| Jérémy Blanc
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Moduli spaces of quadratic rational maps with a marked periodic point of small order
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It is conjectured that quadratic polynomial maps of the affine line, defined over the field of rational numbers, have no periodic point of order more than 3, the cases 4 and 5 being proved not to exist. In this talk, I will describe the analogue case of quadratic endomorphisms of the projective line and explain why the situation is really different, by describing the set of points of order 6.
This is a joint work with J.-K. Canci and N. Elkies.
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| Patrick Ingram
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Arithmetic dynamics of correspondences
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Arithmetic dynamics has, for the most part, been focussed on the iteration of endomorphisms of a variety. More generally, one might consider iteration of a correspondence from the variety to itself. We present a framework for studying the arithmetic dynamics of correspondences, and raise some questions about heights and the action of Galois in this setting.
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| Pär Kurlberg
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Arithmetic geometry applications of cycle lengths mod p
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We will investigate the relationship between periods of iterates of
rational maps (reduced modulo p) and two questions from arithmetic
dynamics, namely dynamical analogues of the Mordell–Lang conjecture (an
infinite intersection of an orbit with a subvariety implies strong
periodicity properties on the subvariety) and the Brauer-Manin problem
(an empty intersection of an orbit with a subvariety follows from the
adelic orbit closure having empty intersection with the subvariety.)
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| Alon Levy
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Several Nonarchimedean Variables, Isolated Periodic Points, and Zhang's Conjecture
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we develop a technique that generalizes hyperbolic dynamics in the nonarchimedean case. Specifically, if \(\phi: \mathbb{P}^n \to \mathbb{P}^n\), and \(x\) is a fixed point of \(\phi\) such that \(\phi\) acts on the tangent space at \(x\) with eigenvalues \(a_1, ..., a_n\), then we develop tools to prove that there exists a fixed analytic subvariety through \(x\) assuming a certain independence condition on the eigenvalues. This condition is satisfied for example when \(a_1, ..., a_r\) have absolute values less than 1 and \(a_{r+1}, ..., a_n\) absolute values 1 or more, and then there will be fixed analytic subvarieties one tangent to \(a_1 = ... = a_r = 0\), and one to \(a_{r+1} = ... = a_n = 0\). This has two applications: first, we prove that non-repelling periodic points are isolated in higher dimension; and second, we prove some cases of Zhang's conjecture that if \(\phi\) is defined over the algebraic numbers, there exists an algebraic point with Zariski-dense orbit.
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| Alina Ostafe
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Polynomial orbits in structural sets
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The underlying motive of the talk is showing various instances of the following principle:
Polynomials have no respect for Law and Order.
More precisely, given a polynomial \(f\) over a field \(K\) and a structural set \(S \subset K\) defined in terms unrelated to \(f\), it is natural to expect that the orbits of \(f\) have a finite intersection with \(S\). For example, if \(S\) is an orbit of another polynomial this is known as a problem about orbit intersections, which has recently been studied by Ghioca, Tucker and Zieve. One can also consider the multivariate generalisation of this question. For example, if the set \(S\) is an algebraic variety, this falls within the so-called dynamical Mordell-Lang conjecture.
We are interested in finiteness results or, failing this, in bounding the frequency of such intersections, both in the zero and positive characteristics, as well as for both univariate and multivariate cases. We shall also discuss several open questions in this direction.
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| Laura Paladino
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Preperiodic points of rational maps defined over a global field in terms of good reduction
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This is a joint work with Jung Kyu Canci
of the University of Basel. Let \(K\) be a global field of characteristic \(p\). Let \(D\) be the degree of \(K\) over the base field (i. e. \(\mathbb{Q}\) when \(p=0\) and \({\mathbb{F}}_p(t)\) when \(p>0\)). Let \(S\) be a set of places of \(K\) containing the archimedean ones.
We show a bound for the cardinality of the set of \(K\)--rational preperiodic points of an endomorphism of the projective line of degree \(d\) with good reduction outside \(S\), depending on \(D, d, p\) and \(|S|\). The result is completely new in the function fields case and it improves the ones known in the number fields case.
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| Tadeusz Pezda
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Periods for polynomials over rings of algebraic integers
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For a set \(X\) and a map \(\Phi:X\rightarrow X\) we define a cycle for \(\Phi\) as a tuple
\(x_0,x_1,...,x_{k-1}\) of distinct elements of \(X\) such that
\(\Phi(x_0)=x_1, \Phi(x_1)=x_2,...,\Phi(x_{k-2})=x_{k-1}, \Phi(x_{k-1})=x_0\).
The number \(k\) is called the length of this cycle.
\medskip
For a ring \(R\) let \({\cal CYCL}(R)={\cal CYCL}(R,1)\) be the set of all cycle-lengths for all polynomials \(\Phi\in R[X]\), lying in \(R\).
One may in a natural way extend this to polynomial mappings \(\Phi:R^N\rightarrow R^N\), and denote the corresponding set by \({\cal CYCL}(R,N)\).
\medskip
We will be interested in \({\cal CYCL}(R,N)\) for \(R=Z_K\), the ring of algebraic integers in a finite extension of the rationals \(K\), and for \(p\)-adic rings.
\medskip
For some \(K\) the sets \({\cal CYCL}(Z_K,1)\) were already established, as well as \({\cal CYCL}(Z_K,2)\) for all quadratic \(K\).
Finding \({\cal CYCL}(Z_K,N)\) is an awesome looking diophantine-like problem.
Nevertheless, I will outline a proof that \({\cal CYCL}(Z_K,N)\) may be effectively found
(at least from a theoretical point of view) for any number field \(K\) and any \(N\ge 1\). The cases \(N=1\) and \(N\ge 2\) require quite different methods.
\medskip
The sets \({\cal CYCL}(Z,N)\) were explicitly found for any \(N\), and we will say something about that.
\medskip
Let \(B(R,N)\) be the maximal element of \({\cal CYCL}(R,N)\), and let
\(B(n,N)=\max_{[K:Q]=n}B(Z_K,N)\). It is known that
\[\lim_{nN\rightarrow \infty, N\ge 2}\frac {\log B(n,N)}{nN}=\log 4.\]
\medskip
The presence of the condition \(N\ge 2\) is a bit annoying, but I will present an argument that
\(\lim_n\frac{\log B(n,1)}n=\log 4\) is highly unlikely.
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| Richard Pink
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Orbit length generating functions of automorphisms of a rooted regular binary tree
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To every automorphism \(w\) of an infinite rooted
regular binary tree we associate a two variable generating function
\(\Phi_w\) that encodes information on the orbit structure of \(w\). We prove that this is a rational function if \(w\) can be described by finitely many recursion relations of a particular form.
We show that this condition is satisfied for all elements of the
discrete iterated monodromy group \(\Gamma\) associated to a
postcritically finite quadratic polynomial over \(\mathbf{C}\). For
such \(\Gamma\) we also prove that there are only finitely many
possibilities for the denominator of \(\Phi_w\), and we describe a procedure to determine their lowest common denominator. |
|
Thomas J. Tucker |
Towards an finite index conjecture for iterated Galois groups
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Let \(f\) be a polynomial over a global field. Let \(G\) denote the
inverse limits of the Galois groups of \(f^n\), where \(f^n\) denotes the \(n\)-th
iterate of \(f\). Boston and Jones have suggested that under reasonable
hypotheses, one might hope that \(G\) has finite index in the full group
of automorphisms on an infinite tree corresponding to roots of
iterates \(f^n\). We will show that such a conjecture holds for cubic
polynomials in characteristic \(0\), assuming certain well-known
diophantine conjectures. We will also suggest a more general finite
index conjecture. |
|
Solomon Vishkautsan |
Scarcity of periodic points for rational functions over a number field
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The Uniform Boundedness Conjecture of Morton and Silverman (1994) states that the number of \(K\)-rational preperiodic points of an endomorphism of projective space defined over a number field \(K\) is bounded by a number depending only on the degree of the number field, the degree of the map, and the dimension of the projective space. This conjecture seems very far from resolution.
By adding the hypothesis that the map has good reduction outside of a finite set of places including all archimedean ones, one can prove uniform boundedness (albeit depending on the number of places of bad reduction). Building upon previous results by W. Narkiewicz, P. Morton, J.H. Silverman, J.K. Canci, Laura Paladino and R.L. Benedetto among others, we provide a bound for the number of \(K\)-rational periodic points of endomorphisms of the projective line that is linear in the degree of the map.
This is a joint work with J.K. Canci.
|
|
Junyi Xie |
Algebraic dynamics of the lifts of Frobenius
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In this talk we study some questions in the algebraic dynamics for endomorphisms of projective spaces with coefficients in a \(p\)-adic field whose reduction in positive characteritic is the Frobenius. Our method is based on the theory of perfectoid spaces introduced by P. Scholze. |
