We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes V for the items, and the allowed matchings depend on the classes, according to a matching graph on V. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is i.i.d., the sequence of buffer-contents is a Markov chain, whose stability is investigated.In particular, we prove that the model may be stable if and only if the matching graph is non-bipartite.

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 for the joint spectral radius of a set of matrices. Résumé. Etant donné un système itéré de fonctions (IFS) topicales, vérifiant certaines conditions, nous montrons que les questions d’optimisation asymptotique de la hauteur sont équivalentes à la recherche des extrema d’une fonctionnelle continue, la hauteur moyenne, sur un certain espace compact de mesures. Nous présentons des résultats généraux permettant de déterminer ces extrema, puis appliquons ces méthodes à deux problèmes concrets. Premièrement, nous redémontrons que les empilements les plus denses de deux pièces de Tetris sont sturmiens. Deuxièmement, nous construisons un contre-exemple effectif à la conjecture de finitude de Lagarias et Wang sur le rayon spectral joint d’un ensemble de matrices.

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