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Abstract. Our main result shows that a continuous map f acting on a compact metric space (X, ρ) with a weaker form of specification property and with a pair of distal points is distributionally chaotic in a very strong sense. Strictly speaking, there is a distributionally scrambled set S dense in X which is the union of disjoint sets homeomorphic to Cantor sets so that, for any two distinct points u, v ∈ S, the upper distribution function is identically 1 and the lower distribution function is zero at some ε > 0. As a consequence, we describe a class of maps with a scrambled set of full Lebesgue measure in the case when X is the k-dimensional cube I k . If X = I, then we can even construct scrambled sets whose complements have zero Hausdorff dimension.

We study local analytic solutions f of the generalized Dhombres functional equationholomorphic in some open neighborhood of 0, depending on f , and f (0) = w 0 . After deriving necessary conditions on ϕ for the existence of nonconstant solutions f with f (0) = w 0 we describe, assuming these conditions, the structure of the set of all formal solutions, provided that w 0 is not a root of 1. If |w 0 | = 1 or if w 0 is a Siegel number we show that all formal solutions yield local analytic ones. For w 0 with 0 < |w 0 | < 1 we give representations of these solutions involving infinite products.

We study holomorphic solutions f of the generalized Dhombres equation f (zf (z)) = ϕ(f (z)), z ∈ C, where ϕ is in the class E of entire functions. We show, that there is a nowhere dense set E 0 ⊂ E such that for every ϕ ∈ E \ E 0 , any solution f vanishes at 0 and hence, satisfies the conditions for local analytic solutions with fixed point 0 from our recent paper. Consequently, we are able to provide a characterization of solutions in the typical case where ϕ ∈ E \ E 0 . We also show that for polynomial ϕ any holomorphic solution on C \ {0} can be extended to the whole of C. Using this, in special cases like ϕ(z) = z k+1 , k ∈ N, we can provide a characterization of the analytic solutions in C.

We consider nonautonomous discrete dynamical systems {fn} n≥1 , where every fn is a surjective continuous map [0, 1] → [0, 1] such that fn converges uniformly to a map f . We show, among others, that if f is chaotic in the sense of Li and Yorke then the nonautonomous system {fn} n≥1 is Li-Yorke chaotic as well, and that the same is true for distributional chaos. If f has zero topological entropy then the nonautonomous system inherits its infinite ω-limit sets.

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