Related papers: The rigidity problem for analytic critical circle …

A rational map $f:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ on the Riemann sphere $\widehat{\mathbb{C}}$ is called critically fixed if each critical point of $f$ is fixed under $f$. In this article, we study the properties of a…

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with…

We consider critical points of the geometric obstacle problem on vectorial maps $u: \mathbb{B}^2 \subset \mathbb{R}^2 \to \mathbb{R}^N$ \[ \int_{\mathbb{B}^2} |\nabla u|^2 \quad \mbox{subject to $u \in \mathbb{R}^N \backslash…

We show that limits for the critical exponent tending to \infty exist in both critical circle homeomorphisms of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the…

Using a renormalization method, we study the critical behavior for intermittency in two coupled one-dimensional (1D) maps. We find two fixed maps of the renormalization transformation. They all have common relevant eigenvalues associated…

We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all have finite forward orbit. Thurston's rigidity theorem states that outside a single well-understood family, there are finitely many such maps over $\mathbb{C}$ of…

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is…

In this paper we build a geometric model for the renormalisation of irrationally indifferent fixed points of holomorphic maps with two critical points. The model incorporates arithmetic properties of the rotation number at the fixed point,…

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…

The key result of this article is key lemma: if a Jordan curve $\gamma$ is invariant by a given C 1+$\alpha$ -diffeomorphism f of a surface and if $\gamma$ carries an ergodic hyperbolic probability $\mu$, then $\mu$ is supported on a…

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…

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely…

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…

For a compact Riemannian manifold with boundary, endowed with a magnetic potential $\alpha$, we consider the problem of restoring the metric $g$ and the magnetic potential $\alpha$ from the values of the Ma\~n\'e action potential between…

We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree $4$ critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for…

A replica-symmetry-breaking phase transition is predicted in a host of disordered media. The criticality of the transition has, however, long been questioned below its upper critical dimension, six, due to the absence of a critical fixed…

There is a well developed renormalization theory of real analytic critical circle maps by de Faria, de Melo, and Yampolsky. In this paper, we extend Yampolsky's result on hyperbolicity of renormalization periodic points to a larger class of…

We provide a complete combinatorial classification of critically fixed anti-Thurston maps, i.e., orientation-reversing branched covers of the 2-sphere that fix every critical point. The first step in the proof, and an interesting result in…

In this paper we establish $C^2$ a-priori bounds for the scaling ratios of critical circle mappings in a form that gives also a compactness property for the renormalization operator.

We prove that if $F$ is a degree $3$ Thurston map with two fixed critical points, then any irreducible obstruction for $F$ contains a Levy cycle. As a corollary, it will be shown that if $f$ and $g$ are two postcritically finite cubic…