arXiv subjects

Christian Camilo Silva Villamil

Publications and source records attributed to Christian Camilo Silva Villamil.

6 recordsLinked to original sources

arXiv2026

Dynamical spectra for skew products with unit circle fiber

The dynamical Markov and Lagrange spectra are subsets of the real line widely studied and that share some similarities with the classical spectra, e.g. typical dynamical spectra, associated to horseshoes with Hausdorff dimension greater than one, have nonempty interior and, in the conservative setting, the intersections of the spectra with half-lines determine a continuous function. Here we study some similar questions in the context of skew products of surface diffeomorphisms with unit circle fiber.

math.DS↗

arXiv2026

On the discontinuities of Hausdorff dimension in generic dynamical Lagrange spectrum

Let $φ_0$ be a $C^2$-conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a mixing horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and some diffeomorphism $φ$, close to $φ_0$, let $\mathcal{L}_{φ, f}$ be the Lagrange spectrum associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that, for generic choices of $φ$ and $f$, if $L_{φ, f}$ is the map that gives the Hausdorff dimension of the set $\mathcal{L}_{φ, f}\cap (-\infty, t)$ for $t\in \mathbb{R}$, then there are at most two points that can be limit of a infinite sequence of discontinuities of $L_{φ, f}$.

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arXiv2025

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

Let $φ_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $φ$ of $φ_0$, let $L_{φ, f}$ and $M_{φ, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ and $M_{φ, f}\cap (-\infty, t)$ 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.

math.DS↗

arXiv2025

Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}^{'}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus \mathcal{B}_r$ is less than the Hausdorff dimension of $\mathcal{B}_r$ and for $t\in \mathcal{B}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in $\mathcal{B}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty,3+ρ)$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $ρ>0$ is small.

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arXiv2024

Concentration of dimension in the Lagrange spectrum

Let $φ$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ$ be a mixing horseshoe of $φ$. Given a smooth real function $f$ defined on $S$, we define for points $η$ in the unstable Cantor set of the pair $(φ,Λ)$, a generalization, $k_{φ,Λ,f}(η)$, of the best constant of Diophantine approximation for irrational numbers. We study the set of points $η$ for which the sets $k_{φ,Λ,f}^{-1}((-\infty,η])$ and $k_{φ,Λ,f}^{-1}(η)$ have the same Hausdorff dimension and when the Hausdorff dimension of $Λ$ is less than one, we describe generically the local Hausdorff dimension of the dynamical Lagrange spectrum, $\mathcal{L}_{φ,Λ,f}$, restricted to this set of points. Finally, we recover the same results for the classical Lagrange spectra.

math.DS↗

arXiv2024

Concentration of dimension in extremal points of left-half lines in the Lagrange spectrum

We prove that for any $η$ that belongs to the closure of the interior of the Markov and Lagrange spectra, the sets $k^{-1}((-\infty,η])$ and $k^{-1}(η)$, which are the sets of irrational numbers with best constant of Diophantine approximation bounded by $η$ and exactly $η$ respectively, have the same Hausdorff dimension. We also show that, as $η$ varies in the interior of the spectra, this Hausdorff dimension is a strictly increasing function.

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