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Register a new userThe Geometric Dynamical Northcott Property For Regular Polynomial Automorphisms of the Affine Plane
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We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $mathbf{K}$ of characteristic zero, improving results of Ingram. For that, we show that when $mathbf{K}$ is the field of rational functions of a smooth complex projective curve, the canonical height of a subvariety is the mass of an appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument.
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We establish the Geometric Dynamical Northcott Property for polarized endomorphisms of a projective normal variety over a function field $mathbf{K}$ of characteristic zero. This extends previous results of Benedetto, Baker and DeMarco in dimension $1$, and of Chatzidakis-Hrushovski in higher dimension. Our proof uses complex dynamics arguments and does not rely on the previous one. We first show that when $mathbf{K}$ is the field of rational functions of a smooth complex projective variety, the canonical height of a subvariety is the mass of the appropriate bifurcation current and that a marked point is stable if and only if its canonical height is zero. We then establish the Geometric Dynamical Northcott Property using a similarity argument. Moving from points to subvarieties, we propose, for polarized endomorphisms, a dynamical version of the Geometric Bogomolov Conjecture, recently proved by Cantat, Gao, Habegger and Xie. We establish several cases of this conjecture notably non-isotrivial polynomial skew-product with an isotrivial first coordinate.
We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an unlikely intersection statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical Andre-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
For every $minmathbb{N}$, we establish the equidistribution of the sequence of the averaged pull-backs of a Dirac measure at any given value in $mathbb{C}setminus{0}$ under the $m$-th order derivatives of the iterates of a polynomials $fin mathbb{C}[z]$ of degree $d>1$ towards the harmonic measure of the filled-in Julia set of $f$ with pole at $infty$. We also establish non-archimedean and arithmetic counterparts using the potential theory on the Berkovich projective line and the adelic equidistribution theory over a number field $k$ for a sequence of effective divisors on $mathbb{P}^1(overline{k})$ having small diagonals and small heights. We show a similar result on the equidistribution of the analytic sets where the derivative of each iterate of a Henon-type polynomial automorphism of $mathbb{C}^2$ has a given eigenvalue.
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In this paper we study principally polarized abelian varieties that admit an automorphism of prime order $p>2$. It turns out that certain natural conditions on the multiplicities of its action on the differentials of the first kind do guarantee that those polarized varieties are not jacobians of curves.
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