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

  • Markus Szymon Fraczek
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2139)

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.

Keywords

Period Function Analytic Continuation Composition Operator Transfer Operator Decimal Place 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Markus Szymon Fraczek
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

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