Fixed Points and Stability of Density Evolution

Density evolution is a dynamic system in a space of probability distributions repre- senting the progress of iterative decoders in the infinite block length limit. In this paper we establish some basic results concering this process. In particular we show that the decoding threshold is equivalent to to appearance of non-trivial fixed point solutions to the density evolution equations. In the case of LDPC codes we prove the sufficiency of the previously published stability condition for stability of the δ∞ fixed point and slightly strengthen the necessity result.