Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2207.08458

Shrinking target for $C^1$ weakly conformal IFS with overlaps

Edouard Daviaud
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article, we study the Hausdor dimension of weakly conformal IFS's shrinking targets with possible overlaps, provided the conformality dimension of the systems and the dimension of the attractor are equal. Those results extends the works of Hill-Velani as well as the results obtained in [6] for self-similar IFS's.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Edouard Daviaud. 2022-07-18. Shrinking target for $C^1$ weakly conformal IFS with overlaps. https://arxiv.org/abs/2207.08458

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Hopf quotients of the infinite-dimensional Gaussian pyramid

We study the infinite-dimensional Gaussian pyramid and its quotients by the global sign flip and the $U(1)$-Hopf action. We resolve affirmatively a long-standing problem posed by Tomohiro Fukaya around 2014: these three limiting geometries are pairwise non-similar, meaning that no positive rescaling makes any two of them coincide.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the space of isometry classes of nonempty compact metric spaces, equipped with the Gromov--Hausdorff distance, is homeomorphic to the real separable infinite-dimensional Hilbert space. We construct a continuous assignment of full-support probability measures that is equivariant under isometries and finite-dimensional local approximations that control all pairwise distances. These approximations yield the absolute retract property for all metrizable spaces. We also prove that any countable family of continuous maps from compact metrizable spaces can be approximated, with respect to a prescribed open cover, by maps whose images form a discrete family.

math.MG↗
arXiv2026

An approximate counterexample to the Barker--Larman problem in dimension $4$

The Barker--Larman problem asks if a convex body $K \subseteq \mathbb R^n$ containing the Euclidean ball $\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $λ_0 = 4 \sqrt{3}π$ and any $N \in \mathbb N$, we obtain the existence of a family of convex bodies $K_{λ,N}$ with $λ\in (λ_0-r_N, λ_0 + r_N)$, such that the sections of $K_{λ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |λ- λ_0|^{N+1}$ of $λ$, while the difference between outradius and inradius of $K_{λ,N}$ is larger than $C |λ- λ_0|$. The bodies $K_{λ,N}$ are constructed via radial functions as \[ρ_{K_{λ,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(λ-λ_0)^n}{n!} φ_n(t) \right)^{-1},\] where $t \in [0,2π), λ\in \mathbb R$ and $φ_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \times 10^{-7}$.

math.MG↗