Regular or stochastic dynamics in higher degree families of unimodal maps

Trevor Clark, Princeton University
Jadwin Hall A09

About fifteen years ago, Palis conjectured that typical dynamical systems should possess good statistical properties. Through the work of Avila, Lyubich, de Melo and Moreira, this has been proven for unimodal maps with a non-degenerate critical point. I will show how to remove the condition on the critical point in analytic families of unimodal maps; along the way proving that that the hybrid classes in the space of unimodal maps yield a lamination near all but countably many maps in the family. The essential difference in the higher degree case is the presence of non-renormalizable maps without "decay of geometry". The key to their study is the use of a generalized renormalization operator, which has much in common with the usual renormalization operator.