L^∞ Eigenvalues of Non-Singular Automorphisms

Abstract

Let (X, Ɓ, m) be a standard probability space and let T be an ergodic non-singular automorphism on (X, Ɓ, m). A complex number λ is said to an L ^∞ eigenvalue of T if there is a non-zero function f _λ ∈ L ^∞(X, Ɓ, m) such that f _λ (Tx) = λf(x) a.e. m. We call any such f _λ an L ^∞ eigenfunction of T corresponding to the eigenvalue λ. Since ||f _λ ∘ T||_∞ = ||f _λ ||_∞ we have |λ| = 1. The collection e(T) of L ^∞ eigenvalues of T forms a subgroup of the circle group. Further

$$\left| {{f_\lambda }\left( {Tx} \right)} \right| = \left| \lambda \right|\left| {{f_\lambda }\left( x \right)} \right| = \left| {{f_\lambda }\left( x \right)} \right|a.e.m.$$

Since T is ergodic |f _λ | is constant a.e. m. The function $\frac{{{f_\lambda }}}{{\left| f \right|}}$ is an eigenfunction of absolute value one, with eigenvalue λ