# An Inner Product on Adelic Measures

# An Inner Product on Adelic Measures

**Start:**Tuesday, June 6, 2023 10:00 am

**End:**Tuesday, June 6, 2023 12:00 pm

Kidder 274

#### Peter Oberly

Oregon State University

We define an inner product on a vector space of adelic measures over a number field. When restricted to the subspace of adelic measures with total mass zero, the norm induced by our inner product agrees with the mutual energy pairing considered by Favre & Rivera-Letelier, and by Fili. We find that the norm of the 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, a notion of dynamical distance between two rational maps is also commensurate with a height.

**Contact:**Carol Murphy (Carol.Murphy@oregonstate.edu)