## Computing the entropy of multidimensional subshifts of finite type

ABSTRACT : Multidimensional subshifts of finite type are discrete dynamical systems as a set of colorings of an infinite regular grid with elements of a finite set A together with the shift action. The set of colorings is defined by forbidding a finite set of patterns all over the grid (also called local rules). The most simple and most considered grids of this type are Z2 and more generally Zd for d = 1. In this case, one can consider a coloring as a bi-dimensional and infinite word on the alphabet A. They are notably involved in statistical physics in the study of so-called lattice models....

Read More## Restrictions on the group of automorphisms preserving a minimal subshift

ABSTRACT : A subshift is a closed shift invariant set of sequences over a finite alphabet. An automorphism is an homeomorphism of the space commuting with the shift map. The set of automorphisms is a countable group generally hard to describe. We will present in this talk a survey of various restrictions on these groups for zero entropy minimal subshifts.

Read More## Decidability of the isomorphism and the factorization for minimal substitution subshifts

ABSTRACT Classification is a central problem in the study of dynamical systems, in particular for families of systems that arise in a wide range of topics. Hence it is important to have algorithms deciding wether a dynamical system have some given property. Let us mention subshifts of finite type that appear, for example, in information theory, hyperbolic dynamics, $C^*$-algebra, statistical mechanics and thermodynamic formalism. The most important and longstanding open problem for this family originates in [Williams:1973] and is stated in [Boyle:2008] as follows : Classify subshifts of...

Read More## On the action of the semigroup of non singular integral matrices on $\R^n$

Abstract. Let Γ be the multiplicative semigroup of all n × n matrices with integral entries and nonzero determinant. Let 1 ≤ p ≤ n−1 and V = Rnp = Rn ⊕···⊕Rn (p copies). Consider the action of Γ on V , given by the natural action on each component, by matrix multiplication on the left. Then for x= (x1, . . . , xp) ∈ V , the Γ-orbit is dense in V if and only if there is no linear combination pj=1 λjxj, with λj ̸= 0 for some j, which is a rational vector in Rn; in fact the assertion holds also for the orbit of the subgroup SL(n, Z) that is contained in Γ. When x is such that the...

Read More## Exponentes de Lyapunov y rigidez para difeomorfismos hiperbólicos y parcialmente hiperbólicos

ABSTRACT Voy a presentar unos resultados de rigidez en termino de exponentes de Lyapunov para difeomorfismos hiperbólicos y parcialmente hiperbólicos. Si un difeomorfismo hiperbólico (o parcialmente hiperbólico) es cerca a un automorfismo lineal (o un skew product sobre un automorfismo lineal), preserva el volumen, y tiene los mismos exponentes de Lyapunov (estables e inestable), entonces es suavemente conjugado al automorfismo lineal (o a un skew product sobre el automorfismo lineal). En el caso de difeomorfismos hiperbólicos el resultado puede ser visto como un análogo a la conjetura de...

Read More## Almost everywhere convergence of ergodic averages

ABSTRACT: In this talk I would like to discuss some of my results concerning almost everywhere convergence of non-conventional ergodic averages of L1 functions. These topics include: divergence of ergodic averages along the squares; convergence along some sequences of zero Banach density; convergence for arithmetic weights: the prime divisor functions ω and Ω.

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