Critical-orbit bound for unstable manifolds of renormalization

Let $\mathcal{R}$ be a compact analytic renormalization operator with a hyperbolic fixed point, and let $\mathcal{W}^u_{\mathrm{loc}}$ be its local unstable manifold. Assume that every map on $\mathcal{W}^u_{\mathrm{loc}}$ admits a global transcendental extension. Critical-orbit bound. The dimension of the local unstable manifold satisfies

$$\dim \mathcal{W}^u_{\mathrm{loc}}\leq \text{number of critical orbits}.$$

This is presented as a general philosophy motivated by the multiple-critical-point extension of the paper's hyperbolicity result; its validity in the stated generality remains open.