Search ResultsShowing 1-20 of 34
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arXiv:2210.14733 (Published 2022-10-26)
An asymptotic for sums of Lyapunov exponents in families
Let f_t be a meromorphic family of endomorphisms of P^N_C of degree at least 2, and let L(f_t) be the sum of Lyapunov exponents associated to f_t. Favre showed that L(f_t)=L(f)\log|t^{-1}|+o(\log|t^{-1}|) as t -> 0, where L(f) is the sum of Lyapunov exponents on the generic fibre, interpreted as an endomorphism of some projective Berkovich space. Under some additional constraints on the family, we provide an explicit error term.
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arXiv:2210.14676 (Published 2022-10-26)
A remark on post-critically finite compositions of polynomials
Categories: math.NTThe second author proved that the set of post-critically finite polynomials of given degree is a set of bounded height, up to change of variables. Motivated by an observation about unicritical polynomials, we complement this by proving that the set of monic polynomials g(z) of given degree with the property that there exists a d > 1 such that g(z^d) is post-critically finite, is also a set of bounded height. Moreover, we establish a lower bound on the critical height of g(z^d).
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arXiv:2207.07206 (Published 2022-07-14)
Explicit canonical heights for divisors relative to endomorphisms of $\mathbb{P}^N$
Categories: math.NTGiven an endomorphism f of projective space, we exhibit explicit bounds on the difference between the naive height of a divisor and its canonical height relative to f.
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arXiv:2104.12877 (Published 2021-04-26)
Variation of the canonical height in a family of polarized dynamical systems
Call and Silverman introduced the canonical height associated to a polarized dynamical system, that is, an endomorphism of a projective variety and an ample line bundle which pulls back to a tensor power of itself. They also presented an asymptotic for the variation of this height in a family over a one-dimensional base in terms of the height on the generic fibre and the height of the parameter. Here we improve this asymptotic, saving a power in the error term. As a corollary, we give an explicit bound on the height of parameters at which the dynamical system specialized to a finite orbit, in the case of endomorphisms of projective space over the projective line.
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arXiv:2011.02975 (Published 2020-11-05)
Solutions to difference equations have few defects
We demonstrate a strong form of Nevanlinna's Second Main Theorem for solutions to difference equations f(z+1)=R(z, f(z)), with the coefficients of R growing slowly relative to f, and R of degree at least 2 in the second coordinate.
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arXiv:2011.02968 (Published 2020-11-05)
Effective finiteness of solutions to certain differential and difference equations
For R(z, w) rational with complex coefficients, of degree at least 2 in w, we show that the number of rational functions f(z) solving the difference equation f(z+1)=R(z, f(z)) is finite and bounded just in terms of the degrees of R in the two variables. This complements a result of Yanagihara, who showed that any finite-order meromorphic solution to this sort of difference equation must be a rational function. We prove a similar result for the differential equation f'(z)=R(z, f(z)), building on a result of Eremenko.
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arXiv:2007.00567 (Published 2020-07-01)
Degree gaps for multipliers and the dynamical Andre-Oort conjecture
We demonstrate how recent work of Favre and Gauthier, together with a modification of a result of the author, shows that a family of polynomials with infinitely many post-critically finite specializations cannot have any periodic cycles with multiplier of very low degree, except those which vanish, generalizing results of Baker and DeMarco, Favre and Gauthier, and Ghioca and Ye.
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arXiv:2006.15365 (Published 2020-06-27)
Minimally critical regular endomorphisms of A^N
We study the dynamics of a class of endomorphisms of A^N which restricts, when N = 1, to the class of unicritical polynomials. Over the complex numbers, we obtain lower bounds on the sum of Lyapunov exponents, and a statement which generalizes the compactness of the Mandelbrot set. Over the algebraic numbers, we obtain estimates on the critical height, and over general algebraically closed fields we obtain some rigidity results for post-critically finite morphisms of this form.
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arXiv:2006.12869 (Published 2020-06-23)
Minimally critical endomorphisms of P^N
We study the dynamics of the map endomorphism of N-dimensional projective space defined by f(X)=AX^d, where A is a matrix and d is at least 2. When d>N^2+N+1, we show that the critical height of such a morphism is comparable to its height in moduli space, confirming a case of a natural generalization of a conjecture of Silverman.
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arXiv:1806.04980 (Published 2018-06-13)
Current Trends and Open Problems in Arithmetic Dynamics
Robert Benedetto, Laura DeMarco, Patrick Ingram, Rafe Jones, Michelle Manes, Joseph H. Silverman, Thomas J. TuckerComments: 67 pages, survey article, comments welcomeArithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
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arXiv:1706.05352 (Published 2017-06-16)
Critical orbits of polynomials with a periodic point of specified multiplier
Categories: math.NTAnswering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space. We also apply the methods over function fields to draw conclusions about algebraically parametrized families, and prove an analogous result for quadratic rational maps.
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arXiv:1610.07904 (Published 2016-10-25)
The critical height is a moduli height
The critical height of a rational function (with algebraic coefficients) is a natural measure of dynamical complexity, essentially an adelic analogue of the Lyapunov exponent. Coordinate-free, it is well-defined on moduli space, but bears no obvious relation to the arithmetic geometry of that space as a variety. At a conference in 2010, Silverman conjectured that this disconnect is superficial, and that in fact the critical height should be at least commensurate to any ample Weil height on the moduli space, except on the Lattes locus (where that has no hope of being true). Here, we prove that conjecture.
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arXiv:1604.05197 (Published 2016-04-18)
p-adic uniformization and the action of Galois on certain affine correspondences
Categories: math.NTGiven two monic polynomials f and g with coefficients in a number field K, and some a in K, we examine the action of the absolute Galois group of K on the directed graph of iterated preimages of a under the correspondence g(y)=f(x), assuming that deg(f)>deg(g) and that gcd(deg(f), deg(g))=1. If a prime of K exists at which f and g have integral coefficients, and at which a is not integral, we show that this directed graph of preimages consists of finitely many Galois-orbits. We obtain this result by establishing a p-adic uniformization of such correspondences, tenuously related to Bottcher's uniformization of polynomial dynamical systems over the complex numbers.
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arXiv:1511.00194 (Published 2015-11-01)
Finite ramification for preimage fields of postcritically finite morphisms
Andrew Bridy et al.Categories: math.NTGiven a finite endomorphism $\varphi$ of a variety $X$ defined over the field of fractions $K$ of a Dedekind domain, we study the extension $K(\varphi^{-\infty}(\alpha)) : = \bigcup_{n \geq 1} K(\varphi^{-n}(\alpha))$ generated by the preimages of $\alpha$ under all iterates of $\varphi$. In particular when $\varphi$ is post-critically finite, i.e., there exists a non-empty, Zariski-open $W \subseteq X$ such that $\varphi^{-1}(W) \subseteq W$ and $\varphi : W \to X$ is \'etale, we prove that $K(\varphi^{-\infty}(\alpha))$ is ramified over only finitely many primes of $K$. This provides a large supply of infinite extensions with restricted ramification, and generalizes results of Aitken-Hajir-Maire in the case $X = \mathbb{A}^1$ and Cullinan-Hajir, Jones-Manes in the case $X = \mathbb{P}^1$. Moreover, we conjecture that this finite ramification condition characterizes post-critically finite morphisms, and we give an entirely new result showing this for $X = \mathbb{P}^1$. The proof relies on Faltings' theorem and a local argument.
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arXiv:1510.08807 (Published 2015-10-29)
Canonical heights and preperiodic points for subhomogeneous families of polynomials
A family $f_t(z)$ of polynomials over a number field $K$ will be called \emph{subhomogeneous} if and only if $f_t(z)=F(z^e, t)$ for some binary homogeneous form $F(X, Y)$ and some integer $e\geq 2$. For example, the family $z^d+t$ is subhomogeneous. We prove a lower bound on the canonical height, of the form \[\hat{h}_{f_t}(z)\geq \epsilon \max\{h_{\mathsf{M}_d}(f_t), \log|\operatorname{Norm}\mathfrak{R}_{f_t}|\},\] for values $z\in K$ which are not preperiodic for $f_t$. Here $\epsilon$ depends only on the number of places at which $f_t$ has bad reduction. For suitably generic morphisms $\varphi:\mathbb{P}^1\to \mathbb{P}^1$, we also prove an absolute bound of this form for $t$ in the image of $\varphi$ over $K$ (assuming the $abc$ Conjecture), as well as uniform bounds on the number of preperiodic points (unconditionally).
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arXiv:1411.1041 (Published 2014-11-04)
Canonical heights for correspondences
Categories: math.NTThe canonical height associated to a polarized endomporhism of a projective variety, constructed by Call and Silverman and generalizing the N\'eron-Tate height on a polarized Abelian variety, plays an important role in the arithmetic theory of dynamical systems. We generalize this construction to polarized correspondences, prove various fundamental properties, and show how the global canonical height decomposes as an integral of a local height over the space of absolute values on the algebraic closure of the field of definition.
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arXiv:1408.5416 (Published 2014-08-22)
Variation of the canonical height for polynomials in several variables
Let K be a number field, X/K a curve, and f/X a family of endomorphisms of projective N-space. It follows from a result of Call and Silverman that the canonical height associated to the family f, evaluated along a section, differs from a Weil height on the base by little-o of a Weil height. In the case where f is a family with an invariant hyperplane, whose restriction to this invariant hyperplane is isotrivial, we improve this by showing that the canonical height along a section differs from a Weil height on the base by a bounded amount.
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arXiv:1310.4114 (Published 2013-10-15)
Rigidity and height bounds for certain post-critically finite endomorphisms of projective space
The holomorphic endomorphism f of projective space is called post-critically finite (PCF) if the forward image of the critical locus, under iteration of f, has algebraic support (i.e., is a finite union of hypersurfaces). In the case of dimension 1, a deep result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattes maps. This note proves a corresponding result in arbitrary dimension, for a certain subclass of morphism. Specifically, we restrict attention to morphisms f of degree at least two, with a totally invariant hyperplane H, such that the restriction of f to H is the dth power map in some coordinates. This condition defines a subvariety of the space of coordinate-free endomorphisms of projective space (of a given dimension). We prove that there are no families of PCF maps in this space, and derive several related arithmetic results.
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The filled Julia set of a Drinfeld module and uniform bounds for torsion
Comments: Minor changesCategories: math.NTIf M is a Drinfeld module over a local function field L, we may view M as a dynamical system, and consider its filled Julia set J. If J^0 is the connected component of the identity, relative to the Berkovich topology, we give a characterisation of the component module J/J^0 which is analogous to the Kodaira-Neron characterisation of the special fibre of a Neron model of an elliptic curve over a non-archimedean field. In particular, if L is the fraction field of a discrete valuation ring, then the component module is finite, and moreover trivial in the case of good reduction. In the context of global function fields, the filled Julia set may be considered as an object over the ring of finite adeles. In this setting we formulate a conjecture about the structure of the (finite) component module which, if true, would imply Poonen's Uniform Boundedness Conjecture for torsion on Drinfeld modules of a given rank over a given global function field. Finally, we prove this conjecture for certain families of Drinfeld modules, obtaining uniform bounds on torsion in some special cases.
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A lower bound for the canonical height associated to a Drinfeld module
Comments: The lower bound in the main result has been significantly improved, and is now essentially sharp for Drinfeld modules with potentially good reduction at all but a bounded number of placesCategories: math.NTDenis associated to each Drinfeld module M over a global function function field L a canonical height function, which plays a role analogous to that of the Neron-Tate height in the context of elliptic curves. We prove that there exist constants \epsilon>0 and C, depending only on the number of places at which M has bad reduction, such that either x in M is a torsion point of bounded order, or else the canonical height of x is bound below by \epsilon max{h(j_M), deg(D_M)}, where j_M is a certain invariant of the isomorphism class of M, and D_M is the minimal discriminant of M. As an application, we make some observations about specializations of one-parameter families of Drinfeld modules.