arXiv subjects
Publications and source records attributed to Carlos Gustavo Moreira.
We show that the non-uniformly hyperbolic horseshoes of Palis and Yoccoz occur in the standard family of area-preserving diffeomorphisms of the two-torus for a set of (large) parameters of positive Lebesgue measure.
In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. Recently, the authors answered positively this question. In this paper, we prove a much stronger version of this result by considering other subsets of $\mathbb{C}$.
In this paper, we shall prove that any subset of $\overline{\mathbb Q}\cap B(0,1)$, which is closed under complex conjugation and which contains the element $0$, is the exceptional set of uncountably many transcendental functions, analytic in the unit ball, with integer coefficients. This solves a strong version of an old question proposed by K. Mahler (1976).
We prove dimension formulas for arihmetic sums of regular Cantor sets, and, more generally, for images of cartesian products of regular Cantor sets by differentiable real maps.
In 1902, P. Stäckel proved the existence of a transcendental function $f(z)$, analytic in a neighbourhood of the origin, and with the property that both $f(z)$ and its inverse function assume, in this neighbourhood, algebraic values at all algebraic points. Based on this result, in 1976, K. Mahler raised the question of the existence of such functions which are analytic in $\mathbb{C}$. In this work, we provide a positive answer for this question by showing the existence of uncountable many of these functions.
We propose a counting dimension for subsets of Z and prove that, under certain conditions on two such subsets E and F, for Lebesgue almost every real λ the counting dimension of E+[λF] is at least the minimum between 1 and the sum of the counting dimensions of E and F. Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E+[λF] has positive upper Banach density for Lebesgue almost every λ. The result has direct consequences when E,F are arithmetic sets, e.g. the integer values of a polynomial with integer coefficients.
We give sufficient conditions for two Cantor sets of the line to be nested for a positive set of translation parameters. This problem occurs in diophantine approximations. It also occurs as a toy model of the parameter selection for non-uniformly hyperbolic attractors of the plane. For natural Cantors sets, we show that this condition is optimal.
We consider a family of planar systems introduced in 1991 by Benedicks and Carleson as a toy model for the dynamics of the so-called Hénon maps. We show that Smale's Axiom A property is $C^1$-dense among the systems in this family, despite the existence of $C^2$-open subsets (closely related to the so-called Newhouse phenomena) where Smale's Axiom A is violated. In particular, this provides some evidence towards Smale's conjecture that Axiom A is a $C^1$-dense property among surface diffeomorphisms. The basic tools in the proof of this result are: 1) a recent theorem of Moreira saying that stable intersections of dynamical Cantor sets (one of the main obstructions to Axiom A property for surface diffeomorphisms) can be destroyed by $C^1$-perturbations; 2) the good geometry of the dynamical critical set (in the sense of Rodriguez-Hertz and Pujals) thanks to the particular form of Benedicks-Carleson toy models.
A permutation sequence is said to be convergent if the density of occurrences of every fixed permutation in the elements of the sequence converges. We prove that such a convergent sequence has a natural limit object, namely a Lebesgue measurable function $Z:[0,1]^2 \to [0,1]$ with the additional properties that, for every fixed $x \in [0,1]$, the restriction $Z(x,\cdot)$ is a cumulative distribution function and, for every $y \in [0,1]$, the restriction $Z(\cdot,y)$ satisfies a "mass" condition. This limit process is well-behaved: every function in the class of limit objects is a limit of some permutation sequence, and two of these functions are limits of the same sequence if and only if they are equal almost everywhere. An ingredient in the proofs is a new model of random permutations, which generalizes previous models and might be interesting for its own sake.
We give an example illustrating that two notions of bounded distortion for $\mathcal C^1$ expanding maps in $\R$ are different.
We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Let $K_1,...,K_n$ Borel subsets of $\mathbb R^{m_1},... ,\mathbb R^{m_n}$ respectively, and $π:\mathbb R^{m_1}\times...\times\mathbb R^{m_n}\to\mathbb R^k$ be a surjective linear map. We set $$\mathfrak{m}:=\min\{\sum_{i\in I}\dim_H(K_i) + \dimπ(\bigoplus_{i\in I^c}\mathbb R^{m_i}), I\subset\{1,...,n\}, I\ne\emptyset\}.$$ Consider the space $Λ_m=\{(t,O), t\in\mathbb R, O\in SO(m)\}$ with the natural measure and set $Λ=Λ_{m_1}\times...\timesΛ_{m_n}$. For every $λ=(t_1,O_1,...,t_n,O_n)\inΛ$ and every $x=(x^1,,x^n)\in\mathbb R^{m_1}\times...\times\mathbb R^{m_n}$ we define $π_λ(x)=π(t_1O_1x^1,...,t_nO_nx^n)$. Then we have $(i)$ If $\mathfrak{m}>k$, then $π_λ(K_1\times...\times K_n)$ has positive $k$-dimensional Lebesgue measure for almost every $λ\inΛ$. $(ii)$ If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times...\times K_n)=\dim_H(K_1)+...+\dim_H(K_n)$, then \dim_H(π_λ(K_1\times...\times K_n))=\mathfrak{m}$ for almost every $λ\inΛ$.
We prove the following variant of Marstrand's theorem about projections of cartesian products of sets: Consider the space $Λ_m=\set{(t,O), t\in\R, O\in SO(m)}$ with the natural measure and set $Λ=Λ_{m_1}\times\ppp\timesΛ_{m_n}$. For every $\la=(t_1,O_1,\ppp,t_n,O_n)\inΛ$ and every $x=(x^1,\ppp,x^n)\in\R^{m_1}\times\ppp\times\R^{m_n}$ we define $π_\la(x)=π(t_1O_1x^1,\ppp,t_nO_nx^n)$. Suppose that $π$ is surjective and set $$\mathfrak{m}:=\min\set{\sum_{i\in I}\dim_H(K_i) + \dimπ(\bigoplus_{i\in I^c}\R^{m_i}), I\subset\set{1,\ppp,n}, I\ne\emptyset}.$$ Then we have {thm*} \emph{(i)} If $\mathfrak{m}>k$, then $π_\la(K_1\times\ppp\times K_n)$ has positive $k$-dimensional Lebesgue measure for almost every $\la\inΛ$. \emph{(ii)} If $\mathfrak{m}\leq k$ and $\dim_H(K_1\times\ppp\times K_n)=\dim_H(K_1)+\ppp+\dim_H(K_n)$, then $\dim_H(π_\la(K_1\times\ppp\times K_n))=\mathfrak{m}$ for almost every $\la\inΛ$. {thm*}
In a paper from 1954 Marstrand proved that if K is a Borel subset of the plane with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a new proof of this theorem, extending the techniques developed in a previous work.
In 1954 Marstrand proved that if K is a subset of R^2 with Hausdorff dimension greater than 1, then its one-dimensional projection has positive Lebesgue measure for almost-all directions. In this article, we give a combinatorial proof of this theorem when K is the product of regular Cantor sets of class C^{1+a}, a>0, for which the sum of their Hausdorff dimension is greater than 1.
We prove that almost every non-regular real quadratic map is Collet-Eckmann and has polynomial recurrence of the critical orbit (proving a conjecture by Sinai). It follows that typical quadratic maps have excellent ergodic properties, as exponential decay of correlations (Keller and Nowicki, Young) and stochastic stability in the strong sense (Baladi and Viana). This is an important step to get the same results for more general families of unimodal maps.
We prove that there is a residual set of families of smooth or analytic unimodal maps with quadratic critical point and negative Schwarzian derivative such that almost every non-regular parameter is Collet-Eckmann with subexponential recurrence of the critical orbit. Those conditions lead to a detailed and robust statistical description of the dynamics. This proves the Palis conjecture in this setting.
In this work, we relate the geometry of chaotic attractors of typical analytic unimodal maps to the behavior of the critical orbit. Our main result is an explicit formula relating the combinatorics of the critical orbit with the exponents of periodic orbits. This connection between topological and smooth invariants is obtained through an analysis of the physical measure. Since the exponents of periodic orbits form a complete set of smooth invariants in this setting, we have ``typical geometric rigidity'' of the dynamics of such chaotic attractors. This unexpected result implies that the lamination structure of spaces of analytic maps (obtained by the partition into topological conjugacy classes, see \cite {ALM}) has an absolutely singular nature.
We obtain estimates relating the phase space and the parameter space of analytic families of unimodal maps. Using those estimates, we show that typical analytic unimodal maps admit a quasiquadratic renormalization. This reduces the study of the statistical properties of typical unimodal maps to the quasiquadratic case which had been studied in \cite {AM2}. The estimates proved here correspond exactly to the Phase-Parameter relation proved in \cite {AM} in the quadratic case, and allows one to obtain sharp estimates on the dynamics of typical unimodal maps which were available only in the quadratic case: as an example we conclude that the exponent of the polynomial recurrence of the critical orbit is exactly one. We also show that those ideas lead to a new proof of a Theorem of Shishikura: the set of non-renormalizable parameters in the boundary of the Mandelbrot set has Lebesgue measure zero. Further applications of those results can be found in the companion paper \cite {AM3}.