Tail field representations and the zero-two law (Q5939285)

Positive linear contractions on \(L_p\), \(1\leq p<\infty\), are considered. A positive contraction \(T\) is said to contain a circle of length \(m\), if there is a nonzero function \(f\) such that the iterated values \(f,Tf,\dots, T^{m-1}f\) have disjoint supports, while \(T^mf= f\). A contraction \(T\) is said to contain a line if for every \(m\) there is a nonzero function \(f\), (which may depend on \(m\)) such that \(f,Tf,\dots, T^{m-1}f\) have disjoint supports. The main result of the paper is theorem 3 asserting that if two above properties of a contraction \(T\) holds, then \(\forall f\in L_p\), \(\lim_{n\to\infty}\|T^n f-T^{n+1} f\|_p= 0\). From this result it follows that any condition on \(T\) which excludes circles and lines must imply the conclusion of the zero-two law. In the proof of the theorem 3 the properties of an associated \(L_p\)-isometry and its tail \(\sigma\)-algebra are considered. The asymptotic circle and line cases for \(T\) are related to corresponding exact circle and line cases which are defined for the isometry. These exact circle and line cases are obtained by constructing Rohlin towers for the nonsingular point transformation which induces the \(L_p\)-isometry.