Chaotic sets of shift and weighted shift maps

Scrambled sets of shift operators

“…Note that some results of the Li-Yorke chaotic sets for the shift operator is already obtained by Fu et al in [2,3]. Our new contribution here is to characterize Li-Yorke chaotic sets by orbit invariants, Furstenberg families and p-scrambled points.…”

“…In [3], Fu and You thought that a possible route to solve Question 1.5 is to re-define the function η(·, ·). Theorem 2.3 shows that this works.…”

“…In [3], Fu and You proved that if for all p ∈ N, x is p-scrambled, then the orb f (x) is a scrambled set. In section 2 below, more results on p-scrambled points will be given.…”

“…At the end of [3], Fu and You also posed the following two open problems on the scrambled sets of σ:…”

Chaotic sets of continuous and discontinuous maps

“…Then α is an orbit invariant on S, while S is not a scrambled set of f . However, in [12] it is shown that when we restrict our discussions to shift maps σ : Σ(N) → Σ(N), N ≥ 2, together with η(x, y) ≡ 1, x, y ∈ S, the existence of an orbit invariant on S is also a sufficient condition for S ⊆ Σ(N) to be a scrambled set of σ . Where η(·, ·) : S × S → [0, 1] is defined as:…”