Tipus de document
Treball de fi de màsterData de publicació
Llicència de publicació
Si us plau utilitzeu sempre aquest identificador per citar o enllaçar aquest document: https://hdl.handle.net/2445/228324
Linearization problems for circle diffeomorphisms and generalized interval exchange transformations
Títol de la revista
Autors
Director/Tutor
ISSN de la revista
Títol del volum
Recurs relacionat
Resum
The goal of this thesis is to study the linearization problem for circle diffeomorphisms and their natural extensions, generalized interval exchange transformations (GIETs). Linearization questions for these systems concern the existence and regularity of conjugacies to their corresponding linear models, rigid rotations in the circle case and piecewise isometries in the GIET setting. This problem lies at the core of modern one-dimensional dynamics and remains an active area of research, with inffuential contributions made by A. Avila, V. Arnold, M. Herman, S. Marmi, P. Moussa, C. Ulcigrai, M. Viana, J.-C. Yoccoz, among many others. The thesis is developed along two complementary directions. First, we investigate obstructions to linearization. For circle diffeomorphisms, this is exempliffed by the construction of the classical Denjoy counterexample, which shows that topological conjugacy to a rotation may fail in low regularity. We generalize this counterexample to GIETs, which to the best of our knowledge is done explicitly for the first time in the literature for this setting. The second direction concerns positive linearization and rigidity results. Here the contrast between the two settings becomes apparent: for example, while suficient smoothness governs the existence of topological conjugacy for circle diffeomorphisms, the situation for GIETs differs and smoothness stops playing a role in the existence of a conjugacy to the linear model. We present a survey of recent developments and discuss the extent to which linearization results persist beyond the classical circle setting.
Descripció
Treballs finals del Màster en Matemàtica Avançada, Facultat de Matemàtiques, Universitat de Barcelona: Any: 2026. Director: Núria Fagella i Kostiantyn Drach
Matèries (anglès)
Citació
Col·leccions
Citació
BAUMEISTER, Maximilian. Linearization problems for circle diffeomorphisms and generalized interval exchange transformations. [consulta: 9 de maig de 2026]. Disponible a: https://hdl.handle.net/2445/228324