Groups of Circle Diffeomorphisms (original) (raw)

Dynamical Systems on the Circle

2019

In this paper we introduce dynamical systems on the circle. Beginning with elementary notions of dynamical systems, we develop several tools to study these systems, leading to the rotation number, a key invariant of circle homeomorphisms. We then show that any non-periodic circle homeomorphism is semiconjugate to the rotation by its rotation number. Finally, we present a proof of Denjoy’s Theorem, which states that any non-periodic circle diffeomorphism with derivative of bounded variation is conjugate to the rotation by its rotation number.

Dynamics of multicritical circle maps

São Paulo Journal of Mathematical Sciences

This paper presents a survey of recent and not so recent results concerning the study of smooth homeomorphisms of the circle with a finite number of non-flat critical points, an important topic in the area of One-dimensional Dynamics. We focus on the analysis of the fine geometric structure of orbits of such dynamical systems, as well as on certain ergodic-theoretic and complex-analytic aspects of the subject. Finally, we review some conjectures and open questions in this field.

Structurally unstable regular dynamics in 1D piecewise smooth maps, and circle maps

Chaos, Solitons & Fractals, 2012

In this work we consider a simple system of piecewise linear discontinuous 1D map with two discontinuity points: X 0 = aX if jXj < z, X 0 = bX if jXj > z, where a and b can take any real value, and may have several applications. We show that its dynamic behaviors are those of a linear rotation: either periodic or quasiperiodic, and always structurally unstable. A generalization to piecewise monotone functions X 0 = F(X) if jXj < z, X 0 = G(X) if jXj > z is also given, proving the conditions leading to a homeomorphism of the circle.

Quasisymmetric orbit-flexibility of multicritical circle maps

Ergodic Theory and Dynamical Systems, 2021

Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in (0,1)(0,1)(0,1) , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this ...

Anosov and Circle Diffeomorphisms

Springer Proceedings in Mathematics, 2011

We present an infinite dimensional space of C 1+ smooth conjugacy classes of circle diffeomorphisms that are C 1+ fixed points of renormalization. We exhibit a one-to-one correspondence between these C 1+ fixed points of renormalization and C 1+ conjugacy classes of Anosov diffeomorphisms.

On dynamics and bifurcations of area-preserving maps with homoclinic tangencies

Nonlinearity, 2015

We study bifurcations of area-preserving maps, both orientable (symplectic) and nonorientable, with quadratic homoclinic tangencies. We consider one and two parameter general unfoldings and establish results related to the appearance of elliptic periodic orbits. In particular, we find conditions for such maps to have infinitely many generic (KAM-stable) elliptic periodic orbits of all successive periods starting at some number.

A group of diffeomorphisms of the interval with intermediate growth

2005

A theory for groups of diffeomorphisms of the interval has been extensively developed by many authors (see for example [16, 18, 19, 24, 26, 28, 29, 30, 33]). One of the most interesting topics of this theory is the interplay between the differentiability class of the diffeomorphisms and the algebraic (as well as dynamical) properties of the group (and the action). For instance, as a consequence of the classical Bounded Distorsion Principle, groups of C2-diffeomorphisms appear to have a very rigid behavior. This is no longer true for subgroups of Diff+([0, 1]), as it is well illustrated in the literature [4, 22, 31]. The aim of this work is to study this phenomenon for a remarkable class of groups, first introduced by R. Grigorchuk.