arXiv subjects

Davi Lima

Publications and source records attributed to Davi Lima.

11 recordsLinked to original sources

arXiv2026

Stability and limit theorems in random dynamical systems

The robust statistical description of dynamical systems under perturbations is a central problem in ergodic theory. In this paper, we investigate the statistical properties of skew-product maps driven by a subshift of finite type with contracting fiber maps, a setting that naturally encompasses Iterated Function Systems (IFS) and Random Dynamical Systems (RDS). Diverging from the classical perturbative frameworks that rely on the compact embedding of anisotropic Banach spaces, we employ a flexible operator approach based on the Lipschitz regularity of the invariant measure's disintegrations with respect to the Wasserstein metric. Our main results are threefold: first, we prove the quantitative statistical stability of the unique invariant measure under admissible deterministic perturbations, obtaining an explicit modulus of continuity of the form $O(R(\delta) \log \delta)$. Second, we establish the exponential decay of correlations on new pair of spaces of observables. Finally, leveraging this exponential decay and Gordin's method, we prove the Central Limit Theorem for the fluctuations of Birkhoff averages of Lipschitz observables.

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arXiv2024

inf(M \ L)=3

The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed that the initial portions of these spectra coincide: indeed, $L\cap (0,3) = M\cap (0,3)$ is a discrete set of explicit quadratic irrationals accumulating only at $3$. In this article, we show that the statement above ceases to be true immediately after $3$: in particular, $L\cap (3,3+\varepsilon)\neq M\cap (3,3+\varepsilon)$ for all $\varepsilon>0$, and thus $\inf(M\setminus L)=3$. In fact, we derive this result as a by-product of lower bounds on the Hausdorff dimension of $(M\setminus L)\cap (3,3+\varepsilon)$ implying that $\liminf\limits_{\varepsilon\to 0} \frac{\dim_H((M\setminus L)\cap(3,3+\varepsilon))}{\dim_H(M\cap (3,3+\varepsilon))}\geq \frac{1}{2}$ and, as it turns out, these bounds are obtained from the study of projections of Cartesian products of almost affine dynamical Cantor sets via an argument of probabilistic flavor based on Baker--W\"ustholz theorem on linear forms in logarithms of algebraic numbers.

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arXiv2024

Counting and Hausdorff measures for integers and $p$-adic integers

In this work, we aim to advance the development of a fractal theory for sets of integers. The core idea is to utilize the fractal structure of $p$-adic integers, where $p$ is a prime number, and compare this with conventional densities and counting measures for integers. Our approach yields some results in combinatorial number theory. The results show how the local fractal structure of a set in $\mathbb{Z}_p$ can provide bounds for the counting measure for its projection onto $\mathbb{Z}$. Additionally, we establish a relationship between the counting dimension of a set of integers and its box-counting dimension in $\mathbb{Z}_p$. Since our results pertain to sets that are projections of closed sets in $\mathbb{Z}_p$, we also provide both necessary and sufficient combinatorial conditions for a set $E\subset \mathbb{Z}$ to be the projection of a closed set in $\mathbb{Z}_p$.

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arXiv2023

Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra

Let $\varphi_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $\Lambda_0$ be a transitive horseshoe of $\varphi_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $\varphi$ of $\varphi_0$, let $L_{\varphi, f}$ and $M_{\varphi, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $\Lambda(\varphi)$ of the horseshoe $\Lambda_0$ and $f$. We show that for generic choices of $\varphi$ and $f$, the Hausdorff dimension of the sets $L_{\varphi, f}\cap (-\infty, t)$ and $M_{\varphi, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension.

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arXiv2020

Dynamical characterization of initial segments of the Markov and Lagrange spectra

We prove that, for every $k\ge 4$, the sets $M(k)$ and $L(k)$, which are Markov and Lagrange dynamical spectra related to conservative horseshoes and associated to continued fractions with coefficients bounded by $k$ coincide with the intersections of the classical Markov and Lagrange spectra with $(-\infty, \sqrt{k^2+4k}]$. We also observe that, despite the corresponding statement is also true for $k = 2$, it is false for $k = 3$.

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arXiv2020

2-Adic Stratification of Totients

In this paper we study the multiplicities and the asymptotic behaviour of the numbers of totients in the strata given by 2-adic valuation.

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arXiv2020

Lipschitz regularity of the invariant measure of random dynamical systems

In this article we derive a regularity result for the disintegration of the invariant measure associated to a class of Random Dynamical Systems - RDS. The results of this work are obtained by constructing a suitable anisotropic normed space defined by the Wasserstein-Kantorovich-like metric and understanding the dynamics of the associated transfer operator in a neighborhood of its fixed point. Precisely, we employ functional analytic techniques to demonstrate a spectral gap for its action on suitable spaces of signed measures. We apply this analysis to prove an exponential decay of correlation statement for Lipschitz observables and statistical properties of the RDS.

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arXiv2019

$M\backslash L$ is not closed

We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ which is accumulated by a sequence of elements of the complement $M\setminus L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\setminus L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.

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arXiv2019

$M\setminus L$ near 3

We construct four new elements $3.11>m_1>m_2>m_3>m_4$ of $M\backslash L$ lying in distinct connected components of $\mathbb{R}\setminus L$, where $M$ is the Markov spectrum and $L$ is the Lagrange spectrum. These elements are part of a decreasing sequence $(m_k)_{k\in\mathbb{N}}$ of elements in $M$ converging to $3$ and we give some evidence towards the possibility that $m_k\in M\setminus L$ for all $k\geq 1$. In particular, this indicates that $3$ might belong to the closure of $M\setminus L$, so that the answer to Bousch's question about the closedness of $M\setminus L$ might be negative.

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arXiv2018

On the distribution of totients 2 mod. 4

In this paper we study the distribution of totients $2$ mod. $4$. We prove that the asymptotic magnitude of such totients with multiplicity two is half of that of prime numbers. As a corollary we obtain that the relative asymptotic density of the number of those totients with multiplicity four over the number of totients with multiplicity two is zero. We also obtain that the set of totients which have, a bigger than one, power of a prime in their pre-images and multiplicity $k$ has relative asymptotic density over the number of all totients of multiplicity $k$ equals to zero. A result on the distribution of consecutive pairs of totients $2$ mod. $4$, which relates to cousin primes, is also provide.

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arXiv2018

Phase Transitions on the Markov and Lagrange Dynamical Spectra

The Markov and Lagrange dynamical spectra, was introduced by Moreira and share several geometric and topological aspects with the classical ones. However, some features of generic dynamical spectra associated to hyperbolic sets can be proved in the dynamical case and we do not know if they are true in classical case. They can be a good source of natural conjectures about the classical spectra: it is natural to conjecture that some properties which hold for generic dynamical spectra associated to hyperbolic maps also holds for the classical Markov and Lagrange spectra. In this paper, we show that, for generic dynamical spectra associated to horseshoes, there is a transition point a in the spectra such that for any $\delta > 0$, the intersection of the spectra with $(a-\delta; a)$ is a countable union of Cantor sets with Hausdorff dimension smaller than 1, while the intersection of the spectra with $(a; a + \delta)$ have non-empty interior. This is still open for the classical Markov and Lagrange spectra.

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