Homeomorphisms with the whole compacta being scrambled sets

A homeomorphism on a metric space (X, d) is completely scrambled if for eac h x not equal y is an element of X, lim sup(n--> + infinity) d (f(n)(x), f( n)(y)) > 0 and lim inf(n-->+infinity) d(f(n)(x), f(n)(y)) 0. We study the b asic properties of completely scrambled homeomorphisms on compacta and show that there are 'many' compacta admitting completely scrambled homeomorphis ms, which include some countable compacta (we give a characterization), the Canter set and continua of arbitrary dimension.