26 research outputs found
Schwarz reflections and the Tricorn
We continue our study of the family of Schwarz reflection maps
with respect to a cardioid and a circle which was started in [LLMM1]. We prove
that there is a natural combinatorial bijection between the geometrically
finite maps of this family and those of the basilica limb of the Tricorn, which
is the connectedness locus of quadratic anti-holomorphic polynomials. We also
show that every geometrically finite map in arises as a conformal
mating of a unique geometrically finite quadratic anti-holomorphic polynomial
and a reflection map arising from the ideal triangle group. We then follow up
with a combinatorial mating description for the "periodically repelling" maps
in . Finally, we show that the locally connected topological model
of the connectedness locus of is naturally homeomorphic to such a
model of the basilica limb of the Tricorn
Singular continuous spectrum is generic
In a variety of contexts, we prove that singular continuous spectrum is
generic in the sense that for certain natural complete metric spaces of
operators, those with singular spectrum are a dense .Comment: 5 page
Gaussian free field and conformal field theory
In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory