Thierry Bousch

Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture

Given an Iterated Function System (IFS) of topical maps verifying some conditions, we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional, the average height, on some compact space of measures. We give general results to determine these extrema, and then apply them to two concrete problems. First, we give a new proof of the theorem that the densest heaps of two Tetris pieces are sturmian. Second, we construct an explicit counterexample to the Lagarias-Wang finiteness conjecture.

La condition de Walters

Abstract We discuss a regularity condition introduced by Walters in 1978 for the needs of the thermodynamic formalism. We show that it is the “right” regularity condition to require in a large class of problems, including the thermodynamic formalism as well as the study of maximising measures, and the search of normal forms modulo coboundaries of continuous functions.

Finite-range topical functions and uniformly topical functions

We introduce a remarkable subclass of the class of topical functions, the class of uniformly topical functions, whose dynamical behaviour is investigated. Every uniformly topical endofunction has a spectral vector, related to some special fixed points (possibly at infinity), about which we establish various properties. In the stochastic case, we prove a multiplicative ergodic theorem, asserting that the stochastic spectral vector exists in all cases.

Une propriete de domination convexe pour les orbites sturmiennes

Let x = (x0, x1, . . .) be a N-periodic sequence of integers (N > 1), and s a sturmian sequence with the same barycenter (and also N-periodic, consequently). It is shown that, for affine functionsα : R NN) → R which are increasing relatively to some order 6 2 on R NN) (the space of all N-periodic sequences), the average of |α| on the orbit of x is greater than its average on the orbit of s.