On the effect of pruning on the singularity structure of zeta functions

Details On the effect of pruning on the singularity structure of zeta functions On the effect of pruning on the singularity structure of zeta functions

1996-06-13

arxiv.org/abs/chao-dyn/9606004v1

Nonlinear Sciences

Chaotic Dynamics

12 pages LaTeX

Scientific paper

10.1063/1.532094

We investigate the topological zeta function for unimodal maps in general and dynamical zeta functions for the tent map in particular. For the generic situation, when the kneading sequence is aperiodic, it is shown that the zeta functions have a natural boundary along its radius of convergence, beyond which the function lacks analytic continuation. We make a detailed study of the function $\prod_{n=0}^{\infty}(1-z^{2^n})$ associated with sequences of period doublings. It is demonstrated that this function has a dense set of poles and zeros on the unit circle, exhibiting a rich number theoretical structure.

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