Date:
Tue, 23/01/202414:00-15:00
Abstract. The problem of when two rational functions share the same Julia set goes back to Fatou, Julia and Ritt's study of commuting rational functions. This problem and more recent one about sharing a maximal measure are well studied. A similar looking problem of characterizing Iterated Function Systems having the same attractor is much more open and a solution is known for some special classes of self-similar sets. We give a solution for self-conformal sets (by this we mean attractors of analytic IFS on the plane with the Strong Separation Condition).
It is based on the following result.
Consider a class of plane compacts K with the property that
fixed points of local holomorphic symmetries of K are dense in K.
This class contains Julia sets and also self-conformal sets. Then
either any non-trivial local holomorphic family of symmetries of such compact K is finite
or the compact is locally "laminar". Proof follows the one for Julia sets.
No knowledge about the Julia set is used except that periodic points are dense.
fixed points of local holomorphic symmetries of K are dense in K.
This class contains Julia sets and also self-conformal sets. Then
either any non-trivial local holomorphic family of symmetries of such compact K is finite
or the compact is locally "laminar". Proof follows the one for Julia sets.
No knowledge about the Julia set is used except that periodic points are dense.