Dynamics in One Dimension

The set of recurrent points of a continuous self-map on compact metric spaces and strong chaos

“…Let f ∈ C 0 (I). If f has a positive topological entropy, then there exists an N > 0 such that f N has a regular shift invariant set ( [1]). From the theorem, we deduce that f N has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.…”

“…Introduction. Throughout this paper, X will denote a compact metric space with metric d, and I is the closed interval [0,1].…”

Counting preimages

“…. , and thus (4) lim inf n→∞ 1 n log Card(f −n (x)) ≥ 1 k log(s/2) = 1 k log s − 1 k log 2. There are increasing sequences (k i ) and (s i ) such that f k i has an s i -horseshoe and (5) h(f ) = lim i→∞ 1 k i log s i (see, e.g., [1]), and therefore by (4) we get lim inf n→∞ 1 n log Card(f −n (x)) ≥ h(f ).…”

“…Clearly, if a map f is in PMM then it is piecewise strictly monotone. It is also known (see, e.g., [4]) that it is locally eventually onto, that is, for every nonempty open set U there is n such that f n (U) = [0, 1].…”

Chaotic behaviour of the map x ↦ ω(x, f)

“…For example, the notions of topological entropy and Li-Yorke chaos make explicit use of the separation of trajectories, while Devaney chaos incorporates sensitivity as well as the requirement that nearby points must generate distinct ω-limit sets. These ideas are developed at some length in [7,14,22].…”