A solution to Schröder's equation in several variables

Robert A. Bridges

doi:10.1016/j.jfa.2016.02.024

A solution to Schröder's equation in several variables

Robert A. Bridges

Data and Software Engineering

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

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 Schröder's equation, Fφ=φ^'(0)^kF with k a 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 (Schröder'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. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schröder's equation.

Original language	English
Pages (from-to)	3137-3172
Number of pages	36
Journal	Journal of Functional Analysis
Volume	270
Issue number	9
DOIs	https://doi.org/10.1016/j.jfa.2016.02.024
State	Published - May 1 2016

Funding

Notice: This research was conducted at Purdue University, West Lafayette, IN, with authorship completed at Oak Ridge National Laboratory. This research was partially supported by NSF Analysis and Cyber-Enabled Discovery Initiative Programs, grant number DMS-1001701 . This manuscript has been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with the U.S. Department of Energy . The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes. The Department of Energy will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan ( http://energy.gov/downloads/doe-public-access-plan ).

Funders	Funder number
National Science Foundation	DE-AC05-00OR22725, DMS-1001701
U.S. Department of Energy

Keywords

Analytic functions
Bergman space
Compact operator
Composition operator
Functional equation
Iteration
Schröder

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10.1016/j.jfa.2016.02.024

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abstract = "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 Schr{\"o}der's equation, Fφ=φ'(0)kF with k a 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 (Schr{\"o}der'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. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schr{\"o}der's equation.",

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AB - 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 Schröder's equation, Fφ=φ'(0)kF with k a 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 (Schröder'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. How exactly resonance can lead to an obstruction of a full rank solution is discussed as well as some consequences of having solutions to Schröder's equation.

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KW - Bergman space

KW - Compact operator

KW - Composition operator

KW - Functional equation

KW - Iteration

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A solution to Schröder's equation in several variables

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