Research Statement -benjamin Hutz Local Information Dynatomic Cycles

Dynatomic cycles for morphisms of projective varieties

We prove the effectivity of the dynatomic cycles for morphisms of projective varieties. We then analyze the degrees of the dynatomic cycles and multiplicities of formal periodic points and apply these results to the existence of periodic points with arbitrarily large primitive periods. 1991 Mathematics Subject Classification. 14C99, 14J99.

Misiurewicz points for polynomial maps and transversality

The behavior under iteration of the critical points of a polynomial map plays an essential role in understanding its dynamics. We study the special case where the forward orbits of the critical points are finite. Thurston’s theorem tells us that fixing a particular critical point portrait and degree leads to only finitely many possible polynomials (up to equivalence) and that, in many cases, th...

A Computational Investigation of Wehler K3 Surfaces

This article examines dynamical systems on a class of K3 surfaces in P×P with an infinite automorphism group. In particular, this article develops an algorithm to find Q-rational periodic points using information over Fp for various primes p. This algorithm is then optimized to examine the growth of the average number of cycles versus p and to determine the number of Fpm -rational points. The p...

Arithmetic and Diophantine Geometry

Numerical Evidence for a Conjecture of Poonen

The purpose of this note is give some evidence in support of conjectures of Poonen, and Morton and Silverman, on the periods of rational numbers under the iteration of quadratic polynomials. Suppose that φc(z) = z 2 + c, where c ∈ Q. We will say that α ∈ P1(Q) is a periodic point with exact period n for φc if φ n c (α) = α, while φ m c (α) 6= α for 0 < m < n. For example, the point at infinity ...