arXiv subjects

Welington de Melo

Publications and source records attributed to Welington de Melo.

8 recordsLinked to original sources

arXiv2026

Blaschke-type models for multimodal circle maps

For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space.

math.DS↗

arXiv2015

Rigidity of 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 $C^{1+α}$ for Lebesgue almost every rotation number.

math.DS↗

arXiv2013

Rigidity of smooth 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 $C^{1+α}$ circle diffeomorphism, for some universal $α>0$.

math.DS↗

arXiv2007

Global hyperbolicity of renormalization for C^r unimodal mappings

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C^r unimodal maps with quadratic critical point. We show that in this space the bounded-type limit sets of the renormalization operator have an invariant hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an intermediate step between Lyubich's results and ours, we prove that the renormalization operator is hyperbolic in a Banach space of real analytic maps. We construct the local stable manifolds and prove that they form a continuous lamination whose leaves are C^1 codimension one, Banach submanifolds of the ambient space, and whose holonomy is C^{1+β} for some beta >0. We also prove that the global stable sets are C^1 immersed (codimension one) submanifolds as well, provided r ge 3+ alpha with alpha close to one. As a corollary, we deduce that in generic, one-parameter families of C^r unimodal maps, the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a Cantor set with Hausdorff dimension less than one.

math.DS↗

arXiv1998

The multipliers of periodic points in one-dimensional dynamics

It will be shown that the smooth conjugacy class of an $S-$unimodal map which does not have a periodic attractor neither a Cantor attractor is determined by the multipliers of the periodic orbits. This generalizes a result by M.Shub and D.Sullivan for smooth expanding maps of the circle.

math.DS↗

arXiv1997

Rigidity of critical circle mappings, I

We prove that two $C^r$ critical circle maps with the same rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$ provided their successive renormalizations converge together at an exponential rate in the $C^0$ sense. The number $α$ depends only on the rate of convergence. We also give examples of $C^\infty$ critical circle maps with the same rotation number that are not $C^{1+β}$ conjugate for any $β>0$.

math.DS↗

arXiv1997

Rigidity of critical circle mappings, II

We prove that any two real-analytic critical circle maps with cubic critical point and the same irrational rotation number of bounded type are $C^{1+α}$ conjugate for some $α>0$.

math.DS↗

arXiv1996

Universal models for Lorenz maps

The existence of smooth families of Lorenz maps exhibiting all possible dynamical behavior is established and the structure of the parameter space of these families is described.

math.DS↗