On the cohomology of measurable sets

TL;DR

This paper investigates the structure of the first cohomology group of ergodic automorphisms, showing it is uncountable and that generically the induced systems are weakly mixing, using spectral and descriptive set theory tools.

Contribution

It provides a new proof of the uncountability of the first cohomology group using spectral genericity, and establishes that generically the induced systems are weakly mixing.

Findings

H is uncountable for ergodic automorphisms

B is meager in A, indicating generic properties

Induced systems are generically weakly mixing

Abstract

If T is an ergodic automorphism of a Lebesgue probability space (X,A,m), the set of coboundries B = db =T(b)+b with symmetric difference + form a subgroup of the set of cocycles A. Using tools from descriptive set theory, Greg Hjorth showed in 1995 that the first cohomology group H=A/B is uncountable. This can surprise, given that in the case of a finite ergodic probability space, H has only 2 elements. Hjorth's proof used descriptive set theory in the complete metric space (A,d(a,b)=m(a+b)), leading to the statement that B is meager in A. We use a spectral genericity result of Barry Simon to establish the same. It leads to the statement noted first by Karl Petersen in 1973 that for a generic a in A, the induced system T_a is weakly mixing, which is slightly stronger than a result of Nate Friedman and Donald Ornstein about density of weakly mixing in the space of all induced systems T_a coming from an ergodic automorphism T.