Related papers: Dynamical Belyi maps and arboreal Galois groups

A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and…

We study the residual monodromy representations associated to the Dwork family in characteristic two. Various applications involving 2-adic and mod 2 Galois representations are discussed. Combining the author's previous work with Tsuzuki…

An operation on species corresponding to the inner plethysm of their associated cycle index series is constructed. This operation, the inner plethysm of species, is generalized to n-sorted species. Polynomial maps on species are studied and…

In this article we introduce the notion of projective isomonodromy, which is a special type of monodromy evolving deformation of linear differential equations, based on the example of the Darboux-Halphen equation. We give an algebraic…

Topological conjugateness of one dimensional unimodal dynamical systems, which are generated by interval [0, 1] into itself maps are studied. We study the smoothness and differentiability of the conjugacy of symmetrical and non-symmetrical…

We prove that the arboreal Galois representation attached to a large class of quadratic polynomials defined over a field of rational functions in characteristic zero has finite index in the full automorphism group of the associated preimage…

We survey some recent developments at the interface of algebraic geometry, surface topology, and the theory of ordinary differential equations. Motivated by "non-abelian" analogues of standard conjectures on the cohomology of algebraic…

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

We give a self-contained introduction to the theory of directed graphs, leading up to the relationship between the Perron-Frobenius eigenvectors of a graph and its autocatalytic sets. Then we discuss a particular dynamical system on a fixed…

We study the inverse problem in the difference Galois theory of linear differential equations over the difference-differential field $\mathbb{C}(x)$ with derivation $\frac{d}{dx}$ and endomorphism $f(x)\mapsto f(x+1)$. Our main result is…

We define vector fields, leaves and trajectories for schemes. With these tools, we are able to give a geometrical interpretation and to generalize several results of differential Galois theory and constructions on differential schemes. We…

A map on a surface whose automorphism group has a subgroup acting regularly on its vertices is called a Cayley map. Here we generalize that notion to maniplexes and polytopes. We define $\mathcal{M}$ to be a \emph{Cayley extension} of…

Counting non-isomorphic tree-like multigraphs that include self-loops and multiple edges is an important problem in combinatorial enumeration, with applications in chemical graph theory, polymer science, and network modeling. Traditional…

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals…

We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal…

Iterated monodromy groups of postcritically-finite rational maps form a rich class of self-similar groups with interesting properties. There are examples of such groups that have intermediate growth, as well as examples that have…

We use Galois closures of finite rational maps between complex projective varieties to introduce a new method for producing varieties such that the holomorphic part of the cup product map has non-trivial kernel. We then apply our result to…

Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincar\'e map on compact…

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat…