F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ? we say that x ∈ 0, 1] is the p-limit point of a sequence (x _n ) _n∈? ? 0, 1] (in symbols, x = p -lim _n∈? x _n ) if for every neighbourhood V of x, {n ∈ ?: x _n ∈ V} ∈ p. For a function f: 0, 1] → 0, 1] the function f ^p : 0, 1] → 0, 1] is defined by f ^p (x) = p -lim _n∈? f ⁿ (x) for each x ∈ 0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of f ^p . For a filter F we also define the ω ^F -limit set of f at x. We consider a question about continuity of the multivalued map x → ω _f ^F (x). We point out some connections between the Baire class of f ^p and tame dynamical systems, and give some open problems.