A compact set satisfies Łojasiewicz-Siciak condition if it is polynomially convex and there exist constants B,β > 0 such that if dist(z,K) ≤ 1. (LS) Here denotes the pluricomplex Green function of the set K. We cite theorems where this condition is necessary in the assumptions and list known facts about sets satisfying inequality (LS).
It is a survey article showing how an enhanced version of the Banach contraction principle can lead to generalizations of attractors of iterated function systems and to Julia type sets.
Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.
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