arXiv subjects

Adam Towsley

Publications and source records attributed to Adam Towsley.

9 recordsLinked to original sources

arXiv2022

Integrality and Thurston Rigidity for Bicritical PCF Polynomials

We give an algebraic proof of an important consequence of Thurston rigidity for bicritical PCF polynomials with periodic critical points under certain mild assumptions. The key result is that when the family of bicritical polynomials is parametrized using dynamical Belyi polynomials, the PCF solutions are integral at certain special primes, which we term ``index divisor free primes.'' We prove the existence of index divisor free primes in all but finitely many cases and conjecture the complete list of exceptions. These primes are then used to prove transversality.

math.DS↗

arXiv2020

New Normal Forms For Degree Three Polynomials and Rational Functions

When studying families in the moduli space of dynamical systems, choosing an appropriate representative function for a conjugacy class can be a delicate task. The most delicate questions surround rationality of the conjugacy class compared to rationality of the defining polynomials of the representation. We give a normal form for degree three polynomials which has the property that the set of fixed points is equal to the set of fixed point multipliers. This normal form is given in terms of moduli space invariants and, hence, has nice rationality properties. We further classify all degree three rational maps which can be conjugated to have a similar relationship between the fixed points and the fixed point multipliers.

math.DS↗

arXiv2019

Permutation polynomials: iteration of shift and inversion maps over finite fields

We show that all permutations in $S_n$ can be generated by affine unicritical polynomials. We use the $\operatorname{PGL}$ group structure to compute the cycle structure of permutations with low Carlitz rank. The tree structure of the group generated by shift and inversion maps is used to study the randomness properties of permutation polynomials.

math.NT↗

arXiv2013

Endomorphisms of Bounded Height and Resultant

Let K be an algebraic number field. For a degree d rational morphism of projective n-space defined over K let R denote its minimal resultant ideal. For a fixed height function on the moduli space of dynamical systems this paper shows that all such morphisms of bounded height and resultant are contained in finitely many PGL(K) equivalence classes. This answers a question of Silverman in the affirmative.

math.NT↗

arXiv2013

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.

math.DS↗

arXiv2013

Evidence for the Dynamical Brauer-Manin Criterion

Let f:X->X be a morphism of a variety over a number field K. We consider local conditions and a "Bruaer-Manin" condition, defined by Hsia and Silverman, for the orbit of a point P in X(K) to be disjoint from a subvariety V of X, i.e., the intersection of the orbit of P with V is empty. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection of the orbit of P with V; this evidence stems from a probabilistic argument as well as unconditional results in the case of etale maps.

math.NT↗

arXiv2012

Newton's Method Over Global Height Fields

Newton's method is used to approximate roots of complex valued functions f by creating a sequence of points that converges to a root of f in the usual topology. For any field K equipped with a set of pairwise inequivalent absolute values satisfying a product formula, we completely describe the conditions under which Newton's method applied to a squarefree polynomial f with K-coefficients will succeed in finding a root of f in the v-adic topology for infinitely many places v of K. Furthermore, we show that if K is a finite extension of the rationals or of the rational function field over a finite field, then the Newton approximation sequence fails to converge v-adically for a positive density of places v.

math.NT↗

arXiv2012

A Hasse Principle for Periodic Points

Let $F$ be a global field, let $\vp \in \Fx$ be a rational map of degree at least 2, and let $\a \in F$. We say that $\a $ is periodic if $\vpn (\a) = \a$ for some $n \geq 1$. A Hasse principle is the idea, or hope, that a phenomenon which happens everywhere locally should happen globally as well. The principle is well known to be true in some situations and false in others. We show that a Hasse principle holds for periodic points, and further show that it is sufficient to know that $\a$ is periodic on residue fields for every prime in a set of natural density density 1 to know that $\a$ is periodic in $F$.

math.NT↗

arXiv2011

Correlation Angles and Inner Products: Application to a Problem from Physics

Covariance is used as an inner product on a formal vector space built on n random variables to define measures of correlation Md across a set of vectors in a d-dimensional space. For d = 1, one has the diameter; for d = 2, one has an area. These concepts are directly applied to correlation studies in climate science.

stat.AP↗