound and an exact expression for the Lyapunov dimension of the smoothened Lozi map. This bound is then tested in simulations of the communication protocol for the observation problem of the smoothened Lozi map.