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Edson de Faria

Publications and source records attributed to Edson de Faria.

At least 19 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↗
arXiv2023

Automorphic measures and invariant distributions for circle dynamics

Let $f$ be a $C^{1+bv}$ circle diffeomorphism with irrational rotation number. As established by Douady and Yoccoz in the eighties, for any given $s>0$ there exists a unique automorphic measure of exponent $s$ for $f$. In the present paper we prove that the same holds for multicritical circle maps, and we provide two applications of this result. The first one, is to prove that the space of invariant distributions of order 1 of any given multicritical circle map is one-dimensional, spanned by the unique invariant measure. The second one, is an improvement over the Denjoy-Koksma inequality for multicritical circle maps and absolutely continuous observables.

math.DS↗
arXiv2023

Dennis Sullivan's Work on Dynamics

In this expository paper, we provide the readers with an overview of Dennis Sullivan's major contributions to the area of Dynamical Systems.

math.DS↗
arXiv2021

Quasisymmetric orbit-flexibility of multicritical circle maps

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 Yoccoz, if $f$ is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. As it follows from the a-priori bounds of Herman and Swiatek, the same holds if $f$ is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if $f$ is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$, then the number of equivalence classes is uncountable (Theorem A). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems B and C).

math.DS↗
arXiv2021

Dynamics of multicritical circle maps

This paper presents a survey of recent and not so recent results concerning the study of smooth homeomorphisms of the circle with a finite number of non-flat critical points, an important topic in the area of One-dimensional Dynamics. We focus on the analysis of the fine geometric structure of orbits of such dynamical systems, as well as on certain ergodic-theoretic and complex-analytic aspects of the subject. Finally, we review some conjectures and open questions in this field.

math.DS↗
arXiv2020

On the Erdős-Sloane and Shifted Sloane Persistence Problems

In this paper, we investigate two variations on the so-called persistence problem of Sloane: the shifted version, which was introduced by Wagstaff; and the nonzero version, proposed by Erdős. We explore connections between these problems and a recent conjecture of de Faria and Tresser regarding equidistribution of the digits of some integer sequences and some of its natural generalizations.

math.NT↗
arXiv2020

There are no $σ$-finite absolutely continuous invariant measures for multicritical circle maps

It is well-known that every multicritical circle map without periodic orbits admits a unique invariant Borel probability measure which is purely singular with respect to Lebesgue measure. Can such a map leave invariant an infinite, $σ$-finite invariant measure which is absolutely continuous with respect to Lebesgue measure? In this paper, using an old criterion due to Katznelson, we show that the answer to this question is no.

math.DS↗
arXiv2018

Generalized Whitney Topologies are Baire

In this paper we show that certain generalizations of the $C^r$-Whitney topology, which include the Hölder-Whitney and Sobolev-Whitney topologies on smooth manifolds, satisfy the Baire property, to wit, the countable intersection of open and dense sets is dense.

math.GT↗
arXiv2018

Dynamics of asymptotically holomorphic polynomial-like maps

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

math.DS↗
arXiv2017

Genericity of Infinite Entropy for Maps with Low Regularity

For bi-Lipschitz homeomorphisms of a compact manifold it is known that topological entropy is always finite. For compact manifolds of dimension two or greater, we show that in the closure of the space of bi-Lipschitz homeomorphisms, with respect to either the Hölder or the Sobolev topologies, topological entropy is generically infinite. We also prove versions of the $C^1$-Closing Lemma in either of these spaces. Finally, we give examples of homeomorphisms with infinite topological entropy which are Hölder and/or Sobolev of every exponent.

math.DS↗
arXiv2016

Beau bounds for multicritical circle maps

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 single critical point.

math.DS↗
arXiv2015

Real bounds and quasisymmetric rigidity of multicritical circle maps

Let $f, g:S^1\to S^1$ be two $C^3$ critical homeomorphisms of the circle with the same irrational rotation number and the same (finite) number of critical points, all of which are assumed to be non-flat, of power-law type. In this paper we prove that if $h:S^1\to S^1$ is a topological conjugacy between $f$ and $g$ and $h$ maps the critical points of $f$ to the critical points of $g$, then $h$ is quasisymmetric. When the power-law exponents at all critical points are integers, this result is a special case of a general theorem recently proved by T.~Clark and S.~van Strien \cite{CS}. However, unlike the proof given in \cite{CS}, which relies on heavy complex-analytic machinery, our proof uses purely real-variable methods, and is valid for non-integer critical exponents as well. We do not require $h$ to preserve the power-law exponents at corresponding critical points.

math.DS↗
arXiv2015

Differentiability of correlations in Realistic Quantum Mechanics

We prove a version of Bell's Theorem in which the Locality assumption is weakened. We start by assuming theoretical quantum mechanics and weak forms of relativistic causality and of realism (essentially the fact that observable values are well defined independently of whether or not they are measured). Under these hypotheses, we show that only one of the correlation functions that can be formulated in the framework of the usual Bell theorem is unknown. We prove that this unknown function must be differentiable at certain angular configuration points that include the origin. We also prove that, if this correlation is assumed to be twice differentiable at the origin, then we arrive at a version of Bell's theorem. On the one hand, we are showing that any realistic theory of quantum mechanics which incorporates the kinematic aspects of relativity must lead to this type of \emph{rough} correlation function that is once but not twice differentiable. On the other hand, this study brings us a single degree of differentiability away from a relativistic von Neumann no hidden variables theorem.

quant-ph↗
arXiv2015

Real bounds and Lyapunov exponents

We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponent with respect to its unique invariant Borel probability measure. Moreover, no critical point of such a map satisfy the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds, without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizable unimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutral measures of certain rational maps of the Riemann sphere.

math.DS↗
arXiv2013

On Sloane's persistence problem

We investigate the so-called persistence problem of Sloane, exploiting connections with the dynamics of circle maps and the ergodic theory of $\mathbb{Z}^d$ actions. We also formulate a conjecture concerning the asymptotic distribution of digits in long products of finitely many primes whose truth would, in particular, solve the persistence problem. The heuristics that we propose to complement our numerical studies can be thought in terms of a simple model in statistical mechanics.

math.DS↗
arXiv2012

Bell inequality violations under reasonable and under weak hypotheses

Given a sequence of pairs of spin-one half particles in the singlet state, assume that Alice measures the normalized projections along some vector of the spins of one vector per pair along that vector while Bob measures the normalized projections along some vector of the spins of the other member of each pair. Then Quantum Mechanics, or QM, lets one evaluate the correlation of the projections along these two vectors as minus the cosinus of the angle between said vectors; we assume that all vectors are chosen in a fixed plane. Assuming Classical Microscopic Realism, or CMR, there exist also normalized projection pairs of the spins of the pairs of particles along some other pair of vectors. Assuming QM and MR, we also have that the correlations of the projections along the other vectors as minus the cosinus of the angle between the extra vectors. Assuming Locality,i.e., the impossibility of any effect of an event on another event when said events are spatially separated, beside QM and MR, the theory of Bell lets one deduce various violations of some inequalities at some choices of quadruplets of the vectors that have been chosen. Our main result is the existence of quadruplets where at least one of the said inequalities is violated if one only assumes QM, MR and some very mild further hypotheses. These weak hypotheses only concern the behavior of correlations that we use near special quadruplets. We thus get versions of Bell's theorem that are strictly stronger than the original one and in particular do not assume Locality.

quant-ph↗
arXiv2011

Analyzing the correlations for spin-1/2 particles and singlet pairs

We revisit the computation of correlations of spin projections onto unit vectors for spin-1/2 particles in Quantum Mechanics. We then choose one of the Boole inequalities that, as we recall, must be obeyed by collections of sequences of normalized spin-1/2 projections onto unit vectors that belong to, say, the plane that is orthogonal to the classical trajectory or to the three dimensional space. Next we define the concept of being a-p for i-indexed sequences of spin-1/2 particles: it turns out that a sequence of spin-1/2 particles is a-p if and only if the particles behave as if they had indeed been prepared (in the usual sense) to have definite spins +1 or -1 along the axis a, the sequence of signs being pre-determined. The a-p property is formulated in a way abstract enough to let one generalize it and we define the concept of being a-p and b-p. However, using our chosen Boole inequality we prove that no particle sequence can be both a-p and b-p when a and b are not co-linear. We next consider two modifications of Quantum Mechanics: - The first one by assuming both Weak Realism, the weakest form of Realism needed to develop any main theorem in Bell's Theory, and Locality. - The second one by assuming Weak Realism and an hypothesis that turns out to be none else than (relativistic) Causality extended to cover as well the observable values that make sense only if one assumes Weak Realism. In both cases we arrive at the conclusion that in the EPRB setting where successive pairs of spin-1/2 particles are prepared in the so-called singlet state, one of these particles sequences is both a-p and b-p and a and b are the opposite of the vectors respectively used to project the spin of the other particle and to get the Realism-based alternate value for a projection. Thus the usual discussions of Bell's inequalities under such hypotheses do not even make physical sense.

quant-ph↗