Linearization problems for circle diffeomorphisms and generalized interval exchange transformations

Abstract

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.