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Title: A solution to Schroder's equation in several variables

For this paper, let φ be an analytic self-map of the n -ball, having 0 as the attracting fixed point and having full-rank near 0. We consider the generalized Schroder's equation, F°φ=φ'(0)kF with ka positive integer and prove there is always a solution F with linearly independent component functions, but that such an F cannot have full rank except possibly when k=1. Furthermore, when k=1 (Schroder's equation), necessary and sufficient conditions on φ are given to ensure F has full rank near 0 without the added assumption of diagonalizability as needed in the 2003 Cowen/MacCluer paper. In response to Enoch's 2007 paper, it is proven that any formal power series solution indeed represents an analytic function on the whole unit ball. Finally, how exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schroder's equation.
  1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computational Sciences and Engineering Division
Publication Date:
OSTI Identifier:
Grant/Contract Number:
Accepted Manuscript
Journal Name:
Journal of Functional Analysis
Additional Journal Information:
Journal Volume: 270; Journal Issue: 9; Journal ID: ISSN 0022-1236
Research Org:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING Schroder; functional equation; composition operator; iteration; analytic functions; Bergman space; compact operator