Finding Rational Periodic Points on Wehler K3 Surfaces

This article examines dynamical systems on a class of K3 surfaces in $\mathbb{P}^{2} \times \mathbb{P}^{2}$ with an infinite automorphism group. In particular, this article develops an algorithm to find $\mathbb{Q}$-rational periodic points using information modulo $p$ for various primes $p$. The algorithm is applied to exhibit K3 surfaces with $\mathbb{Q}$-rational periodic points of primitive period $1,...,16$. A portion of the algorithm is then used to determine the Riemann zeta function modulo 3 of a particular K3 surface and find a family of K3 surfaces with Picard number two.Comment: to appear New Zealand Journal of Mathematic

Good reduction and canonical heights of subvarieties

We bound the length of the periodic part of the orbit of a preperiodic rational subvariety via good reduction information. This bound depends only on the degree of the map, the degree of the subvariety, the dimension of the projective space, the degree of the number field, and the prime of good reduction. As part of the proof, we extend the corresponding good reduction bound for points proven by the author for non-singular varieties to all projective varieties. Toward proving an absolute bound on the period for a given map, we study the canonical height of a subvariety via Chow forms and compute the bound between the height and canonical height of a subvariety. This gives the existence of a bound on the number of preperiodic rational subvarieties of bounded degree for a given map. An explicit bound is given for hypersurfaces.Comment: 14 page

Good Reduction of Periodic Points

We consider the dynamical system created by iterating a morphism of a projective variety defined over the field of fractions of a discrete valuation ring. We study the primitive period of a periodic point in this field in relation to the primitive period of the reduced point in the residue field, the order of the action on the cotangent space, and the characteristic of the residue field.Comment: to appear Illinois Journal of Mat

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

Rational Periodic Points for Degree Two Polynomial Morphisms on Projective Space

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 $\P^N$ where each component is a degree 2 homogeneous form in $N+1$ variables which has a \Q-periodic point of primitive period $\frac{(N+1)(N+2)}{2} + \lfloor \frac{N-1}{2}\rfloor$. This result is then used to show that for $N$ large enough there exists morphisms of $\P^N$ with \Q-rational periodic points with primitive period larger that $c(k)N^k$ for any $k$ and some constant $c(k)$.Comment: to appear in Acta Arithmetic

Determination of all rational preperiodic points for morphisms of PN

For a morphism $f:\P^N \to \P^N$, 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 dynatomic zero-cycle is generalized to preperiodic points. Along with examining their basic properties, these generalized dynatomic cycles are shown to be effective.Comment: 18 pages. To appear in Mathematics of Computation. Sage implementation of the algorithm is Sage Trac Ticket #1421

The field of definition for dynamical systems on P^N

Let Hom^N_d be the set of morphisms of degree d from P^N to itself. For f an element of PGL_{N+1}, let phi^f represent the conjugation action f^{-1} phi f. Let M^N_d = Hom_d^N/PGL_{N+1} be the moduli space of degree d morphisms of P^N. A field of definition for class of morphisms is a field over which at least one morphism in the class is defined. The field of moduli for a class of morphisms is the fixed field of the set of Galois elements fixing that class. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a field of definition for morphisms whose stabilizer group is trivial

Almost Newton, sometimes Latt\`es

Self-maps everywhere defined on the projective space $\P^N$ over a number field or a function field are the basic objects of study in the arithmetic of dynamical systems. One reason is a theorem of Fakkruddin \cite{Fakhruddin} (with complements in \cite{Bhatnagar}) that asserts that a "polarized" self-map of a projective variety is essentially the restriction of a self-map of the projective space given by the polarization. In this paper we study the natural self-maps defined the following way: $F$ is a homogeneous polynomial of degree $d$ in $(N+1)$ variables $X_i$ defining a smooth hypersurface. Suppose the characteristic of the field does not divide $d$ and define the map of partial derivatives $\phi_F = (F_{X_0},...,F_{X_N})$. The map $\phi_F$ is defined everywhere due to the following formula of Euler: $\sum X_i F_{X_i} = d F$, which implies that a point where all the partial derivatives vanish is a non-smooth point of the hypersuface F=0. One can also compose such a map with an element of \PGL_{N+1}. In the particular case addressed in this article, N=1, the smoothness condition means that $F$ has only simple zeroes. In this manner, fixed points and their multipliers are easy to describe and, moreover, with a few modifications we recover classical dynamical systems like the Newton method for finding roots of polynomials or the Latt\`es map corresponding to the multiplication by 2 on an elliptic curve.Comment: 11 page

Misiurewicz Points for Polynomial Maps and Transversality

The behavior under iteration of the critical points of polynomial maps 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 portrait and degree leads to only finitely may possible polynomials (up to equivalence) and that, in many cases, their defining equations intersect transversely. We provide explicit algebraic formulae for the parameters where the critical points of the unicritical polynomials and bicritical cubic polynomials have a specified exact period. We pay particular attention to the parameters where the critical orbits are strictly preperiodic, called Misiurewicz points. Our main tool is the generalized dynatomic polynomial. We also study the discriminants of these polynomials to examine the failure of transversality in positive characteristic for unicritical polynomials

On Poonen's Conjecture Concerning Rational Preperiodic Points of Quadratic Maps

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. In particular, Poonen conjectured that there are at most 9 periodic points defined over the rational numbers for any map in the family x^2 + c for c rational. We verify this conjecture for c values up to height 10^8. For quadratic number fields, we provide evidence that the upper bound on the exact period of Q-rational periodic point is 6.Comment: revised. too appear in Rocky Mountain Journal of Mat