Continuously generated fixed points

Abstract

Fixed points are solutions to equations of the form ƒ(x)=x. A fixed-point operator is a functional F:[ D→ D]→ D such that F(ƒ)=ƒ(F(ƒ)) for any ƒ in [ D → D]. Fixed-point operators, like the least fixed-point operator Y, are used to obtain solutions for recursively defined functions and domain equations. We work within the category of domains. There we characterize the fixed points of a continuous function that can be obtained via a continuous fixed-point operator: the continuously generated fixed points of ƒ. We define a continuous functional Cgn:[ D→ D]→[ D→ D] that maps any continuous function f into a retraction Cgn( f) whose retract Cgn(ƒ)( D) is precisely the domain of continuously generated fixed points of ƒ. We then investigate the continuously generated fixed points of β r. r° r to obtain the domain of retractions of D whose retracts, r( D), are themselves domains. We end by showing that F is a continuous fixed-point operator if and only if it is a continuously generated fixed-point of λyλƒ.ƒ(y(ƒ)), i.e. iff F= Cgn(λyλƒ.ƒ(y(ƒ)))(F) .