WAP Systems and Labeled Subshifts

The main object of this work is to present a powerful method of construction of subshifts which we use chiefly to construct WAP systems with various properties. Among many other applications of this so called labeled subshifts, we obtain examples of null as well as non-null WAP subshifts, WAP subshifts of arbitrary countable (Birkhoff) height, and completely scrambled WAP systems of arbitrary countable height. We also construct LE but not HAE subshifts, and recurrent non-tame subshifts.