Dynamics of random products of positive contractions on L 1
We study a trajectorywise analogue of the classical zero–two phenomenon for products
of positive contractions on L 1 -spaces. Motivated by the strong zero–two law for a single positive
contraction, we ask whether a uniform bound strictly below 2 for the limiting successive differences
along a fixed realization forces these differences to converge to zero. We show that this is false
in general. More precisely, for every c ∈ (0,2) we construct two rank-one positive contractions on
L 1 ([0,1]) and an alternating realization ω such that
kT n+1
ω
f − T n
ω fk 1 = c
for every n ≥ 1 and every nonnegative f with kfk 1 = 1. Hence the hypothesis is satisfied for a
suitable ε > 0, while the conclusion fails. In fact, the same equality holds in operator norm. Thus
a direct pathwise extension of the deterministic zero–two criterion to non-autonomous, and hence
random, products is impossible without additional assumptions.
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