Seminar Algebra and Geometry: Salomon Vishkautsan (ERC postdoc at Scuola Normale Superiore di Pisa)

Residual periodicity in arithmetic dynamics

In this talk we present a local-global property in arithmetic dynamics called strong residual periodicity, as defined by Bandman, Grunewald and Kunyavskii in 2010. We start with a dynamical system induced by an endomorphism of a quasi projective variety defined over a number field. This system can be reduced mod p for “almost all” primes in the ring of integers of the number field. We can then ask how the dynamics of the global system relate to the dynamics of the system reduced mod p for almost all primes p. Strong residual periodicity occurs when points of small period exist modulo almost every prime,but “cannot be explained” by the dynamics of the global system. The aim of this talk is to present many motivating examples and raise some interesting questions to encourage further research on this topic.

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