Let X be a unary relation on the domain of (ω,<). The degree spectrum of X on (ω,<) is the set of Turing degrees of the image of X in all computable presentations of (ω,<). Many results are known about the types of degree spectra that are possible for relations forming infinite and coinfinite c.e. sets, high c.e. sets and non-high c.e. sets on the standard copy. We show that if the degree spectrum of X contains the computable degree then its degree spectrum is precisely the set of Δ_2 degrees. The structure ζ can be viewed as a copy of ω* followed by a copy of ω and, for this reason, the degree spectrum of X on ζ can be largely understood from the work on ω. A helpful correspondence between the degree spectra on ω and ζ is presented and the known results for degree spectra on the former structure are extended to analogous results for the latter.

Let X be a unary relation on the domain of (ω,<). The degree spectrum of X on (ω,<) is the set of Turing degrees of the image of X in all computable presentations of (ω,<). Many results are known about the types of degree spectra that are possible for relations forming infinite and coinfinite c.e. sets, high c.e. sets and non-high c.e. sets on the standard copy. We show that if the degree spectrum of X contains the computable degree then its degree spectrum is precisely the set of Δ_2 degrees. The structure ζ can be viewed as a copy of ω* followed by a copy of ω and, for this reason, the degree spectrum of X on ζ can be largely understood from the work on ω. A helpful correspondence between the degree spectra on ω and ζ is presented and the known results for degree spectra on the former structure are extended to analogous results for the latter.

Cette thèse étudie les structures dont la théorie au second ordremonadique est décidable, et en particulier la hiérarchie à pile. Onpeut définir celle-ci comme la hiérarchie pour $n$ des graphesd'automates à piles imbriquées $n$ fois ; une définition externe, partransformations de graphes, est également disponible. Nous nousintéressons à l'exemple des ordinaux. Nous montrons que les ordinauxplus petits que $epsilon_0$ sont dans la hiérarchie, ainsi que des graphesporteurs de plus d'information, que l'on appelle "graphecouvrants''. Nous montrons ensuite l'inverse : tous les ordinaux de lahiérarchie sont plus petits que $epsilon_0$. Ce résultat utilise le fait queles ordres d'un niveau sont en fait isomorphes aux structures desfeuilles des arbres déterministes dans l'ordre lexicographique, aumême niveau. Plus généralement, nous obtenons une caractérisation desordres linéaires dispersés dans la hiérarchie. Dans un troisièmetemps, nous resserons l'intérêt aux ordres de type $omega$ --- les mots infinis --- pour montrer que les mots du niveau 2 sont les motsmorphiques, ce qui nous amène à une nouvelle extension au niveau 3 / This thesis studies the structures with decidable monadic second-ordertheory, and in particular the pushdown hierarchy. The latter can bedefined as the family for $n$ of pushdown graphs with $n$ timesimbricated stacks ; another definition is by graph transformations. Westudy the example of ordinals. We show that ordinals smaller that $epsilon_0$are in the hierarchy, along with graphs called "covering graphs'', which carry more data than ordinals. We show then the converse : allordinals of the hierarchy are smaller than $epsilon_0$. This result uses thefact that linear orders of a level are actually isomorphic to thestructure of leaves of deterministic trees by lexicographic ordering, at the same level. More generally, we obtain a characterisation ofscattered linear orders in the hierarchy. We finally focus on the caseof orders of type $omega$ --- infinite words --- and show that morphicwords are exactly words of the second level of the hierarchy. Thisleads us to a new definition of words for level 3

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