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

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.Comment: 12 pages LaTe

The role of singularities in chaotic spectroscopy

We review the status of the semiclassical trace formula with emphasis on the particular types of singularities that occur in the Gutzwiller-Voros zeta function for bound chaotic systems. To understand the problem better we extend the discussion to include various classical zeta functions and we contrast properties of axiom-A scattering systems with those of typical bound systems. Singularities in classical zeta functions contain topological and dynamical information, concerning e.g. anomalous diffusion, phase transitions among generalized Lyapunov exponents, power law decay of correlations. Singularities in semiclassical zeta functions are artifacts and enters because one neglects some quantum effects when deriving them, typically by making saddle point approximation when the saddle points are not enough separated. The discussion is exemplified by the Sinai billiard where intermittent orbits associated with neutral orbits induce a branch point in the zeta functions. This singularity is responsible for a diverging diffusion constant in Lorentz gases with unbounded horizon. In the semiclassical case there is interference between neutral orbits and intermittent orbits. The Gutzwiller-Voros zeta function exhibit a branch point because it does not take this effect into account. Another consequence is that individual states, high up in the spectrum, cannot be resolved by Berry-Keating technique.Comment: 22 pages LaTeX, figures available from autho

Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.Comment: 36 page