Groups of Circle Diffeomorphisms

Abstract

The original version (in Spanish) of this text was prepared for a mini-course in Antofagasta, Chile. Subsequently, enlarged and revised versions were published in the series Monografías del IMCA (Perú) and Ensaios Matemáticos (Brasil). This translation arose from the necessity of making this text accessible to a larger audience. I would like to thank Juan Rivera-Letelier for his invitation to the II Workshop on Dynamical Systems ( 2001), for which the original version was prepared, and Roger Metzger for his invitation to IMCA (2006), where part of this material was presented. I would also like to thank both Étienne Ghys and Maria Eulália Vares for motivating me and allowing me to publish the revised Spanish version in Brasil, as well as all the people who strongly encouraged me to conclude this English version.This work was partially supported by CONICYT and PBCT via the Research Network on Low Dimensional Dynamics. I would also to acknowledge the support of both the UMPA Department of the École Normale Supérieure de Lyon, where the idea of writing this text was born during my PhD thesis, and the Institut des Hautes Études Scientifiques, where the original notes started taking its definite form while I benefited from a one-year postdoctoral position.This work owes a lot to many of my colleges. Several remarks spread throughout the text and the content of some examples and exercises were born during fruitful and quite stimulating discussions. It is then a pleasure to thank Sylvain Crovisier (Proposition 4.2.25),

On spaces of Conradian group orderings

“…(Here j à ðLebÞ is de-fined by j à ðLebÞðAÞ ¼ LebðjðAÞÞ.) Since Leb is a Radon measure without atoms, this is also the case for n. Finally, the uniqueness of n up to a scalar multiple is an easy exercise (see for instance [10 ) Thus b à ðnÞ is a measure that is invariant under 5a6. The uniqueness of the 5a6-invariant measure up to scalar factor yields b à ðnÞ ¼ ln for some l > 0.…”

The Poisson boundary of a locally discrete group of diffeomorphisms of the circle

“…C 3 ), then the germ of diffeomorphism which conjugates l to a linear map is also of class C 2 (resp. C 3 ), see [23] or [20,Theorem 3.6.2]. Thus, we can suppose that x = 0 and l(y) = αy for y ∈ I, the action being C 2 (resp.…”

“…If the group G does not preserve a probability measure on the circle, then there exists a point x in the limit set and an element l ∈ G such that l(x) = x and α = l ′ (x) < 1, see [6, Théorème F] or [20]. By the C 1+τ -version of Sternberg linearization theorem [4], there is a germ of diffeomorphism ϕ : (S 1 , x) → (R, 0) of class C 1+τ such that ϕ • l = l ′ (x)ϕ.…”

On the chaos game of iterated function systems

“…a) Forward but not backward minimal IFSs on the circle: Consider a group G of homeomorphisms of the circle. Then, there can occur only one of the following three options [32,20]:…”