The fixed points of the transfer operator
Mathematicians love to find groups in unexpected places, and the discovery of Frobenius that the set of phases of the eigenvalues of maximal modulus of a given positive matrix is the finite union of finite abelian groups with multiplication equal to multiplication of phases is a case in point. This discovery has had all kinds of wonderful applications, for example to symbolic dynamics and computational mathematics. But the result does not carry over to the infinite-dimensional version of the Perron-Frobenius theory which you saw in the last chapter. So the next best thing is to look instead for a related property which still has the implications that you want for wavelet analysis. The group property that Frobenius discovered has its origin in the study of inner products u* υ where u and υ are eigenvectors for some given positive matrix R corresponding to a pair of peripheral eigenvalues, say λ u and λ v .
KeywordsPure State Spectral Function Convex Combination Transfer Operator Haar Measure
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