On marginal distributions and isomorphisms of stationary processes

Abstract

LetT be an invertible ergodic aperiodic measure preserving transformation of a Lebesgue space, letA be a finite alphabet, and let π be a probability measure onA n which admits a mixing shift-invariant measureμ π onΩ=A ℤ such that the marginals of anyn successive coordinates are π and the entropyh(T) ofT is smaller than the entropy of the shift in (Ω,μ π). Then there exists a shift invariant measure νπ in Ω which also has marginals π and for whichT is isomorphic to the shift in (Ω, νπ). This contains Krieger's finite generator theorem and strengthens the measure theoretic part of his approximation theorem for shift-invariant measures by showing that the preassigned marginal π can not only be achieved up to an e>0 but exactly. Our result also contains an as yet unpublished theorem of Krieger, which says thatT can be embedded in an arbitrary mixing subshift of finite type, as long as the entropy of the subshift under the measure with maximal entropy exceeds that ofT. In the final section we show that the method can be extended to yield also exact marginals for the generator in the Jewett-Krieger theorem, i.e.T is shown to be isomorphic to a shift in (Ω, νπ) where νπ has exact marginals π and the shift is uniquely ergodic on the support of νπ.