Let X be the unit interval with
the Lebesgue structure and let m be Lebesgue measure. A Markov operator with
invariant measure m is an operator T on L∞(X,m) such that T1 = 1 and
∫Tfdm =∫fdm for all f in L∞(X,m). If 𝜃 is a measure-preserving transformation
on X, then 𝜃f = f ∘𝜃 defines a Markov operator. Each such 𝜃 is an extreme point in
the convex set of Markov operators.
Let Ω(f) be the set of all g ∈ L1(X,m) such that Tf = g for some Markov
operator T. This convex set is called the orbit of f. The extreme points of
Ω(f) are equimeasurable to f and arise from Markov operators of the form
𝜃σ∗. This paper shows the connection between extreme points of the set
of Markov operators and the extreme points of Ω(f). The set of Markov
operators which carry f to a given extreme point of Ω(f) is shown to contain an
extreme Markov operator. The Markov operators of the from 𝜃σ∗ are shown
to be extreme when 𝜃 is invertible. It is also shown that not all extreme
operators factor into 𝜃σ∗ and that there are 𝜃 and σ such that 𝜃σ∗ is not
extreme.
