ERGODIC THEORY FROM HUMSAN TO VIENNA
by Anatole Katok
This text represents a slightly edited version of a set of notes for the talk at
the opening day of the rigidity conference at the Erwin Schroedinger Institute
in Vienna in February 1997 organized by Gregory Margulis, Klaus Schmidt
and myself. Naturally, it reflects my point-of-view at the time which may
not coincide with the present one.
I will mostly concentrate on the earlier part of the period. I will not
make an attempt to assess the mathematical developments during the
last decade or so and restrict myself for that period to comments only
on ”organizational” developments.
1. Ergodic theory at 1965 (from the Moscow vantage
Basic entropy theory has taken shape (Kolmogorov, Sinai, Rokhlin,
Parry, Adler) The isomorphism problem for Bernoulli shifts is very
much on the agenda. The top achievement in that direction was Sinai
weak isomorphism theorem. Mechalkin example.
Basic results of ergodic theory of hyperbolic dynamical systems (mostly
Anosov and Sinai): theory of stable and unstable manifolds, absolute
continuity, ”Hopf argument” for ergodicity, K-property. Development
of topological aspects of hyperbolic dynamics by smale and his asso-
ciates. Kushnirenko inequality. Introduction of topological entropy by
Adler, Konheim and McAndrew.
Theory of Gaussian dynamical systems; Gaussian systems as a source
of examples (Kolmogorov, Girsanov, Vershik) mostly unsuccessful attmpts
to use the Gaussian paradigm as a basis for a general theory (Sinai;
my own students’ work)
New trends: Revival of interest to orbit equivalence and relations
to operator algebras (Kirillov, going back to von Neumann but ap-
parently not aware of the more recent work of H.Dye). Rediscovery
of combinatorial constructions (Oseledec; Katok,Stepin) study of sim-
ple extension of the rotation and interval exchange transformations
which later grew into the periodic approximation method). Stepin’s
counterexample to the ”group property” for the asymptotically cyclic