Support overlapping L1 contractions and exact non-singular transformations

Let T be a positive linear contraction of L1 of a σ-finite measure space (X, Σ, µ) which overlaps supports. In general, T need not be completely mixing, but it is in the following cases:
(i) T is the Frobenius–Perron operator of a non-singular transformation φ (in which case complete mixing is equivalent to exactness of φ).
(ii) T is a Harris recurrent operator.
(iii) T is a convolution operator on a compact group.
(iv) T is a convolution operator on a LCA group