On the Generalised Transfer Operators of the Farey Map with Complex Temperature

Claudio Bonanno ()
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Claudio Bonanno: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Mathematics, 2022, vol. 11, issue 1, 1-16

Abstract: We consider the problem of showing that 1 is an eigenvalue for a family of generalised transfer operators of the Farey map. This is an important problem in the thermodynamic formalism approach to dynamical systems, which in this particular case is related to the spectral theory of the modular surface via the Selberg Zeta function and the theory of dynamical zeta functions of maps. After briefly recalling these connections, we show that the problem can be formulated for operators on an appropriate Hilbert space and translated into a linear algebra problem for infinite matrices. This formulation gives a new way to study numerically the spectrum of the Laplace–Beltrami operator and the properties of the Selberg Zeta function for the modular surface.

Keywords: transfer operators; Gauss and Farey maps; spectral theory of the modular surface; Laguerre polynomials (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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