Eigenvalues of some almost periodic functions

Abstract

Let B U {B_U} be the set of real valued functions on R R which are bounded and uniformly continuous. For f , g ∈ B U f,g \in {B_U} , put \[ d ( f , g ) = sup t ∈ R | f ( t ) − g ( t ) | . d(f,g) = \sup \limits _{t \in R} |f(t) - g(t)|. \] Then B U {B_U} becomes a metric space. On B U {B_U} we define a flow η \eta by η ( f , t ) = f t \eta (f,t) = {f_t} for ( f , t ) ∈ B U × R (f,t) \in {B_U} \times R . We denote the restriction of η \eta to the hull of f ∈ B U f \in {B_U} by η f {\eta _f} . If f f is almost periodic, then the set of eigenvalues of η f {\eta _f} coincides with the module of f f (see J. Egawa, Eigenvalues of compact minimal flows, Math. Seminar Notes (Kobe Univ.), 10 (1982), 281-291. In this paper, we extend this result to almost periodic functions with some additional properties.