On Devaney Chaos and Dense Periodic Points: Period 3 and Higher Implies Chaos

The chaotic behavior on the unit circle

“…The surprising equivalence on the interval is because transitivity implies dense periodic points, but the converse is not necessarily true [7]. The result cannot be generalized to higher dimensions or the unit circle because the proof of this result uses the ordering in ℝ in an essential way.…”

“…Definition 7 [7]: Let : → be a continuous map on a compact metric space . The function is said to poses a strong dense periodicity property whenever the is dense for all integer and is a collection of periodic points of prime period larger than .…”

Some Chaos Notions on Dendrites

“…x is periodic point of period greater than or equal to m} is dense. The works in [6][7][8][9][10][11] discussed the relation between strong dense periodicity and other notions of chaos on intervals, circles and shift of finite space.…”

Dense periodicity property and Devaney chaos on shifts spaces

“…Definition 1.6. ( [5]) For a space , we say that it has dense for all ∈ ℕ if the set of periodic points in is dense, where the set is defined as = { ∈ : is a periodic point of prime period for some ≥ }.…”

“…This notion was firstly introduced in 2015 [5] and as far as we know, there is no other stronger dense periodicity property introduced to describe chaos. The identity map has dense periodic points but is not transitive.…”