MASSON , Benoît

In this thesis we study several discrete dynamical systems which can simulate the formation of sandpiles. The behavior of the basic models SPM and IPM(k) is well-known under specific initial conditions. We extend these results to arbitrary initial conditions. Moreover, we introduce the model SSPM which adds symmetry to these models and improves...

We prove fixed points results for sandpiles starting with arbitrary initial conditions. We give an effective algorithm for computing such fixed points, and we refine it in the particular case of SPM.

We introduce a new model for simulating natural phenomena. We address several issues: topology, basic set properties like injectivity and surjectivity, reversibility, and decidability questions about a special kind of conservation law called grain conservation and ultimate periodicity.

A symmetric version of the well-known SPM model for sandpiles is introduced. We prove that the new model has fixed-point dynamics. Although there might be several fixed points, a precise description of the fixed points is given. Moreover, we provide a simple closed formula for counting the number of fixed points originated by initial conditions...

A symmetric version of the well-known SPM model for sandpiles is introduced. We prove that the new model has fixed point dynamics. Although there might be several fixed points, a precise description of the fixed points is given. Moreover, we provide a simple closed formula for counting the number of fixed points originated by initial conditions...

We prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of nilpotency is undecidable. We believe that the proof technique used for this last result might reveal useful for many other results in...