|Location:||MSRI: Online/Virtual, Simons Auditorium|
COMD Stony Brook + MSRI Seminar Series: "Conjugacy Classes Of Real Analytic Maps Pt I: Manifold Structure And Connectedness"
To participate in this seminar, please register HERE.
Avila-Lyubich-de Melo proved that the topological conjugacy classes of unimodal real-analytic maps are real-analytic manifolds, which laminate a neighbourhood of any such mapping without a neutral cycle. Their proof that the manifolds are analytic, uses holomorphic motions, and crucially depends on the fact that they have codimension-one in the space of unimodal mappings.
In joint work with Trevor Clark, we show how to construct a “pruned polynomial-like mapping" associated to a real analytic mapping. This gives a new complex extension of a real-analytic mapping.
The additional structure provided by this extension, makes it possible to generalize this result of Avila-Lyubich-de Melo to interval mappings with several critical points. Thus we show that the conjugacy classes are complex analytic manifolds whose codimension is determined by the number of critical points. Moreover we show that these manifolds are connected and even contractible.
We will discuss applications of this to families of polynomials and of entire maps.
In part II of this talk we will discuss the question whether in the space of unimodal mappings, the conjugacy classes laminate a neighbourhood of every mapping, aiming to answer a question of Avila-Lyubich-de Melo.No Notes/Supplements Uploaded