Related papers: Rigidity of critical circle maps
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 $\alpha>0$.
We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$.
We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$…
We prove that if two analytic multicritical circle maps with the same bounded type rotation number are topologically conjugate by a conjugacy which matches the critical points of the two maps while preserving the orders of their…
We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^2$-topology, are $C^1$ conjugate.
It is shown that if $f$ and $g$ are any two analytic critical circle mappings with the same irrational rotation number, then the conjugacy that maps the critical point of $f$ to that of $g$ has regularity $C^{1+\alpha}$ at the critical…
Given $C^2$ infinitely renormalizable unimodal maps $f$ and $g$ with a quadratic critical point and the same bounded combinatorial type, we prove that they are $C^{1+\alpha}$ conjugate along the closure of the corresponding forward orbits…
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…
A general ansatz in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the…
In this paper we study homeomorphisms of the circle with several critical points and bounded type rotation number. We prove complex a priori bounds for these maps. As an application, we get that bi-cubic circle maps with same bounded type…
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.
Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be…
We find conditions for two piecewise C^{2+\nu} homeomorphisms f and g of the circle to be C^1 conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary…
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…
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…
Criteria for piecewise linear circle homeomorphisms to be conjugate to a rigid rotation, $x\to x+\omega~({\rm mod}~1)$, with rational rotation number $\omega$ are given. The consequences of the existence of such maps in families of maps is…
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…
We construct, for each irrational number $\alpha$, a minimal $C^1$-diffeomorphism of the circle with rotation number $\alpha$ which admits a measur
We consider order preserving $C^3$ circle maps with a flat piece, Fibonacci rotation number, critical exponents $(\ell_1, \ell_2)$ and negative shwarzian derivative. This paper treat the geometry characteristic of the non-wondering (cantor…
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…