9 papers
Plane Cremona maps preserving a Euclidean volume form
Marc Abboud, Jérémy Blanc, Stéphane Lamy
We give a presentation by generators and relations of the group of birational transformations of the plane preserving a differential form with poles along a triple line. This group…
Dynamical degrees of twisted rational maps
Marc Abboud, Junyi Xie
Twisted rational maps arise naturally in relative algebraic dynamics: if a rational self-map preserves a fibration, then the induced map on the generic fiber is usually not an ordi…
Uniform bound on common periodic points for families of regular plane polynomial automorphisms
Marc Abboud, Yugang Zhang
Given two one-dimensional families and of regular plane polynomial automorphisms parameterised by an algebraic curve , all defined over some number field , such that…
A dynamical characterisation of smooth cubic affine surfaces of Markov type
Marc Abboud
Using valuative techniques, we show that a smooth affine surface with a non-elementary automorphism group and completable by a cycle of rational curves is either the algebraic toru…
A local version of the arithmetic Hodge index theorem over quasiprojective varieties
Marc Abboud
We define a local intersection number for metrised line bundles over quasiprojective varieties with compact support and show the local arithmetic Hodge index theorem for this inter…
Intersection of orbits of loxodromic automorphisms of affine surfaces
Marc Abboud
We show the following result: If is an affine surface over a field and are two loxodromic automorphisms with an orbit meeting infinitely many times, then and $…
Rigidity of periodic points for loxodromic automorphisms of affine surfaces
Marc Abboud
We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same peri…
Unlikely intersections problem for automorphisms of Markov surfaces
Marc Abboud
We study the problem of unlikely intersections for automorphisms of Markov surfaces of positive entropy. We show for certain parameters that two automorphisms with positive entropy…
On the dynamics of endomorphisms of affine surfaces
Marc Abboud
In [FJ07], Favre and Jonsson developed tools from valuative theory to study the dynamics of a dominant endomorphism of the complex affine plane. We extend this theory to the case o…