We prove that there do not exist positively expansive cellular automata defined on the full k-ary tree shift (for k ≥ 2). Moreover, we investigate some topological properties of these automata and their relationships, namely permutivity, surjectivity, preinjectivity, right-closingness and openness.

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 ultimate periodicity is undecidable. We believe that the proof technique used for this last result might turn out to be useful for many other results in the same context.

Contents 1. Introduction and notations 1 2. Remarks on the proofs of theorems in [12] 3 3. An isomorphism theorem for semi 1-coronae 5 3.1. Bump lemma: surjectivity of cohomology 6 3.2. Approximation 8 4. Extension of coherent sheaves and analytic subsets 13 References 15

‎In the present paper‎, ‎we are going to use geometric and topological concepts‎, ‎entities and properties of the‎ ‎integral curves of linear vector fields‎, ‎and the theory of differential equations‎, ‎to establish a representation for some groups on $R^{n} (ngeq 1)$‎. ‎Among other things‎, ‎we investigate the surjectivity and faithfulness of the representation‎. At the end‎, ‎we give some app...