Ergodic Theory and Dynamical Systems

Homeomorphisms with the whole compacta being scrambled sets

a1 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People's Republic of China (e-mail:


A homeomorphism on a metric space (X,d) is completely scrambled if for each x\not= y\in X, \lim sup_{n\longrightarrow +\infty} d(f^n(x),f^n(y))>0 and \lim inf_{n\ longrightarrow +\infty}d(f^n(x),f^n(y))=0. We study the basic properties of completely scrambled homeomorphisms on compacta and show that there are ‘many’ compacta admitting completely scrambled homeomorphisms, which include some countable compacta (we give a characterization), the Cantor set and continua of arbitrary dimension.

(Received January 20 1999)
(Revised December 13 1999)