Donald Samuel Ornstein

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30 July 1934
New York, USA

Don Ornstein is an American mathematician working in the area of ergodic theory.


Donald Ornstein is known to his friends and colleagues as Don. His parents were Harry Ornstein and Rose Wisner. He was educated at Swarthmore College, Pennsylvania, and then entered the University of Chicago where he undertook research towards his Ph.D. advised by Irving Kaplansky. He was initiated as a member of the Chicago Chapter of Sigma Xi in autumn 1956. He submitted his thesis Dual Vector Spaces to the University of Chicago and was awarded a Ph.D. in 1957. However, by the time he was awarded his doctorate, Ornstein was already at Institute for Advanced Study at Princeton where he spent two academic years 1956-58. Following these years when he concentrated on research, he was appointed as an Instructor in Mathematics at the University of Wisconsin, spending two academic years 1958-60 in this post.

Ornstein's first publication appeared in 1959 in the Annals of Mathematics. Not surprisingly, this first paper was based on his Ph.D. thesis and had the same title Dual Vector Spaces. However, his second paper, written jointly with R V Chacon, was on ergodic theory, the topic for which Ornstein is now best known. The paper A general ergodic theorem (1960) contained a proof of a conjecture that had been made by Eberhard Hopf. Two further publications appeared in 1960, namely The differentiability of transition functions and On invariant measure. Both were single authored and, in the second paper, he gave a negative answer to a long-standing question in the theory of measurable transformations. In 1960 Ornstein was appointed as an Assistant professor at Stanford University. He continued to hold a position at Stanford for the rest of his career, being promoted to an associate professor in 1963 and also being a Sloan fellow during 1963-65. During this period he married Shari Richman in 1964; they had two sons and one daughter. He continued as an associate professor at Stanford during 1965-66 and he was promoted to full professor in 1966. During the academic year 1967-68 he was a visiting professor at Cornell University and at New York University's Courant Institute.

The year 1968 was an important one for Ornstein for in that year he began proving a number of exceptionally important results which led to major advances in ergodic theory in the following years. He was invited to write the paper [2] describing these advances and to give the James K Whittemore Lectures in Mathematics given at Yale University describing his breakthroughs. The Yale lectures were published as the monograph Ergodic theory, randomness, and dynamical systems in 1974. Also in 1974 the American Mathematical Society awarded Ornstein their Bôcher Memorial Prize:-
... in recognition of his paper "Bernoulli shifts with the same entropy are isomorphic" (1970).
To understand what Ornstein proved in this papers, we quote from his own description in [1]:-
Ergodic theory, the study of measure-preserving transformations or flows, arose from the study of the long-term statistical behaviour of dynamical systems. Consider, for example, a billiard ball moving at constant speed on a rectangular table with a convex obstacle. The state of the system (the position and velocity of the ball), at one instant of time, can be described by three numbers or a point in Euclidean 3-dimensional space, and its time evolution by a flow on its state space, a subset of 3-dimensional space. The Lebesgue measure of a set does not change as it evolves and can be identified with its probability. One can abstract the statistical properties (e.g., ignoring sets of probability 0) and regard the state-space as an abstract measure space. Equivalently, one says that two flows are isomorphic if there is a one-to-one measure-preserving (probability-preserving) correspondence between their state spaces so that corresponding sets evolve in the same way (i.e., the correspondence is maintained for all time). It is sometimes convenient to discretize time (i.e., look at the flow once every minute), and this is also referred to as a transformation. Measure-preserving transformations (or flows) also arise from the study of stationary processes. The simplest examples are independent processes such as coin tossing. The outcome of each coin tossing experiment (the experiment goes on for all time) can be described as a doubly-infinite sequence of heads H and tails T. The state space is the collection of these sequences. Each subset is assigned a probability. For example, the set of all sequences that are H at time 3 and T at time 5 gets probability 14\large\frac{1}{4}\normalsize. The passage of time shifts each sequence to the left (what used to be time 1 is now time 0). (This kind of construction works for all stochastic processes, independence and discrete time are not needed.) The above transformation is called the Bernoulli shift B(½, ½). If, instead of flipping a coin, one spins a roulette wheel with three slots of probability p1,p2,p3p_{1}, p_{2}, p_{3} , one would get the Bernoulli shift B(p1,p2,p3p_{1}, p_{2}, p_{3}).

Bernoulli shifts play a central role in ergodic theory, but it was not known until 1958 whether or not all Bernoulli shifts are isomorphic. A N Kolmogorov and Ya G Sinai solved this problem by introducing a new invariant for measure-preserving transformations: the entropy, which they took from Shannon's theory of information. They [used entropy to prove] that not all Bernoulli shifts are isomorphic. The simplest case of the Ornstein isomorphism theorem (1970), states that two Bernoulli shifts of the same entropy are isomorphic.
Paul C Shields, reviewing Ornstein's book Ergodic theory, randomness, and dynamical systems (1974) writes:-
In 1969, the author solved one of the most difficult problems in ergodic theory when he showed that two Bernoulli shifts with the same entropy are isomorphic. This immediately led to many new results about measure-preserving transformations including proofs that many transformations of physical and mathematical interest are just disguised versions of Bernoulli shifts. In these lectures the author gives a clear and thorough discussion of the ideas which are the basis for this growing new branch of ergodic theory. ... Since the publication in 1974 of this book the subject has continued to grow. There is now an extensive theory of isomorphism of flows, group actions (including Ising models), and even communications channels.
Ornstein published other significant papers in 1970 in addition to Bernoulli shifts with the same entropy are isomorphic for which he received the Bôcher Memorial Prize, although it is clear that it was this paper which was the starting point for all that followed. These other 1970 papers were Factors of Bernoulli shifts are Bernoulli shifts and Imbedding Bernoulli shifts in flows which contained a generalisation of the results of his prize-winning paper. He continued to produce papers in the following couple of years which pushed understanding of the topic significantly forward: A Kolmogorov automorphism that is not a Bernoulli shift; A K-automorphism with no square root and Pinsker's conjecture; and The isomorphism theorem for Bernoulli flows. The paper [2] gives a very clear overview of this work.

After the publication of this outstanding work, Ornstein was a visiting professor at the Hebrew University in Jerusalem in 1975-76 and at the University of California's Mathematical Sciences Research Institute at Berkeley in 1983-84.

Perhaps the greatest honours that Ornstein has received was his election to the National Academy of Sciences in 1981 and to the American Academy of Arts and Sciences in 1991.

References (show)

  1. D S Ornstein, Ornstein isomorphism theorem, in Michiel Hazewinkel (ed.), Encyclopaedia of Mathematics (Springer-Verlag, Berlin-Heidelberg-New York, 2002).
  2. D S Ornstein, An Application of Ergodic Theory to Probability Theory, The Annals of Probability 1 (1) (1973), 43-58.
  3. Donald Samuel Ornstein, The International Who's Who 2004 (Routledge, 2003).

Additional Resources (show)

Honours (show)

Honours awarded to Donald Ornstein

  1. AMS Bôcher Prize 1974

Written by J J O'Connor and E F Robertson
Last Update March 2011