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The joint spectral radius measures the worst exponential growth under arbitrary products from a finite family of matrices. Submultiplicativity gives its existence and invariance under changes of the chosen norm. Every fixed-length product $B$ satisfies $\rho (B{)}^{1/p}\le \hat{\rho }(\mathcal{A})$, while the maximum norm of all length-$p$ products to the power $1/p$ is an upper bound. A repeated product gives the first inequality; grouping an arbitrary long word into blocks proves the second. The spectral radius of each individual matrix does not control growth of all switched products.

There are no numerical continuity bounds determined by $w$ and $a$ alone: the subdivision mask coefficients matter. First check constant reproduction, ${\sum }_j{M}_{ϵ-aj}=1$ for every residue $ϵ$, and convergence. For the usual stationary scalar scheme, two useful calculations are a uniform contraction test and a spectral obstruction test. Higher smoothness is tested with derived difference subdivision masks.

Write $M(z)={\sum }_j{M}_j{z}^j$ and $s_a(z)=1+z+\cdots +z^{a-1}$. Using backward differences $\Delta P_i=P_i-P_{i-1}$, the generating-function identityproves the subdivision difference scheme relation $\Delta S_M=S_{M/s_a}\Delta$. Iterating gives ${\Delta }^r{S}_M=S_{M/s_a^r}{\Delta }^r$. To test a candidate $C^r$ limit, suppose the requisite sum rules hold, so $M/s_a^{r+1}$ is a finite Laurent polynomial. The scaled derivative controls $q_i^{\ell }=a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are refined by the subdivision maskwhose residue sums are one. Their first differences are refined byThe powers of $a$ are essential because each refinement shrinks parameter intervals by $a$.

A lower continuity bound comes from proving uniform contraction. The infinity norm of one difference-refinement step is bounded byIf $q<1$, adjacent derivative controls contract geometrically at every location. More generally, form the $p$-step subdivision maskand test ${\max }_{ϵ\mathrm{mod}{a}^p}{\sum }_j|(B_r^{[p]}{)}_{ϵ-a^{p}j}|<1$. This proves convergence of the constant-preserving derivative scheme to a continuous limit. One justification is to compare consecutive piecewise-linear interpolants: their difference is bounded by a constant times the maximum adjacent-control difference, and the geometrically decreasing bound is summable. To identify the limit as $P^{(r)}$, use fixed-degree Cardinal B-spline interpolants of the original refined controls. Their $r$th derivatives are sums of the scaled finite differences; convergence of both functions and derivatives gives $P\in C^r$. Testing successive $r$ supplies a guaranteed lower bound.

An upper continuity bound comes from an observable noncontracting shape mode. Build finite local subdivision matrices for $B_r$ on a refinement-invariant control neighborhood. For a digit $ϵ$ their entries are $(A_{ϵ}{)}_{ij}=(B_r{)}_{ϵ+i-aj}$, since the new index is $ak+ϵ+i$. Restrict to the realizable difference space and discard modes that do not affect the limit. A word ${ϵ}_1,\dots ,{ϵ}_p$ corresponds to a nested parameter location. If its product has an observable eigenvalue $\lambda$ with $|\lambda |>1$, repetition of this word makes some scaled derivative differences grow rather than tend to zero. For a stable limit representation, $C^r$ regularity would force these differences to vanish; hence general-position data cannot have $C^r$ continuity. A modulus-one mode requires examination of whether it actually persists, rather than an automatic rounding rule.

The two calculations can be organized through the joint spectral radius $\hat{\rho }$ of these restricted matrices:These norm and spectral bounds for subdivision regularity make the test computable from the subdivision mask. The right bound certifies all-word contraction when it is less than one; the left bound detects a repeated-word obstruction when it exceeds one. Longer products improve the tests. A single matrix's spectrum is not a substitute for controlling arbitrary products. The product estimates also appear in Charina's analysis of subdivision regularity; the strict contraction and observable-mode arguments above explain their respective roles.