4,255 research outputs found
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 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
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