Piotr Szuca

Density versions of Schur's theorem for ideals generated by submeasures

We characterize ideals of subsets of natural numbers for which some versions of Schurs theorem hold. These are similar to generalizations shown by Bergelson (1986) in [1] and Frankl, Graham and Rodl (1990) in [7]. Additionally, we prove a generalization of an iterated version of Ramseys theorem.

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 (xn)n∈ℕ ⊂ [0, 1] (in symbols, x = p -limn∈ℕxn) if for every neighbourhood V of x, {n ∈ ℕ: xn ∈ V} ∈ p. For a function f: [0, 1] → [0, 1] the function fp: [0, 1] → [0, 1] is defined by fp(x) = p -limn∈ℕfn(x) for each x ∈ [0, 1]. This map is rarely continuous. In this note we study properties which are equivalent to the continuity of fp. For a filter F we also define the ωF-limit set of f at x. We consider a question about continuity of the multivalued map x → ωfF(x). We point out some connections between the Baire class of fp and tame dynamical systems, and give some open problems.