Algebraic Number Theory: Juan Rivera-Letelier, University of Rochester

Counting Non-Repelling Cycles of p-Adic Rational Maps

A classical theorem of Shishikura, building on Fatou's work,
asserts that a complex rational map of degree d can have at most 2d -
2 non-repelling cycles. The analogous statement fails for p-adic
rational maps, since they can have infinitely many non-repelling
cycles. I will survey results that bound the number of non-repelling
cycles when it is finite, including results on the action of rational
maps on Berkovich space. These complement Bottiglione's recent result
counting repelling cycles and determining their asymptotic
distribution. I'll conclude by proposing open problems.