相关论文: $L^q$-spectra of box-like graph-directed self-affi…

We study $L^q$-spectra of planar self-affine measures generated by diagonal systems with an emphasis on providing closed form expressions. We answer a question posed by Fraser in 2016 in the negative by proving that a certain natural closed…

We study the dimension theory of a class of planar self-affine multifractal measures. These measures are the Bernoulli measures supported on box-like self-affine sets, introduced by the author, which are the attractors of iterated function…

For self-similar measures with overlaps, closed formulas of the $L^q$-spectrum have been obtained by Ngai and the author for measures that are essentially of finite type in [J. Aust. Math. Soc. \textbf{106} (2019), 56--103]. We extend the…

We study the $L^q$-spectrum of measures in the plane generated by certain nonlinear maps. In particular we consider attractors of iterated function systems consisting of maps whose components are $C^{1+\alpha}$ and for which the Jacobian is…

In this paper a sponge in $\mathbb{R}^d$ is the attractor of an iterated function system consisting of finitely many strictly contracting affine maps whose linear part is a diagonal matrix. A suitable separation condition is introduced…

We conduct the multifractal analysis of self-affine measures for "almost all" family of affine maps. Besides partially extending Falconer's formula of $L^q$-spectrum outside the range $1< q\leq 2$, the multifractal formalism is also…

We establish matrix representations for self-similar measures on $\mathbb{R}^d$ generated by equicontractive IFSs satisfying the finite type condition. As an application, we prove that the $L^q$-spectrum of every such self-similar measure…

We study the L^q -dimensions of self-affine measures and the Kaenmaki measure on a class of self-affine sets similar to the class considered by Hueter and Lalley. We give simple, checkable conditions under which the Lq -dimensions are equal…

We study the box dimensions of self-affine sets in $\mathbb{R}^3$ which are generated by a finite collection of generalised permutation matrices. We obtain bounds for the dimensions which hold with very minimal assumptions and give rise to…

We present a self-contained proof of a formula for the $L^q$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $L^q$ norm of convolutions). This is a special…

For any self-similar measure $\mu$ in $\mathbb{R}$, we show that the distribution of $\mu$ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the IFS. This generalizes the net…

We establish a generic formula for the generalised q-dimensions of measures supported by almost self-affine sets, for all q>1. These q-dimensions may exhibit phase transitions as q varies. We first consider general measures and then…

We consider a class of planar self-affine sets which we call "box-like". A box-like self-affine set is the attractor of an iterated function system (IFS) of affine maps where the image of the unit square, [0,1]^2, under arbitrary…

We study equilibrium measures (K\"aenm\"aki measures) supported on self-affine sets generated by a finite collection of diagonal and anti-diagonal matrices acting on the plane and satisfying the strong separation property. Our main result…

We establish an Expander Mixing Lemma for directed graphs in terms of the eigenvalues of an associated asymmetric transition probability matrix, extending the classical spectral inequality to the asymmetric setting. As an application, we…

Let $X=\bigcup\varphi_{i}X$ be a strongly separated self-affine set in $\mathbb{R}^2$ (or one satisfying the strong open set condition). Under mild non-compactness and irreducibility assumptions on the matrix parts of the $\varphi_{i}$, we…

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and…

Franco, Galloni, Penante, and Wen have proposed a boundary measurement map for a graph on any closed orientable surface with boundary. We consider this boundary measurement map which takes as input an edge weighted directed graph embedded…

Using methods from ergodic theory along with properties of the Furstenberg measure we obtain conditions under which certain classes of plane self-affine sets have Hausdorff or box-counting dimensions equal to their affinity dimension. We…

We study the spectral properties of certain non-self-adjoint matrices associated with large directed graphs. Asymptotically the eigenvalues converge to certain curves, apart from a finite number that have limits not on these curves.