Codes from iterated maps

The authors consider codes of the following type. Let S (the signal set) be a subset of n-dimensional Euclidean space Rⁿ. Let f:S→S be a continuous mapping. The code C(S,f) consists of those bi-infinite sequences x=...x_-l,x₀,x₁ ,x₂,...∈S^𝒵 that satisfy x_t =f(x_t-1) for all t∈𝒵. Note that the “future” of each codeword is completely determined by its “past”. At first sight, it might seem that the information rate (i.e. the number of information bits per code symbol) of any such code must be zero. However, as the example shows, this need not be so if S is an infinite set. It can also be shown that codes of this type can have an arbitrarily large minimum distance, which dispels any lingering suspicion that such codes are somehow inherently “bad”