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Abstracts of talks

A tale of two entropies

Roy Adler, IBM Yorktown Heights adler@watson.ibm.com

For the purposes of this talk we divide the subject of abstract dynamical systems into two areas: ergodic theory and topological dynamics. In ergodic theory, a dynamical system is endowed with a probability measure so that one can study distribution of orbits in phase space; in topological dynamics, a metric so that one can study, stability, almost periodicity, density of orbits, etc. Ideas originating in Shannon's work in information theory have played a crucial role in isomorphism theories of dynamical systems. Shannon used notions of entropy and channel capacity to determine the amount of information which can be transmitted through channels. Kolmogorov and Sinai made his idea into a powerful invariant for ergodic theory; and then Ornstein proved that entropy completely classifies special systems called Bernouli shifts. Adler, Konheim, and McAndrew introduce into topological dyanamics an analogous invariant called topological entropy which turns out to be a generalization of channel capacity. Then Adler and Marcus developed an isomorphism theory in which this invariant completely classifies symbolic systems called shifts of finite type. The surprise is that this abstract theory has engineering applications: namely their methods of proof lead to construction of encoders and decoders for data transmission and storage. As a result the history of Shannon's ideas have come full circle. He used them to study transmission of information. Mathematicians took up his ideas to classify dynamical systems, first in ergodic theory and then in topological dynamics. Results in thesecond area then lead to methods for optimizing transmission and storage of data. For references, see [1, 2, 61].

(Editor's note: due to personal reasons which arose at the last minute, Roy Adler was unfor tunately unable to attend the workshop.)

Non-commutative dynamical entropy

Robert Alicki, University of Gdansk fizra@univ.gda.pl

The concept of dynamical entropy, called also Kolmogorov-Sinai invariant, proved to be very useful in the classical ergodic theory in particular for studying systems displaying chaos . Since twenty years several attempts to generalize this notion to non-commutative algebraic dynamical systems have been undertaken. In the present lecture a brief survey of the recent investigations based on the new definition of non-commutative dynamical entropy introduced by M.Fannes and the author is presented. Several examples of dynamical systems including classical systems, shift on quantum spin chain, quasi-free Fermion automorphisms, Powers Price binary shifts, Stormer free shift and quantum Arnold cat map are discussed. The new dynamical entropy is compared with the alternative formalisms of Connes-Narnhofer-Thirring and Voiculescu. For references, see [3, 4, 22].

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