Shape Measures and Typical Growth in the Markoff Tree

The positive integer solutions of the Markoff equation are generated by Vieta involutions and form the classical Markoff tree. This paper studies a layer-uniform asymptotic problem which is different from the uniqueness conjecture and from the endpoint Lyapunov spectrum: what is the typical logarithmic shape of a large-level Markoff triple, and what is the typical growth rate of its largest coordinate? For an ordered Markoff triple \((x,y,z)\) with \(x \le y \le z\), set \(u = (\log x)/(\log z)\) and \(v = (\log y)/(\log z)\). We prove the sharp shape-band estimate \[ u + v = 1 - \frac{c(x,y,z)}{\log z}, \qquad \log 2 \le c(x,y,z) \le \log 3. \] Thus every large-level weak shape limit is carried by the line \(u + v = 1\). On this line the one-dimensional layer shape converges, with a quantitative Wasserstein estimate, to the unique invariant measure of the two ordinary Möbius maps \[ a(u) = \frac{u}{1+u}, \qquad b(u) = \frac{1-u}{2-u}. \] Under the change of coordinate \(r = u/(1-u)\), this limiting shape measure is exactly the pushforward of Minkowski's question-mark measure. Hence it is continuous, non-atomic, and singular with respect to Lebesgue measure, and its distribution function is \(q(u/(1-u))\), where \(q\) denotes Minkowski's question-mark function. Finally, we prove an almost sure typical double-exponential growth law. If \(M_n\) is the largest coordinate on a Bernoulli-uniform branch at level \(n\), then \[ \frac{1}{n}\log\log M_n \longrightarrow l_{\mathrm{sh}} \] almost surely, where \[ l_{\mathrm{sh}} = \frac{1}{2}\int_0^{1/2} \log((1+u)(2-u)) \, d h_{\mathrm{sh}}(u) = \frac{1}{2}\int_0^1 \log\left(\frac{(1+2r)(2+r)}{(1+r)^2}\right) d h_q(r). \] Numerically, the shape mean is about \(0.32075\), the shape standard deviation is about \(0.09385\), and \(l_{\mathrm{sh}}\) is about \(0.39618\). The source package contains exact enumeration data and code reproducing these constants.