Iteration of analytic multifunctions

Abstract

It is shown that iteration of analytic set-valued functions can be used to generate composite Julia sets in ${\bf C}^N$. Then it is shown that the composite Julia sets generated by a finite family of regular polynomial mappings of degree at least 2 in ${\bf C}^N$, depend analytically on the generating polynomials, in the sense of the theory of analytic set-valued functions. It is also proved that every pluriregular set can be approximated by composite Julia sets. Finally, iteration of infinitely many polynomial mappings is used to give examples of pluriregular sets which are not composite Julia sets and on which Markov's inequality fails.