Instytut Matematyczny Polskiej Akademii Nauk / pl / Wydawnictwa / Banach Center Publications / Wszystkie tomy

Streszczenie

We give a discussion of the classical Bowen–Series coding and, in particular, its application to the study of zeta functions and their zeros. In the case of compact surfaces of constant negative curvature $\kappa = -1$ the analytic extension of the Selberg zeta function to the entire complex plane is classical, and can be achieved using the Selberg trace formula. However, an alternative dynamical approach is to use the Bowen–Series coding on the boundary at infinity to construct a piecewise analytic expanding map from which the extension of the zeta function can be obtained using properties of the associated transfer operator. This latter method has the advantage that it also applies in the case of infinite area surfaces provided they don’t have cusps. For such examples the location of the zeros is somewhat more mysterious. However, in particularly simple cases there is a striking structure to the zeros when we take appropriate rescaling. We give some insight into this phenomenon.