Knowlpedia

Theorem

Circle reparametrization by a positive density

Integrating a positive normalized periodic density gives an orientation-preserving circle diffeomorphism.

Statement

Let ρ\rho be a smooth positive 11-periodic function with ∫01ρ(s) ds=1\int_0^1\rho(s)\,ds=1. Define

Φ(θ)=∫0θρ(s) ds.\Phi(\theta)=\int_0^\theta\rho(s)\,ds.

Then Φ′=ρ>0\Phi'=\rho>0 and Φ(θ+1)=Φ(θ)+1\Phi(\theta+1)=\Phi(\theta)+1, so Φ\Phi induces a smooth invertible map of the with a smooth inverse.

Changing an average

For g(ϕ)=f(Φ−1(ϕ))g(\phi)=f(\Phi^{-1}(\phi)), substitution yields

∫01g(ϕ) dϕ=∫01f(θ)ρ(θ) dθ.\int_0^1g(\phi)\,d\phi=\int_0^1 f(\theta)\rho(\theta)\,d\theta.

Thus the parameter speed changes how long a loop spends at each of its values. If the density depends smoothly on a compact parameter set and has a common positive lower bound, the inverse reparametrizations depend smoothly on those parameters, with bounded derivatives at every fixed order. A vanishing density would require a separate inverse-regularity analysis.