A uniform derivative bound certifies contraction of relaxed density mixing

Given: A nonempty closed convex $K\subseteq {\mathcal{D}}_r$, a $C^1$ map $\mathrm{\Phi }$ defined on a neighborhood of $K$ with $\mathrm{\Phi }(K)\subseteq {\mathcal{P}}_r$, a fixed $\alpha \in [0,1]$, invariance $T_{\alpha }(K)\subseteq K$, and ${\sup }_{D\in K}\mathrm{\parallel }D{T}_{\alpha }(D){\mathrm{\parallel }}_{F\to F}\le q<1$.

The response map is $\mathrm{\Phi }(D)={\mathrm{\Pi }}_r(F(D))$ wherever $F(D)$ has a strict $r$th gap (The occupied-projector response map on its strict-gap domain).

At a strict gap, the response map has a unique rank-$r$ projector value (A strict occupied gap defines a unique rank-r spectral projector).

The gapped block projector is $C^1$ and has the contour derivative on its common-gap neighborhood (A separated block spectral projector is Frechet differentiable with a resolvent bound).

A continuously differentiable vector-valued map with derivative norm bounded by $q$ on an interval is $q$-Lipschitz (If $f:[a,b]\to {\mathbb{R}}^m$ is differentiable with integrable $f^{\mathrm{\prime }}$ then ${\int }_a^b{f}^{\mathrm{\prime }}=f(b)-f(a)$; and a bounded derivative makes $f$ Lipschitz).

A contraction of a nonempty complete metric space has a unique fixed point and convergent iterates (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point).

The composition derivative is given by the chain rule (The chain rule for total derivatives: $D(g\circ f)(a)=Dg(f(a))\circ Df(a)$).

A $C^1$ map has a continuous Frechet derivative on its open domain (Continuously differentiable maps, local inverses, and local diffeomorphisms).

The admissible density domain is complete in Frobenius norm (The admissible Hermitian density domain is closed, convex, and complete).