Bernard Host

Substitutional dynamical systems, Bratteli diagrams and dimension groups

The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit, and render an effective method to compute an invariant of (ordered)

Convergence of polynomial ergodic averages

K

Mixing of all orders and pairwise independent joinings of systems with singular spectrum

-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

Asymptotic pairs in positive-entropy systems

We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials.

Nombres normaux, entropie, translations

We prove that every pairwise independent joining of weakly mixing systems with purely singular spectrum is independent; it follows that every mixing system with purely singular spectrum is mixing of all orders.

Convergence of Conze–Lesigne averages

We show that in a topological dynamical system

Multiple recurrence and convergence for sequences related to the prime numbers

(X,T)

Continuous and Measurable Eigenfunctions of Linearly Recurrent Dynamical Cantor Systems

of positive entropy there exist proper (positively) asymptotic pairs, that is, pairs

Ergodic averages of commuting transformations with distinct degree polynomial iterates

(x,y)

Nil-Bohr sets of integers

such that