We apply set-theoretical ideas to an iterationproblem of dynamical systems. Among other results,we prove that these iterations never stabiliselater than the first uncountable ordinal; forevery countable ordinal we give examples inBaire space and in Cantor space of an iterationthat stabilises exactly at that ordinal; we givean example of an iteration with recursive data which stabilises exactly at the first non-recursive ordinal; and we find new examplesof complete analytic sets simplydefinable from concepts of recurrence.

2000 Mathematics Subject Classification:primary 03E15, 37B20, 54H05;secondary 37B10, 37E15.