29 jan 2019 -- 16:00
Aula D'Antoni, Dip.Matematica, Università "Tor Vergata", Roma
Abstract.
The moduli space of all degree D rational maps is an orbifold of dimension 2D−2. We present a language for describing dynamically natural subspaces, for example, the loci of maps having
• specified critical orbit relations,
• cycles of specified period and multiplier,
• parabolic cycles of specified degeneracy and index,
• Herman ring cycles of specified rotation number,
or some combination thereof.
We present a methodology for proving the smoothness and transversality of such loci. The natural setting for the discussion is a family of deformation spaces arising functorially from first principles in Teichmuller theory. Transversality ¨ flows from an infinitesimal rigidity principle (following Thurston), in the corresponding variational theory viewed cohomologically (following Kodaira-Spencer). Results for deformation spaces may then be transferred to moduli space. Moreover, the deformation space formalism and associated transversality principles apply more generally to finite type transcendental maps.