Numerical Results for Spectra and Traces of the Transfer Operator for Character Deformations

Abstract

In Chap. 7 we discussed how to evaluate the Selberg zeta function $Z^{\left (n\right )}(\beta,\chi )$ by computing the spectrum of the transfer operators

$$\displaystyle{\tilde{\mathcal{L}}_{\beta,\chi }^{\left (n\right )} = \left (\begin{array}{cc} 0 &\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )} \\ \mathcal{L}_{\beta,-1,\chi }^{\left (n\right )} & 0 \end{array} \right ),\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}\mathcal{L}_{\beta,-1,\chi }^{\left (n\right )}\text{ and }\mathcal{P}_{ k}\mathcal{L}_{\beta,+1,\chi }^{\left (n\right )}.}$$

To obtain a numerical approximation of the spectrum of the transfer operator $\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}$ in Proposition  7.5

$$\displaystyle\begin{array}{rcl} \left [\mathcal{L}_{\beta,\varepsilon,\chi }^{\left (n\right )}\vec{f}\left (z\right )\right ]_{ i}& =& \sum _{k=0}^{\infty }\sum _{ s=0}^{\infty }\sum _{ m=1}^{n}\sum _{ j=1}^{\mu _{n} }\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j}\frac{f_{j}^{\left (k\right )}\left (1\right )} {k!} \sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} {}\\ & & \frac{1} {s!} \frac{\varGamma (2\beta +t+s)} {\varGamma (2\beta + t)} \varPhi \!\left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta +t+s, \frac{m+1} {n} \right )\!\left (z-1\right )^{s}, {}\\ \end{array}$$

we approximate this operator by the matrix $\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}$ in Proposition  7.7

$$\displaystyle\begin{array}{rcl} \left [\left (\mathcal{M}_{\beta,\varepsilon,\chi }^{\left (n\right ),N}\right )_{ s,k}\right ]_{i,j}& =& \frac{1} {s!}\sum _{t=0}^{k}\binom{k}{t}\frac{\left (-1\right )^{k-t+s}} {n^{2\beta +t+s}} \frac{\varGamma (2\beta + t + s)} {\varGamma (2\beta + t)} \sum _{m=1}^{n}\left [U^{\chi }\left (ST^{m\varepsilon }\right )\right ]_{ i,j} {}\\ & & \varPhi \left (\chi \left (r_{j}^{\left (n\right )}T^{n\varepsilon }\left (r_{ j}^{\left (n\right )}\right )^{-1}\right ),2\beta + t + s, \frac{m + 1} {n} \right ) {}\\ \end{array}$$

and compute its spectrum.