Donald Samuel Ornstein

Donald S. Ornstein, 1970

Donald Samuel Ornstein (born July 30, 1934 in New York ) is an American mathematician who works in the field of ergodic theory .

Ornstein received his doctorate in 1957 from the University of Chicago under the direction of Irving Kaplansky (topic: Dual Vector Spaces ). During his career at Stanford University , he oversaw the doctoral theses of twenty-three students. From 1963 he was a Sloan Research Fellow .

He became famous in 1968 for his work on the isomorphism of Bernoulli maps (Bernoulli Shifts), for which he was awarded the Bôcher Memorial Prize in 1974. Bernoulli maps are stochastic processes that generalize shifts in the space of the sequences of coin tosses. In each time step, a value i from a pool of N values ​​is assumed with probability . Ornstein proved the equivalence (in terms of their stochastic behavior) of Bernoulli shifts if their Kolmogorov entropy, defined by , is equal. The equivalence is defined by size-maintaining isomorphisms (one-to-one mapping of the state spaces), for which the Kolmogorov entropy is an invariant, as Andrei Kolmogorow and Jakow Sinai showed in 1958. According to Kolmogorow and Sinai, Bernoulli shifts are not all equivalent to one another, but as Ornstein showed they are equivalent if the entropy is the same. ${\displaystyle p_{i}}$ ${\displaystyle h=-\sum _{i=1}^{N}p_{i}\log p_{i}}$

He has been a member of the National Academy of Sciences since 1981 and of the American Academy of Arts and Sciences since 1991 . In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( Entropy is enough to classify Bernoulli shifts but not K-automorphisms ). He is a fellow of the American Mathematical Society .

Fonts

• Ergodic theory, randomness, and dynamical systems , Yale University Press, 1974
• With Benjamin Weiss Statistical properties of chaotic systems , Bulletin AMS, Vol. 24, 1991, p. 1
• Ornstein Ornstein Isomorphism Theorem
• with Benjamin Weiss, Daniel J. Rudolph Equivalence of measure preserving transformations , American Mathematical Society 1982

Individual evidence

1. ↑ Donald Samuel Ornstein in the Mathematics Genealogy Project (English)
2. ↑ Ornstein Bernoulli shifts with the same entropy are isomorphic , Advances in Mathematics, Vol. 4, 1970, pp. 337-352
3. ↑ with ${\displaystyle \sum _{i}p_{i}=1}$