Explanipedia

Iterated Relation Systems on Riemannian Manifolds Open

Jie Liu, Sze-Man Ngai, Lei Ouyang · 2025

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as these fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we…

Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets Open

Sze-Man Ngai, Shun Zhou · 2025

This paper extends the Hodge-de Rham theory of Aaron \textit{et al.} [Commun. Pure Appl. Anal. {\bf 13} (2014)] to higher-dimensional level-$l$ Sierpinski gaskets $SG_{\ell}^{n},$ providing a framework for analyzing differential forms and …

Iterated relation systems on Riemannian manifolds Open

Jie Liu, Sze-Man Ngai, Lei Ouyang · 2024

For fractals on Riemannian manifolds, the theory of iterated function systems often does not apply well directly, as fractal sets are often defined by relations that are multivalued or non-contractive. To overcome this difficulty, we intro…

Hodge Theorem for Krein-Feller operators on compact Riemannian manifolds Open

Sze-Man Ngai, Lei Ouyang · 2024

For an open set in a compact smooth oriented Riemannian n-manifold and a positive finite Borel measure with support contained in the closure of the open set, we define an associated Krein-Feller operator on k-forms by assuming the Poincare…

Schrödinger equations defined by a class of self-similar measures Open

Sze-Man Ngai, Wei Tang · 2023

We study linear and non-linear Schrödinger equations defined by fractal measures. Under the assumption that the Laplacian has compact resolvent, we prove that there exists a unique weak solution for a linear Schrödinger equation, and then …

Krein-Feller operators on Riemannian manifolds and a compact embedding theorem Open

Sze-Man Ngai, Lei Ouyang · 2023

For a bounded open set Omega in a complete oriented Riemannian n-manifold and a positive finite Borel measure mu with support contained in the closure of Omega, we define an associated Dirichlet Laplacian Delta_mu by assuming the Poincare …

A class of self-affine tiles in $\mathbb{R}^d$ that are $d$-dimensional tame balls Open

Guotai Deng, Chuntai Liu, Sze-Man Ngai · 2022

We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to $\mathbb{R}^3}$ by the authors. …

Existence of $L^q$-dimension and entropy dimension of self-conformal measures on Riemannian manifolds Open

Sze-Man Ngai, Yangyang Xu · 2022

Peres and Solomyak proved that on $\mathbb R^n$, the limits defining the $L^q$-dimension for any $q\in(0,\infty)\setminus\{1\}$, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By int…

Orthogonal Polynomials Defined by Self-Similar Measures with Overlaps Open

Sze-Man Ngai, Wei Tang, Anh T. Tran, Shuai Yuan · 2020

Here, we study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Gar…

Spectral asymptotics of Laplacians related to one-dimensional graph-directed self-similar measures with overlaps Open

Sze-Man Ngai, Yuanyuan Xie · 2020

For the class of graph-directed self-similar measures on $\\mathbf{R}$, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined …

Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates Open

Qingsong Gu, Jiaxin Hu, Sze-Man Ngai · 2019

We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite …

Spectral Asymptotics of Laplacians Defined by Fractal Measures and Some Applications Open

Sze-Man Ngai · 2019

We report some results concerning spectral asymptotics of fractal Laplacians defined one-dimensional self-similar measures with overlaps. We also discuss some applications of the theory, including heat kernel estimates and wave propagation…

Spectral asymptotics of one-dimensional fractal Laplacians in the absence of second-order identities Open

Sze-Man Ngai, Wei Tang, Yuanyuan Xie · 2018

We observe that some self-similar measures defined by finite or infinite iterated function systems with overlaps are in certain sense essentially of finite type, which allows us to extract useful measure-theoretic properties of iterates of…

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