SIGN CHANNELS, DIRICHLET L-VALUES AND A NON-PERTURBATIVE HAUSDORFF DIMENSION ON THE BRANCH TREE OF THE EXPONENTIAL

Let $log_{k}(y)=\log y + 2\pi ik$ with Im $\log \in [0, 2\pi)$, and let $J_{\infty}$ be the limit set of the associated conformal iterated function system on the branch tree of the exponential map. We show that the arithmetic of this system is organised by a two-state sign channel. Its transfer matrix has entries $(2\pi)^{-s}\zeta(s,\frac{5}{4})$ and $(2\pi)^{-s}\zeta(s,\frac{7}{4})$ and closed eigenvalues with $\lambda_+(s)=(\frac{\pi}{2})^{-s}[(1-2^{-s})\zeta(s)-1-3^{-s}]$ and $\lambda_-(s)=(\frac{\pi}{2})^{-s}[\beta(s)-1+3^{-s}]$ where $\beta=L(\cdot,\chi_{-4})$ so that at $s=3$ the symmetric channel is $(7\zeta(3)-\frac{224}{27})/\pi^{3}$ and the antisymmetric one is $\frac{1}{4}-\frac{208}{27\pi^{3}}$. Apéry's constant governs the sign-blind channel and $\beta(3)=\pi^{3}/32$ the sign-sensitive one. We prove a Hadamard factorisation for the fixed-point spectrum of $\exp$ giving $\Sigma^{\prime}z_{k}^{-1}=\frac{1}{2}$ and rational higher power sums that determine the Lyapunov ground state $\theta=1.37455701$; we extend the holomorphy of the one-letter partition function to Re $s > -1$ with $Z_{1}(0)=-\frac{3}{2}$ and a closed finite part at $s=1$, and we obtain an exact finite-part integral for the sign-odd cubic spectral sum. We then introduce a hierarchy of cylinder transfer matrices and a Nyström discretisation in $t=\log n$ which computes the dimension of the infinite-alphabet system directly, without extrapolation in the alphabet size: $\delta_{\infty}=1.207283(3)$ and $\Lambda_{\infty}=6.689$. Twisting by the near-trivial characters $\chi_{\alpha}$ shows that at $K=\infty$ the spectral gap closes as $\vert{}\alpha\vert{}^{\delta_{\infty}-1}$ rather than quadratically, that the centred winding number obeys a symmetric stable law of index $\delta_{\infty}-1=0.2074$, and that the leading coefficient is $\vert{}\Gamma(1-s)\vert{}\cos(\pi(s-1)/2)/\pi$ at the sign-channel level. Finally, the rotated trees of $e^{z+i\tau}$ realise the shift parameters $\frac{1}{4}\mp\frac{\tau}{2\pi}$ and produce $L(s,\chi_{-3})$, $L(s,\chi_{-4})$ and $L(s,\chi_{-8})$ as one-letter partition functions of genuine dynamical systems, with $\delta_{\infty}(\tau)$ increasing from 1.207284 to 1.207961.