Complex bounds for critical circle maps

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 rotation number.

Let $f: S^1\to S^1$ be a $C^3$ homeomorphism without periodic points having a finite number of critical points of power-law type. In this paper we establish real a-priori bounds, on the geometry of orbits of $f$, which are beau in the sense of Sullivan, i.e. bounds that are asymptotically universal at small scales. The proof of the beau bounds presented here is an adaptation, to the multicritical setting, of the one given by the second author and de Melo, for the case of a si...

In this paper, we prove that for any post-critically finite rational map $f$ on the Riemann sphere $\overline{\mathbb{C}}$ and for each sufficiently large integer $n$, there exists a finite and connected graph $G$ in the Julia set of $f$, such that $f^n(G)\subset G$, $G$ contains all post-critical points in the Julia set and every component of $\overline{\mathbb{C}}\setminus G$ contains at most one post-critical point in the Fatou set. The proof is based on the cluster-\Sie\ ...

We prove that two topologically conjugate bi-critical circle maps whose signatures are the same, and whose renormalizations converge together exponentially fast in the $C^1$-topology, are $C^1$ conjugate.

We prove the existence of fixed points of p-tupling renormalization operators for interval and circle mappings having a critical point of arbitrary real degree r > 1. Some properties of the resulting maps are studied: analyticity, univalence, behavior as $r$ tends to infinity.

We present an application of quasiconformal (QC) surgery for holomorphic maps fibered over an irrational rotation of the unit circle, also known as quasiperiodically forced (QPF) maps. It consists of modifying the fibered multiplier of an attracting invariant curve for a QPF hyperbolic polynomial. This is the analogue of the classical change of multiplier of an attracting cycle in the one-dimensional iteration case, to parametrize hyperbolic components of the Mandelbrot set. ...

We study C^2 weakly order preserving circle maps with a flat interval. In particular we are interested in the geometry of the mapping near to the singularities at the boundary of the flat interval. Without any assumption on the rotation number we show that the geometry is degenerate when the degree of the singularities is less than or equal to two and becomes bounded when the degree goes to three. As an example of application, the result is applied to study Cherry flows.

We prove an extension results for the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4^n/p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic p...

We formulate and prove $\textit{a priori}$ bounds for the renormalization of H\'enon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of the main results, including renormalization convergence, finite-time checkability of the required regularity conditions and regular...

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 $\alpha>0$.