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Let K be a number field, let φ ∈ K (t) be a rational map of degree at least 2, and let α, β ∈ K . We show that if α is not in the forward orbit of β, then there is a positive proportion of primes p of K such that α mod p is not in the forward orbit of β mod p. Moreover, we show that a similar result holds for several maps and several points. We also present… (More)

- Benjamin A. Hutz
- 2008

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… (More)

- Benjamin A. Hutz
- 2008

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.

- BENJAMIN HUTZ
- 2009

We consider dynamical systems arising from iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. Our goal is to obtain information about the dynamical system over the field of fractions by studying the dynamical system over the residue field. In particular, we aim to bound the possible primitive… (More)

- Benjamin Hutz
- Math. Comput.
- 2015

For a morphism f : PN → PN , the points whose forward orbit by f is finite are called preperiodic points for f . This article presents an algorithm to effectively determine all the rational preperiodic points for f defined over a given number field K. This algorithm is implemented in the open-source software Sage for Q. Additionally, the notion of a… (More)

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 <… (More)

- BENJAMIN HUTZ
- 2009

This article addresses the existence of Q-rational periodic points for morphisms of projective space. In particular, we construct an infinitely family of morphisms on P where each component is a degree 2 homogeneous form in N+1 variables which has a Q-periodic point of primitive period (N+1)(N+2) 2 + ̈ N−1 2 ̋ . This result is then used to show that for N… (More)

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… (More)

- Benjamin A. Hutz
- 2009

We consider the dynamical system created by iterating a morphism of a projective variety over the field of fractions of a discrete valuation ring. In the case of good reduction, we study the primitive period of a periodic point on the residue field. We start by defining good reduction, examine the behavior of primitive periods under good reduction, and end… (More)

- Benjamin Hutz
- 2006

In Morton and Silverman [8], the authors give an upper bound on the possible primitive periods of periodic points for rational maps on P (and also automorphisms of P ). I generalize their main theorem to nondegenerate morphisms on smooth, irreducible projective varieties and provide a similar upper bound on primitive periods based on good reduction… (More)