arXiv subjects

Liang-yi Huang

Publications and source records attributed to Liang-yi Huang.

3 recordsLinked to original sources

arXiv2025

Topology automaton and Hölder equivalence of Barański carpets

The study of Lipschitz equivalence of fractals is a very active topic in recent years. In 2023, Huang \emph{et al.} (\textit{Topology automaton of self-similar sets and its applications to metrical classifications}, Nonlinearity \textbf{36} (2023), 2541-2566.) studied the Hölder and Lipschitz equivalence of a class of p.c.f. self-similar sets which are not totally disconnected. The main tool they used is the so called topology automaton. In this paper, we define topology automaton for Barański carpets, and we show that the method used in Huang \emph{et al.} still works for the self-affine and non-p.c.f. settings. As an application, we obtain a very general sufficient condition for Barański carpets to be Hölder (or Lipschitz) equivalent.

math.MG↗

arXiv2024

Locally measure preserving property of bi-Lipschitz maps between Moran sets

In literature it is shown that bi-Lipschitz maps between self-similar sets or self-affine sets enjoy a locally measure preserving property, namely, if $f:(E,μ)\to (F,ν)$ is a bi-Lipschitz map, then the Radon-Nykodym derivative $df^*ν/dμ$ is a constant function on a subset $E'\subset E$ with $μ(E')>0$, where $f^*ν(\cdot)=ν(f(\cdot))$. Indeed, this measure preserving property plays an important role in Lipschitz classification of fractal sets. In this paper, we show that such measure preserving property also holds for bi-Lipschitz maps between two Moran sets in a certain class.

math.GT↗

arXiv2022

Box-counting measure of metric spaces

In this paper, we introduce a new notion called the \emph{box-counting measure} of a metric space. We show that for a doubling metric space, an Ahlfors regular measure is always a box-counting measure; consequently, if $E$ is a self-similar set satisfying the open set condition, then the Hausdorff measure restricted to $E$ is a box-counting measure. We show two classes of self-affine sets, the generalized Lalley-Gatzouras type self-affine sponges and Barański carpets, always admit box-counting measures; this also provides a very simple method to calculate the box-dimension of these fractals. Moreover, among others, we show that if two doubling metric spaces admit box-counting measures, then the multi-fractal spectra of the box-counting measures coincide provided the two spaces are Lipschitz equivalent.

math.MG↗