Graduate Thesis Or Dissertation
 

An inner product on adelic measures

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https://ir.library.oregonstate.edu/concern/graduate_thesis_or_dissertations/3t946055v

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  • We define an inner product on a vector space of adelic measures over a number field $K$. We find that the norm induced by this inner product governs weak convergence at each place of $K$. The canonical adelic measure associated to a rational map is in this vector space, and the square of the norm of the difference of two such adelic measures is the Arakelov-Zhang pairing from arithmetic dynamics. We prove a sharp lower bound on the norm of adelic measures with points of small adelic height. We find that the norm of a canonical adelic measure associated to a rational map is commensurate with a height on the space of rational functions with fixed degree. As a consequence, we deduce that the Arakelov-Zhang pairing of two rational maps $f$ and $g$ is commensurate with the height of $f$. As an application of our theory, we use these height comparisons to show that the canonical measure of $f \circ g^n$ converges weakly at each place of $K$ to the canonical measure of $g$. In the course of this proof, we also find some equivalences between the canonical adelic measures of $f$ and $g$ and their compositions $f\circ g$, $g\circ f$. These equivalences provide a new proof of a special case of a result due to Bell, Huang, Peng, and Tucker.
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