F-limit points in dynamical systems defined on the interval

Given a free ultrafilter p on ℕ we say that x ∈ [0, 1] is the p-limit point of a sequence (x n ) n∈ℕ ⊂ [0, 1] (in symbols, x = p -lim n∈ℕ x n ) if for every neighbourhood V of x, {n ∈ 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∈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.