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An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈$A$,$G$^{α}may not cover$G$_{α}as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about the$G$^{α}/G_{α}is that each is an extension of a trivially ordered subgroup by a subgroup of$R$. It$B$is a plenary subset of$A$, then there exists a$v$-isomorphism μ of$G$into$V(B, G$^{β}/G_{β}), but whether or not μ is an$o$-isomorphism is not known.

Suppose that$X$is a vector field on a manifold$M$whose flow, exp$tX$, exists for all time. If μ is a measure on$M$for which the induced measures$μ$_{$t$}≡(exp$tX$)_{*}$μ$are absolutely continuous with respect to μ, it is of interest to establish bounds on the$L$^{$p$}(μ) norm of the Radon-Nikodym derivative$dμ$_{$t$}/$dμ$. We establish such bounds in terms of the divergence of the vector field$X$. We then specilize$M$to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples on$C$^{m}and on the Riemann surface for$z$^{1/$n$}.