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# Autosimilarite dans les Systemes Isometriques par Morceaux

Abstract : In this thesis, we introduce a piecewise isometric system generalising a known example. It is shown to display an infinite number of periodic points following a selfsimilar structure. The first-return map into one of its atom is a piecewise isometry defined over a selfsimilar partition with an infinite number of atoms having exponentially increasing return-times. The displayed self-similarity describes only a part of the dynamics, but it is stable under continuous variations of the main parameter. This property enables us to identify the geometric and algebraic conditions involved in the birth of a self-similar scheme. We then give general assumptions which imply not only the existence of families of periodic cells whose codes follow a substitutive scheme, but the existence of a non-empty set of aperiodic points. This set is fractal, it can be described by a graph-directed construction and its Hausdorff dimension can be computed explicitly. Moreover, we show that the geometric structure leads naturally to measure-theoretic conjugate the dynamics with a Vershik map over a stationary Bratteli diagram, which is uniquely ergodic under natural primitivity conditions. This coding can be translated'' into the regular coding of the piecewise isometry, leading to a substitution dynamical system.\\ The given framework is general enough to handle many of the piecewise isometries studied up to now. We use it to show that the system mentioned above has an invariant measure with an infinite number of ergodic components.
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https://tel.archives-ouvertes.fr/tel-00005473
Contributor : Guillaume Poggiaspalla <>
Submitted on : Friday, March 26, 2004 - 7:09:07 PM
Last modification on : Thursday, September 13, 2018 - 12:08:03 PM
Long-term archiving on: : Wednesday, September 12, 2012 - 2:20:24 PM

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• HAL Id : tel-00005473, version 1

### Citation

Guillaume Poggiaspalla. Autosimilarite dans les Systemes Isometriques par Morceaux. Mathématiques [math]. Université de la Méditerranée - Aix-Marseille II, 2003. Français. ⟨tel-00005473⟩

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