The three versions of distributional chaos☆

Families, filters and chaos

“…The functions Φ xy (t) and Φ * xy (t) can be used to define distributional chaos whose definition was introduced by Schweizer and Smítal [20] as a property equivalent to positive topological entropy in the context of continuous maps from the unit interval into itself. Presently, there are at least three versions of distributional chaos in the literature [4]. Furthermore, DC1 is completely independent of the value of topological entropy in general (there are examples of maps with entropy zero but DC1 and vice versa).…”

Distributional Chaos and Dendrites

“…A pair of two different points (x, y) ∈ X 2 is scrambled or Li-Yorke if lim inf k→∞ d(f k (x), f k (y)) = 0 (1) and lim sup k→∞ d(f k (x), f k (y)) > 0.…”

“…As we already mentioned, at the beginning, there was a definition of one kind of DC (see [12]), this type is called DC1 in these days, later ( [1]) that type was divided into 3 different types DC1, DC2 and DC3, different in general, but the same in the interval. It also turned out, that DC3 can be a really weak and unstable type of chaos, so in [2] appeared a new kind of DC, namely DC2 1 2 which as was shown, fixed those problems, but in general it is essentially weaker than DC2.…”

On distributional chaos in non-autonomous discrete systems

“…The concept of distributional chaos (it was called strong chaos then) was introduced by Schweizer and Smital from the perspective of probability theory for a continuous map in a compact interval [23]. Since then, it has evolved into three mutually nonequivalent versions of distributional chaos for a map in a metric space: DC1, DC2, and DC3 [3,28]. DC1 was the original version of distributional chaos introduced in [23], and it is the strongest one among these three definitions, while DC2 and DC3 are its generalizations.…”