A problem in convexity leading to the analysis of two functional equations

Author:
John V. Ryff

Journal:
Trans. Amer. Math. Soc. **305** (1988), 377-396

MSC:
Primary 46A55; Secondary 28D05, 39B10, 47B38

MathSciNet review:
920165

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Abstract | References | Similar Articles | Additional Information

Abstract: Transformation semigroups can often be studied effectively by examining their orbit structure. If the class of transformations has a special quality, such as convexity, it is generally reflected in the orbits. This work is concerned with such a circumstance. The goal is to examine the behavior of transformations on extreme points of orbits through the construction of a class of extreme operators. The construction leads naturally to the study of two functional equations which are analyzed in detail. Information about solutions is obtained through different -methods depending on whether or not two basic parameters are rational or irrational. In two cases all solutions are classified. In a third an example of a spanning set of solutions is obtained. Techniques of harmonic analysis and ergodic theory are used to study the functional equations.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0920165-1

Keywords:
Functional equations,
convexity,
extreme points,
extreme operators,
doubly stochastic,
ergodic mappings,
measure-preserving transformations

Article copyright:
© Copyright 1988
American Mathematical Society