On the spectrum of Farey and Gauss maps

A thermodynamic formalism approach to period functions for Maass forms and generalisations

In this paper we study one-parameter families of signed transfer operators P± q associated to the Farey map, as well as two-parameter families of operators for the Gauss map, obtained from the Farey map by inducing. We characterise all the analytic eigenfunctions of P± q with eigenvalue λ not embedded in the continuous spectrum, which for λ = 1 are the period functions for Maass forms studied b...

Farey Sections in the Fields of Gauss and Eisenstein

Jump transformations and an embedding of O ∞ into O 2

A measurable map T on a measure space induces a representation ΠT of a Cuntz algebra ON when T satisfies a certain condition. For such two maps τ and σ and representations Πτ and Πσ associated with them, we show that Πτ is the restriction of Πσ when τ is a jump transformation of σ. Especially, the Gauss map τ1 and the Farey map σ1 induce representations Πτ1 of O∞ and that Πσ1 of O2, respectivel...

Fine Costs for the Euclid Algorithm on Polynomials and Farey Maps

This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq . The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficie...

Fine costs for Euclid’s algorithm on polynomials and Farey maps

This paper studies digit-cost functions for the Euclid algorithm on polynomials with coefficients in a finite field, in terms of the number of operations performed on the finite field Fq . The usual bit-complexity is defined with respect to the degree of the quotients; we focus here on a notion of ‘fine’ complexity (and on associated costs) which relies on the number of their non-zero coefficie...