Related papers: Commuting circle diffeomorphisms with their deriva…

We prove that if d is an integer number bigger than 1 and f_1,...,f_d are commuting circle diffeomorphisms respectively of class C^(1+\tau_k), where \tau_1 + ... + \tau_k > 1, then these maps are simultaneously conjugate to rotations…

We show that a finite number of commuting diffeomorphisms with simultaneously Diophantine rotation numbers are smoothly conjugated to roations.

This paper is a step towards the complete topological classification of {\Omega}-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically…

Classical results by Poincar\'e 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…

We prove that a $C^{2+\alpha}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\delta<\alpha\le1$, is $C^{1+\alpha-\delta}$-smoothly conjugate to a rigid rotation. We also derive…

We prove that any two $C^3$ critical circle maps with the same irrational rotation number of bounded type and the same odd criticality are conjugate to each other by a $C^{1+\alpha}$ circle diffeomorphism, for some universal $\alpha>0$.

Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point, with the same irrational rotation number of bounded type, the same size of the break $c$ and satisfying a certain Zygmund type smoothness condition depending on a…

In this paper we answer positively a question of whether it is possible for a circle diffeomorphism with breaks to be smoothly conjugate to a rigid rotation in the case when its breaks are lying on pairwise distinct trajectories. An example…

We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+\alpha}$ for Lebesgue almost every…

We prove that a $C^{3+\beta}$-smooth orientation-preserving circle diffeomorphism with rotation number in Diophantine class $D_\delta$, $0<\beta<\delta<1$, is $C^{2+\beta-\delta}$-smoothly conjugate to a rigid rotation.

We show that a $C^{1+bv}$ circle diffeomorphism with absolutely continuous derivative and irrational rotation number can be conjugated to diffeomorphisms that are $C^{1+bv}$ arbitrary close to the corresponding rotation. This improves a…

In this paper we prove the $C^1$-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.

By introducing the modulus of continuity, we first establish the corresponding cross-ratio distortion estimates under $ C^2 $ smoothness, and further derive a Denjoy-type inequality, which is almost optimal for dealing with circle…

We extend rotation theory of circle maps to tiling spaces. Specifically, we consider a 1-dimensional tiling space $\Omega$ with finite local complexity and study self-maps $F$ that are homotopic to the identity and whose displacements are…

We prove a rigidity theorem for dominated H\"{o}lder cocycles with values on diffeomorphism groups of a compact manifold over hyperbolic homeomorphisms. More precisely, we show that if two such cocycles have equal periodic data, then they…

Let $\mathcal{F}$ and $\mathcal{K}$ be commuting $C^\infty$ diffeomorphisms of the cylinder $\mathbb{T}\times\mathbb{R}$ that are, respectively, close to $\mathcal{F}_0 (x, y)=(x+\omega(y), y)$ and $T_\alpha (x, y)=(x+\alpha, y)$, where…

Let $f$ and $g$ be two circle endomorphisms of degree $d\geq 2$ such that each has bounded geometry, preserves the Lebesgue measure, and fixes $1$. Let $h$ fixing $1$ be the topological conjugacy from $f$ to $g$. That is, $h\circ f=g\circ…

Let f_1, f_2 be holomorphic endomorphisms of P^k, of degrees d_1\geq 2, d_2\geq 2. Assume that f_1\circ f_2=f_2\circ f_1 and that d_1^{n_1}\not =d_2^{n_2} for all integers n_1, n_2. We then show that f_j are critically finite. Moreover…

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 $C^{2+\alpha}$ smooth circle…

We consider deformations of a group of circle diffeomorphisms with H\"older continuous derivatives in the framework of quasiconformal Teichm\"uller theory and show certain rigidity under conjugation by symmetric homeomorphisms of the…