An infinite sequence of s and 1s, defined in any of the following (equivalent) ways. In each case, an obvious "limit" exists (and can be made rigorous), which is the Morse sequence

  1. Write down a 0. Then, at each stage, copy the previous stage's word, then copy it "inverted", switching 0's and 1's.
  2. Define a morphism by
    0 → 01
    1 → 10
    and extend it to words; now repeatedly apply it to the word "0".

In each case, the same sequence is obtained:


This sequence is aperiodic, but still strongly recurrent.

Method 2 above is an example of an L-system which converges to a single limit.