Special 𝛼-limit points for maps of the interval

Abstract

Abstract.The notion of a special «-limit point is defined. For maps of the interval, it is shown that a point is a special a-limit point if and only if it is an element of the attracting center.

When is the beginning the end? On full trajectories, limit sets and internal chain transitivity

“…Before turning our attention to Question 1.2, and in keeping with the precedent set in [23], we will briefly highlight a corollary concerning γ-limit sets. These were introduced by Hero [27] who studied them for interval maps, γ-limit sets have since been further examined by Sun et al in [46] and [45] for graph maps and dendrites respectively. The γ-limit set of a point x, denoted γ(x), is defined by saying that, for any y ∈ X, y ∈ γ(x) if and only if y ∈ ω(x) and there exists a sequence y i ∞ i=1 in X and a strictly increasing sequence n i ∞ i=1 in N such that f n i (y i ) = x for each i and lim i→∞ y i = y.…”

“…This presents a difficulty when attempting to give a suitable definition for α f (x). Various approaches to this have been taken [2,16,17,27,45,46]. For a discussion on this, we refer the reader to [23].…”

“…We define the limit sets of such a sequence in the natural way: α f ( x i i∈Z ) = α f ( x i i≤0 ) and ω f ( x i i∈Z ) = ω f (x 0 ). Although α-limit sets have not been studied quite as extensively as their ω counterparts, interest in them has been growing (see, for example, [2,16,17,23,27,28]).…”

On backward attractors of interval maps*

“…But when we study continuous mappings f : X → X (not necessarily invertible), a point x may have many preimages (or none at all), and we must clarify what kind of backward limit set we wish to speak of. Several definitions have been proposed including conical limit sets, α-limit sets, branch α-limit sets, special α-limit sets, and others [1,7,11,15,16,20].…”

An example of a homeomorphism of the plane where the set of 𝜔-limit points is not a closed set

“…The attracting center A°° is defined to be the intersection of the sets A" . In [7] it is shown that for self-maps / of the interval A°° = SA(f,I). Proposition 3.…”