F-limit points in dynamical systems defined on the interval
Tóm tắt
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
(x). We point out some connections between the Baire class of f
p
and tame dynamical systems, and give some open problems.
Từ khóa
Tài liệu tham khảo
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