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Ultrafilter-limit points in metric dynamical systems

Salvador García-Ferreira, Manuel Sanchis (2007)

Commentationes Mathematicae Universitatis Carolinae

Given a free ultrafilter p on ℕ and a space X , we say that x ∈ X is the p -limit point of a sequence ( x n ) n ∈ ℕ in X (in symbols, x = p - lim n → ∞ x n ) if for every neighborhood V of x , { n ∈ ℕ : x n ∈ V } ∈ p . By using p -limit points from a suitable metric space, we characterize the selective ultrafilters on ℕ and the P -points of ℕ * = β ( ℕ ) ∖ ℕ . In this paper, we only consider dynamical systems ( X , f ) , where X is a compact metric space. For a free ultrafilter p on ℕ * , the function f p : X → X is defined by f p ( x ) = p - lim n → ∞ f n ( x ) for each x ∈ X . These functions are not continuous in general. For a...

Uncountable ω-limit sets with isolated points

Chris Good, Brian E. Raines, Rolf Suabedissen (2009)

Fundamenta Mathematicae

We give two examples of tent maps with uncountable (as it happens, post-critical) ω-limit sets, which have isolated points, with interesting structures. Such ω-limit sets must be of the form C ∪ R, where C is a Cantor set and R is a scattered set. Firstly, it is known that there is a restriction on the topological structure of countable ω-limit sets for finite-to-one maps satisfying at least some weak form of expansivity. We show that this restriction does not hold if the ω-limit set is uncountable....

Une version feuilletée équivariante du théorème de translation de Brouwer

Patrice Le Calvez (2005)

Publications Mathématiques de l'IHÉS

The Brouwer’s plane translation theorem asserts that for a fixed point free orientation preserving homeomorphism f of the plane, every point belongs to a Brouwer line: a proper topological embedding C of R, disjoint from its image and separating f(C) and f–1(C). Suppose that f commutes with the elements of a discrete group G of orientation preserving homeomorphisms acting freely and properly on the plane. We will construct a G-invariant topological foliation of the plane by Brouwer lines. We apply...

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