A solution to Schroder's equation in several variables
For this paper, let φ be an analytic selfmap of the n ball, having 0 as the attracting fixed point and having fullrank near 0. We consider the generalized Schroder's equation, F _{°}φ=φ'(0) ^{k}F 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.
 Authors:

^{[1]}
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computational Sciences and Engineering Division
 Publication Date:
 Grant/Contract Number:
 AC0500OR22725
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Functional Analysis
 Additional Journal Information:
 Journal Volume: 270; Journal Issue: 9; Journal ID: ISSN 00221236
 Publisher:
 Elsevier
 Research Org:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; Schroder; functional equation; composition operator; iteration; analytic functions; Bergman space; compact operator
 OSTI Identifier:
 1286688
 Alternate Identifier(s):
 OSTI ID: 1467156
Bridges, Robert A. A solution to Schroder's equation in several variables. United States: N. p.,
Web. doi:10.1016/j.jfa.2016.02.024.
Bridges, Robert A. A solution to Schroder's equation in several variables. United States. doi:10.1016/j.jfa.2016.02.024.
Bridges, Robert A. 2016.
"A solution to Schroder's equation in several variables". United States.
doi:10.1016/j.jfa.2016.02.024. https://www.osti.gov/servlets/purl/1286688.
@article{osti_1286688,
title = {A solution to Schroder's equation in several variables},
author = {Bridges, Robert A.},
abstractNote = {For this paper, let φ be an analytic selfmap of the n ball, having 0 as the attracting fixed point and having fullrank 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.},
doi = {10.1016/j.jfa.2016.02.024},
journal = {Journal of Functional Analysis},
number = 9,
volume = 270,
place = {United States},
year = {2016},
month = {3}
}