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On some basic aspects of transfer operator methods for coupled cell systems / Mirko Hessel-von Molo. 2008
Inhalt
Abstract
Zusammenfassung
Introduction
Mathematical background
Coupled Cell Systems
Coupled Cell Networks
From Networks to Systems
Prerequisites from the Representation Theory for Groups
Basics
Description of the irreducible representations of Sd
Description of isotypic decompositions of Vd
Measure Theoretical Prerequisites
Basic concepts
Specific concepts
Probabilistic concepts
Transfer Operators
Transfer Operators for Dynamical Systems
Numerical Approximation of the Transfer Operator
A Structural Theorem
Transfer Operators for Coupled Cell Systems
Consequences of Independence -- Block decompositions of Pf
Coarse Block Decompositions of Pf
A Finer Block Decomposition of Pf
Consequences of Groupoid Equivariance for the Transfer Operator
Symmetry in coupled cell networks
Equivariance
Combining Independence and Equivariance Structure
Numerical Approximations of Coupled Cell Transfer Operators
What prevents an algorithmic use of the structure results
The dependence on the choice of basis
The standard procedure
Possible ways out
Efficient Numerical Approximation
Conclusion
What can be concluded
What else could be done
Relative symmetry groups and the groupoid
Symmetry-adapted bases
Row-wise approximation of the transfer operator
Bibliography
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