We investigate the curves in the complex plane which are generated by sequences of real numbers being the lifts of the points on the orbit of an orientation preserving circle homeomorphism. Geometrical properties of these curves such as boundedness, superficiality, local discrete radius of curvature are linked with dynamical properties of the circle homeomorphism which generates them: rotation number and its continued fraction expansion, existence of a continuous solution of the corresponding cohomological equation and displacement sequence along the orbit.
Let $G$ be the group ${\rm PAff}_+({\bf S}^1)$ of piecewise--affine circle homeomorphisms or the group ${\Diff}^{\infty}(\mathbb R/\mathbb Z)$ of smooth circle diffeomorphisms. A constructive proof that all irrational rotations are distorted in $G$ is given.
The rigidity theory for circle homeomophisms with breaks was studied intensively in the last 20 years. It was proved that under mild conditions of the Diophantine type on the rotation number any two $C^{2+α}$ smooth circle homeomorphisms with a break point are $C^1$ smoothly conjugate to each other, provided that they have the same rotation number and the same size of the break. In this paper we prove that the conjugacy may not be $C^{2-ν}$ even if the maps are analytic outside of the break points. This result shows that the rigidity theory for maps with singularities is very different from th…
In this work we study the following realization problem: given a piecewise homeomorphism $Φ: S^1 \rightarrow S^1$, which geometrical and dynamical conditions on $Φ$ are sufficient to realize it as a Bowen-Series-like map associated to a surface group, seen as a discrete subgroup of ${\rm Homeo}^+ (S^1)$?
Classical results by Poincaré and Denjoy show that two orientation-preserving $C^2$ diffeomorphisms of the circle are topologically conjugate if and only if they have the same rotation number. We show that there is no possibility of getting such a complete numerical Borel invariant for the conjugacy relation of orientation-preserving circle diffeomorphisms by homeomorphisms with higher degree of regularity. For instance, we consider conjugacy by Hölder homeomorphisms or by $C^k$-diffeomorphisms with $k\in \mathbb{Z}^+ \cup \{\infty\}$. The proof combines techniques from Descriptive Set Theory …
We develop a renormalization theory for analytic homeomorphisms of the circle with two cubic critical points. We prove a renormalization hyperbolicity theorem. As a basis for the proofs, we develop complex a priori bounds for multi-critical circle maps.
In this paper we present a generalization of Poincaré's Rotation Theory of homeomorphisms of the circle to the case of one-dimensional compact abelian groups which are solenoidal groups, {\it i.e.}, groups which fiber over the circle with fiber a Cantor abelian group. We define rotation elements, \emph{à la} Poincaré and discuss the dynamical properties of translations on these solenoidal groups. We also study the semiconjugation problem when the rotation element generates a dense subgroup of the solenoidal group. Finally, we comment on the relation between Rotation Theory and entropy for thes…
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. As it follows from the a-priori bounds of Herman and Swiatek, 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 ro…
