We show that there exists a continuous function from the unit Lebesgue interval to itself such that for any $\epsilon \geq 0$ and any natural number k, any point in its domain has an $\epsilon $ -neighbourhood which, when feasible, contains k mutually disjoint extremally scrambled sets of identical Lebesgue measure, homeomorphic to each other. This result enables a satisfying generalisation of Li–Yorke (topological) chaos and suggests an open (difficult) problem as to whether the result is valid for piecewise linear functions.

We study the independence density for finite families of finite tuples of sets for continuous actions of discrete groups on compact metrizable spaces. We use it to show that actions with positive naive entropy are Li–Yorke chaotic and untame. In particular, distal actions have zero naive entropy. This answers a question of Lewis Bowen.

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

We prove that if f:I=[0,1]→I is a C3-map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R(f) has full Lebesgue measure λ; (ii) both S(f) and Scramb(f) have positive measure. Here R(f), S(f), and Scramb(f) respectively stand for the set of approximately periodic points of f, the set of sensitive points to the initial conditions of f, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set for f.Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R(f))=1 or λ(S(f))>0, and provide examples of maps g,h of this type satisfying S(g)=S(h)=I such that, on the one hand, λ(R(g))=0and λ2 (Scramb (g))=0 , and, on the other hand, λ(R(h))=1 .