Point Spectra of Non-Singular Flows

Abstract

An example is given of a family of non-singular transformations whose point spectra are uncountable. § 1. Definition of the Point Spectrum Let (X, 2,P) be a Lebesgue measure space and let {7Y, £eU} be a one-parameter group of null-measure- preserving transformations of X onto itself, where the map Xx.R-^X((x,t)-^Ttx) is measurable. The non-singular flow {Tt t^R] is said to be ergodic if every measurable function f with f(Ttx)=f(x) a.e.x for each t^R is constant a.e.. A real number 0 is called an element of the point spectrum of {Tt;t^R} if there exists a non-zero measurable function (Ttx) =eidt<p(x) a.e.x for each £eB. If {Tt;t^R} is ergodic we may consider \cp(x) = 1. We denote by $p({Tt}) the point spectrum of {Tt;t^R}. It is well known that if {Tt;tE-R} has a finite invariant measure equivalent to P Sp({Tt}) is {0} or a countable subset of the set of all real numbers.