368 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
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