A monotone isomorphism theorem

In the simple case of a Bernoulli shift on two symbols, zero and one, by permuting the symbols, it is obvious that any two equal entropy shifts are isomorphic. We show that the isomorphism can be realized by a factor that maps a binary sequence to another that is coordinatewise smaller than or equal to the original sequence.Comment: 22 page

A monotone Sinai theorem

Sinai proved that a nonatomic ergodic measure-preserving system has any Bernoulli shift of no greater entropy as a factor. Given a Bernoulli shift, we show that any other Bernoulli shift that is of strictly less entropy and is stochastically dominated by the original measure can be obtained as a monotone factor; that is, the factor map has the property that for each point in the domain, its image under the factor map is coordinatewise smaller than or equal to the original point.Comment: Published at http://dx.doi.org/10.1214/14-AOP968 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Finitary isomorphisms of Brownian motions

Ornstein and Shields (Advances in Math., 10:143-146, 1973) proved that Brownian motion reflected on a bounded region is an infinite entropy Bernoulli flow and thus Ornstein theory yielded the existence of a measure-preserving isomorphism between any two such Brownian motions. For fixed h >0, we construct by elementary methods, isomorphisms with almost surely finite coding windows between Brownian motions reflected on the intervals [0, qh] for all positive rationals q.Comment: Published at https://doi.org/10.1214/19-AOP1412 in the Annals of Probability by the Institute of Mathematical Statistic

The orbital equivalence of Bernoulli actions and their Sinai factors

Given a countable amenable group G and 0 < L < 1, we give an elementary construction of a type-III:L Bernoulli group action. In the case where G is the integers, we show that our nonsingular Bernoulli shifts have independent and identically distributed factors.Comment: 45 pages; minor revision

Weak mixing suspension flows over shifts of finite type are universal

Let S be an ergodic measure-preserving automorphism on a non-atomic probability space, and let T be the time-one map of a topologically weak mixing suspension flow over an irreducible subshift of finite type under a Holder ceiling function. We show that if the measure-theoretic entropy of S is strictly less than the topological entropy of T, then there exists an embedding from the measure-preserving automorphism into the suspension flow. As a corollary of this result and the symbolic dynamics for geodesic flows on compact surfaces of negative curvature developed by Bowen and Ratner, we also obtain an embedding from the measure-preserving automorphism into a geodesic flow, whenever the measure-theoretic entropy of S is strictly less than the topological entropy of the time-one map of the geodesic flow.Comment: 27 pages, 1 figur

Insertion and deletion tolerance of point processes

We develop a theory of insertion and deletion tolerance for point processes. A process is insertion-tolerant if adding a suitably chosen random point results in a point process that is absolutely continuous in law with respect to the original process. This condition and the related notion of deletion-tolerance are extensions of the so-called finite energy condition for discrete random processes. We prove several equivalent formulations of each condition, including versions involving Palm processes. Certain other seemingly natural variants of the conditions turn out not to be equivalent. We illustrate the concepts in the context of a number of examples, including Gaussian zero processes and randomly perturbed lattices, and we provide applications to continuum percolation and stable matching

Deterministic Thinning of Finite Poisson Processes

Let Pi and Gamma be homogeneous Poisson point processes on a fixed set of finite volume. We prove a necessary and sufficient condition on the two intensities for the existence of a coupling of Pi and Gamma such that Gamma is a deterministic function of Pi, and all points of Gamma are points of Pi. The condition exhibits a surprising lack of monotonicity. However, in the limit of large intensities, the coupling exists if and only if the expected number of points is at least one greater in Pi than in Gamma.Comment: 16 pages; 1 figur