Composite Julia sets generated by infinite polynomial arrays.
Let \({\mathcal F}\) be a family of proper polynomial mappings \(P: \mathbb{C}^N\to \mathbb{C}^N\) with Lojasiewicz exponent \(> 1\). Then it is known, the set function \[ {\mathcal H}_{\mathcal F}:{\mathcal R}\ni K\mapsto \Biggl(\bigcup_{P\in{\mathcal F}} P^{-1}(K)\Biggr)^\Lambda\in{\mathcal R}\tag{1} \] becomes a contraction of \({\mathcal R}\) and hence has a unique fixed point. Here, \({\mathcal R}\) denotes the family of all polynomial convex pluriregular compact subsets of \(\mathbb{C}^N\) and the exponent \(\Lambda\) denotes the operator taking the polynomially convex hull of the set enclosed in the big parentheses. In this paper, the authors show that under suitable assumptions the mapping (1) remains a contraction of \({\mathcal R}\) even for some class of \({\mathcal F}\) containing infinitely many polynomial mappings. Moreover, they show that mappings of this type preserve the Hölder continuity property of compact sets.
scientific article; zbMATH DE number 2032711