Homomorphisms between multidimensional constant-shape substitutions

Substitutions on Multidimensional Sequen

Consistency of Multidimensional Combinatorial Substitutions

Multidimensional combinatorial substitutions are rules that replace symbols by finite patterns of symbols in Zd. We focus on the case where the patterns are not necessarily rectangular, which requires a specific description of the way they are glued together in the image by a substitution. Two problems can arise when defining a substitution in such a way: it can fail to be consistent, and the p...

Multidimensional Euclidean algorithms, numeration and substitutions

The aim of this survey is to discuss multidimensional continued fraction and Euclidean algorithms from the viewpoint of numeration systems, substitutions, and the symbolic dynamical systems they generate. We will mainly focus on two types of multidimensional algorithms, namely, unimodular Markovian ones which include the most classical ones like e.g. Jacobi-Perron algorithm, and algorithms issu...

Homomorphisms between Diffeomorphism Groups

For r ≥ 3, p ≥ 2, we classify all actions of the groups Diffc(R) and Diff+(S) by Cdiffeomorphisms on the line and on the circle. This is the same as describing all nontrivial group homomorphisms between groups of compactly supported diffeomorphisms on 1-manifolds. We show that all such actions have an elementary form, which we call topologically diagonal. As an application, we answer a question...

Graph Homomorphisms between Trees

In this paper we study several problems concerning the number of homomorphisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization and a dual of Bollobás and T...