Abstract: In this paper, we summarize our recent work on some complexity problems in dynamical
systems related to chaos, entropy and recurrence properties under the ideas of localization (pairs or
tuples). We solve a long open problem by proving that Devaney chaos implies Li-Yorke one. We show
that there are ’many’ compacta admitting completely scrambled homeomorphisms, which include some
countable compacta, the cantor set and continua of arbitrary dimension. Using the local notions of
entropy: entropy tuples and sequence entropy pairs, we characterize the structures of a topological K-
system and a topological null system. Finally, we give a ner classi cation of recurrence properties in
terms of weak disjointness, complexity function of an open cover, and access time set.

Key words: Complexity, tuple, chaos, entropy, recurrence property