arXiv subjects

Jian-Ci Xiao

Publications and source records attributed to Jian-Ci Xiao.

8 recordsLinked to original sources

arXiv2026

Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios

We prove that any non-degenerate Bedford-McMullen carpet does not admit oblique self-embedding similitudes; that is, if $f$ is a similitude sending the carpet into itself, then the image of the $x$-axis under $f$ must be parallel to one of the principal axes. This result leads to a logarithmic commensurability result on the contraction ratios of such embeddings, completing a previous study by Algom and Hochman [Ergod. Th. & Dynam. Sys. 39 (2019), 577-603] on Bedford-McMullen carpets generated by multiplicatively independent exponents. Our approach also provides a new proof of their non-obliqueness statement that avoids analyzing the tangent sets. For the self-similar case, however, we construct a generalized Sierpiński carpet that is symmetric with respect to an appropriate oblique line and hence admits a reflectional oblique self-embedding. As a complement, we prove that if a generalized Sierpiński carpet satisfies the strong separation condition and permits an oblique rotational self-embedding similitude, then the tangent of the rotation angle takes values $\pm 1$.

math.CA↗

arXiv2025

From Lipschitz embedding to Lipschitz equivalence between dust-like self-similar sets

Let $K,F\subset\mathbb{R}^d$ be two dust-like self-similar sets sharing the same Hausdorff dimension. We consider when the mere existence of a Lipschitz embedding from $K$ to $F$ already implies their Lipschitz equivalence. Our main result is threefold: (1) if the Lipschitz image of $K$ intersects $F$ in a set of positive Hausdorff measure, then $K$ admits a Lipschitz surjection onto $F$; (2) if $F$ is in addition homogeneous, then the generating iterated function systems of $K, F$ should have algebraically dependent ratios and consequently, $K$ and $F$ are Lipschitz equivalent; (3) the Lipschitz equivalence can fail without the homogeneity assumption. This answers two questions in Balka and Keleti [Adv. Math. 446 (2024), 109669].

math.CA↗

arXiv2025

The connectedness of Sierpiński sponges with rotational and reflectional components and associated graph-directed systems

We provide two methods to characterize the connectedness of all $d$-dimensional generalized Sierpiński sponges whose corresponding IFSs are allowed to have rotational and reflectional components. Our approach is to reduce it to an intersection problem between the coordinates of graph-directed attractors. More precisely, let $(K_1,\ldots,K_n)$ be a Cantor-type graph-directed attractor in $\mathbb{R}^d$. By creating an auxiliary graph, we provide an effective criterion for whether $K_i\cap K_j$ is empty for every pair of $1\leq i,j\leq n$. Moreover, the emptiness can be checked by examining only a finite number of geometric approximations of the attractor. The approach is also applicable to more general graph-directed systems.

math.GN↗

arXiv2024

A self-similar set with non-locally connected components

Luo, Rao and Xiong [Topol. Appl. 322 (2022), 108271] conjectured that if a planar self-similar iterated function system with the open set condition does not involve rotations or reflections, then every connected component of the attractor is locally connected. We create a homogeneous counterexample of Lalley-Gatzouras type, which disproves this conjecture.

math.GN↗

arXiv2023

On a self-embedding problem of self-similar sets

Let $K\subset\mathbb{R}^d$ be a self-similar set generated by an iterated function system $\{φ_i\}_{i=1}^m$ satisfying the strong separation condition and let $f$ be a contracting similitude with $f(K)\subset K$. We show that $f(K)$ is relative open in $K$ if all $φ_i$'s share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question in Elekes, Keleti and M{á}th{é} [Ergodic Theory Dynam. Systems 30 (2010)]. As a byproduct of our argument, when $d=1$ and $K$ admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction parts of opposite signs, we show that $K$ is symmetric. This partially answers a question in Feng and Wang [Adv. Math. 222 (2009)].

math.DS↗

arXiv2022

On the existence of cut points of connected generalized Sierpinski carpets

In a previous work joint with Dai and Luo, we show that a connected generalized Sierpiński carpet (or shortly a GSC) has cut points if and only if the associated $n$-th Hata graph has a long tail for all $n\geq 2$. In this paper, we extend the above result by showing that it suffices to check a finite number of those graphs to reach a conclusion. This criterion provides a truly "algorithmic" solution to the cut point problem of connected GSCs. We also construct for each $m\geq 1$ a connected GSC with exactly $m$ cut points and demonstrate that when $m\geq 2$, such a GSC must be of the so-called fragile type.

math.GN↗

arXiv2022

Connectedness and local cut points of generalized Sierpinski carpets

We investigate a homeomorphism problem on a class of self-similar sets called generalized Sierpinski carpets (or shortly GSCs). It follows from two well-known results by Hata and Whyburn that a connected GSC is homeomorphic to the standard Sierpinski carpet if and only if it has no local cut points. On the one hand, we show that to determine whether a given GSC is connected, it suffices to iterate the initial pattern twice. On the other hand, we obtain two criteria: (1) for a connected GSC to have cut points, (2) for a connected GSC with no cut points to have local cut points. With these two criteria, we characterize all GSCs that are homeomorphic to the standard Sierpinski carpet. Our results on cut points and local cut points hold for Baranski carpets, too. Moreover, we extend the connectedness result to Baranski sponges. Thus, we also characterize when a Baranski carpet is homeomorphic to the standard GSC.

math.GN↗

arXiv2020

Fractal squares with finitely many connected components

In this paper, we present an effective method to characterize completely when a disconnected fractal square has only finitely many connected components. Our method is to establish some graph structures on fractal squares to reveal the evolution of the connectedness during their geometric iterated construction. We also prove that every fractal square contains either finitely or uncountably many connected components. A few examples, including the construction of fractal squares with exactly $m\geqslant 2$ connected components, are added in addition.

math.GN↗