Concluding Remarks

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


The results in the previous chapters describe several phenomena for the transfer operator and the Selberg zeta function \(Z^{\left (n\right )}(\beta,\chi )\) which seem thus far to be unknown even to the experts. Clearly, one of our basic results is the symmetries of the transfer operator in Sect.  7.7 whose existence we found by investigating a new form of the transfer operator that we derived in Lemma  7.1. This form allows us to write down explicitly the action of the transfer operator on every component of a vector-valued function for any given group \(\varGamma _{0}\left (n\right )\) with a character χ, (see, e.g., Appendix D for the form of the transfer operator for \(\varGamma _{0}\left (8\right )\)).


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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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