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

- Write down a 0. Then, at each stage, copy the previous stage's word, then copy it "inverted", switching 0's and 1's.
- 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:

0
01
0110
01101001
0110100110010110
...

This sequence is aperiodic, but still strongly recurrent.

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