We give a short proof of a theorem of Beleznay asserting that the set $L2$ of reals coding linear orders of the form $I + I$ is complete analytic.

We consider a real-valued function f defined on the set of infinite branches X of a countably branching pruned tree T. The function f is said to be a limsup function if there is a function $u \colon T \to \mathbb {R}$ such that $f(x) = \limsup _{t \to \infty } u(x_{0},\dots ,x_{t})$ for each $x \in X$. We study a game characterization of limsup functions, as well as a novel game characterization of functions of Baire class 1.

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.